SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Modulational instability</title>

</head>

Modulational instability

</h1>

</header>

In the fields of <a href="Nonlinear_optics" title="Nonlinear optics">nonlinear optics</a> and <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, modulational instability or sideband instability is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of <a href="Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">spectral</a>-sidebands and the eventual breakup of the waveform into a train of <a href="Wave_packet" title="Wave packet">pulses</a>.<a href="#cite_note-BenjaminFeir-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-agrawal-3">[3]</a>

It is widely believed that the phenomenon was first discovered − and modeled − for periodic <a href="Surface_gravity_wave" class="mw-redirect" title="Surface gravity wave">surface gravity waves</a> (<a href="Stokes_wave" title="Stokes wave">Stokes waves</a>) on deep water by <a href="T._Brooke_Benjamin" class="mw-redirect" title="T. Brooke Benjamin">T. Brooke Benjamin</a> and Jim E. Feir, in 1967.<a href="#cite_note-4">[4]</a> Therefore, it is also known as the Benjamin−Feir instability. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965,<a href="#cite_note-5">[5]</a> and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.<a href="#cite_note-6">[6]</a> Modulation instability is a possible mechanism for the generation of <a href="Rogue_wave" title="Rogue wave">rogue waves</a>.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Initial_instability_and_gain">Initial instability and gain</h2></div>

Modulation instability only happens under certain circumstances. The most important condition is anomalous group velocity <a href="Dispersion_relation" title="Dispersion relation">dispersion</a>, whereby pulses with shorter <a href="Wavelength" title="Wavelength">wavelengths</a> travel with higher <a href="Group_velocity" title="Group velocity">group velocity</a> than pulses with longer wavelength.<a href="#cite_note-agrawal-3">[3]</a> (This condition assumes a focusing <a href="Kerr_nonlinearity" class="mw-redirect" title="Kerr nonlinearity">Kerr nonlinearity</a>, whereby refractive index increases with optical intensity.)<a href="#cite_note-agrawal-3">[3]</a>

The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, while at other frequencies, a perturbation will <a href="Exponential_growth" title="Exponential growth">grow exponentially</a>. The overall <a href="Gain_(electronics)" title="Gain (electronics)">gain</a> spectrum can be derived <a href="Analytical_expression" class="mw-redirect" title="Analytical expression">analytically</a>, as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.

The tendency of a perturbing signal to grow makes modulation instability a form of <a href="Amplifier" title="Amplifier">amplification</a>. By tuning an input signal to a peak of the gain spectrum, it is possible to create an <a href="Optical_amplifier" title="Optical amplifier">optical amplifier</a>.

<div class="mw-heading mw-heading3"><h3 id="Mathematical_derivation_of_gain_spectrum">Mathematical derivation of gain spectrum</h3></div>

The gain spectrum can be derived <a href="#cite_note-agrawal-3">[3]</a> by starting with a model of modulation instability based upon the <a href="Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">nonlinear Schrödinger equation</a>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial A}{\partial z}}+i\beta _{2}{\frac {\partial ^{2}A}{\partial t^{2}}}=i\gamma |A|^{2}A,}</annotation>

</semantics>

</math><img src="./e028466f3dbc065a27b1663baf37e8939ac28177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.531ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial A}{\partial z}}+i\beta _{2}{\frac {\partial ^{2}A}{\partial t^{2}}}=i\gamma |A|^{2}A,}" loading="lazy"></dd></dl>

which describes the evolution of a <a href="Complex_number" title="Complex number">complex-valued</a> <a href="Slowly_varying_envelope_approximation" title="Slowly varying envelope approximation">slowly varying envelope</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> with time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> and distance of propagation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">. The <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{2}=-1.}</annotation>

</semantics>

</math><img src="./52ca91534578075175b2088069fb6a9a414eaa7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.573ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1.}" loading="lazy"> The model includes <a href="Group_velocity" title="Group velocity">group velocity</a> dispersion described by the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{2}}</annotation>

</semantics>

</math><img src="./8d30285b40d7488ae6caef3beb7106142869fbea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{2}}" loading="lazy">, and <a href="Kerr_effect" title="Kerr effect">Kerr nonlinearity</a> with magnitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma .}</annotation>

</semantics>

</math><img src="./f423d4c0d1a3f651562797e2198c75a3f65e09fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.909ex; height:2.176ex;" alt="{\displaystyle \gamma .}" loading="lazy"> A <a href="Periodic_function" title="Periodic function">periodic</a> waveform of constant power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is assumed. This is given by the solution

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A={\sqrt {P}}e^{i\gamma Pz},}</annotation>

</semantics>

</math><img src="./dfa02b3114b3bb0198332329ac4edc7f6859af7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.949ex; height:3.009ex;" alt="{\displaystyle A={\sqrt {P}}e^{i\gamma Pz},}" loading="lazy"></dd></dl>

where the oscillatory <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\gamma Pz}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{i\gamma Pz}}</annotation>

</semantics>

</math><img src="./2a27d0f328098f1ddd2ae4387e3711ad679df022.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.779ex; height:2.676ex;" alt="{\displaystyle e^{i\gamma Pz}}" loading="lazy"> <a href="Wave_phase" class="mw-redirect" title="Wave phase">phase</a> factor accounts for the difference between the linear <a href="Refractive_index" title="Refractive index">refractive index</a>, and the modified <a href="Refractive_index" title="Refractive index">refractive index</a>, as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=\left({\sqrt {P}}+\varepsilon (t,z)\right)e^{i\gamma Pz},}</annotation>

</semantics>

</math><img src="./c494e171af65926b206f93398b16e230faaa3732.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.161ex; height:3.343ex;" alt="{\displaystyle A=\left({\sqrt {P}}+\varepsilon (t,z)\right)e^{i\gamma Pz},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (t,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon (t,z)}</annotation>

</semantics>

</math><img src="./ae17d38f48e9153c970ec2f55f420bd2a09bbb5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.854ex; height:2.843ex;" alt="{\displaystyle \varepsilon (t,z)}" loading="lazy"> is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">). Substituting this back into the nonlinear Schrödinger equation gives a <a href="Perturbation_theory" title="Perturbation theory">perturbation equation</a> of the form

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varepsilon }{\partial z}}+i\beta _{2}{\frac {\partial ^{2}\varepsilon }{\partial t^{2}}}=i\gamma P\left(\varepsilon +\varepsilon ^{*}\right),}</annotation>

</semantics>

</math><img src="./9372d53f2459d33699a8ca8fba08814675daaae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.773ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \varepsilon }{\partial z}}+i\beta _{2}{\frac {\partial ^{2}\varepsilon }{\partial t^{2}}}=i\gamma P\left(\varepsilon +\varepsilon ^{*}\right),}" loading="lazy"></dd></dl>

where the perturbation has been assumed to be small, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\varepsilon |^{2}\ll P.}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\varepsilon |^{2}\ll P.}</annotation>

</semantics>

</math><img src="./8de47917f547859c399a0b98ed7f63b6b736ab0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.438ex; height:3.343ex;" alt="{\displaystyle |\varepsilon |^{2}\ll P.}" loading="lazy"> The <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> is denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ^{*}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon ^{*}.}</annotation>

</semantics>

</math><img src="./bc89ded9747a4ca9cdfc1ca7a43925bceedf7c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.785ex; height:2.343ex;" alt="{\displaystyle \varepsilon ^{*}.}" loading="lazy"> Instability can now be discovered by searching for solutions of the perturbation equation which grow exponentially. This can be done using a trial function of the general form

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon =c_{1}e^{ik_{m}z-i\omega _{m}t}+c_{2}e^{-ik_{m}^{*}z+i\omega _{m}t},}</annotation>

</semantics>

</math><img src="./1a5032fa9f5f6e0cd024de99985414949e01988f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.354ex; height:3.176ex;" alt="{\displaystyle \varepsilon =c_{1}e^{ik_{m}z-i\omega _{m}t}+c_{2}e^{-ik_{m}^{*}z+i\omega _{m}t},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{m}}</annotation>

</semantics>

</math><img src="./25d16171af1c8efc54dfb43a8c83893cf7516f01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.886ex; height:2.509ex;" alt="{\displaystyle k_{m}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{m}}</annotation>

</semantics>

</math><img src="./238a329c04974a4de95ddcfc91409a2269aca64c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.121ex; height:2.009ex;" alt="{\displaystyle \omega _{m}}" loading="lazy"> are the <a href="Wavenumber" title="Wavenumber">wavenumber</a> and (real-valued) <a href="Angular_frequency" title="Angular frequency">angular frequency</a> of a perturbation, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1}}</annotation>

</semantics>

</math><img src="./77b7dc6d279091d354e0b90889b463bfa7eb7247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{2}}</annotation>

</semantics>

</math><img src="./0b30ba1b247fb8d334580cec68561e749d24aff2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{2}}" loading="lazy"> are constants. The nonlinear Schrödinger equation is constructed by removing the <a href="Carrier_wave" title="Carrier wave">carrier wave</a> of the light being modelled, and so the frequency of the light being perturbed is formally zero. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{m}}</annotation>

</semantics>

</math><img src="./238a329c04974a4de95ddcfc91409a2269aca64c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.121ex; height:2.009ex;" alt="{\displaystyle \omega _{m}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{m}}</annotation>

</semantics>

</math><img src="./25d16171af1c8efc54dfb43a8c83893cf7516f01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.886ex; height:2.509ex;" alt="{\displaystyle k_{m}}" loading="lazy"> don't represent absolute frequencies and wavenumbers, but the difference between these and those of the initial beam of light. It can be shown that the trial function is valid, provided <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}=c_{1}^{*}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{2}=c_{1}^{*}}</annotation>

</semantics>

</math><img src="./5a1e411384158235d3e6b070fbab704afa7177a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.221ex; height:2.843ex;" alt="{\displaystyle c_{2}=c_{1}^{*}}" loading="lazy"> and subject to the condition

<msub>

</mrow>

</msub>

<msqrt>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{m}=\pm {\sqrt {\beta _{2}^{2}\omega _{m}^{4}+2\gamma P\beta _{2}\omega _{m}^{2}}}.}</annotation>

</semantics>

</math><img src="./b5f76d83ffc7bb3fd8ad23fb9f18db4b5f6b66a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.777ex; height:4.843ex;" alt="{\displaystyle k_{m}=\pm {\sqrt {\beta _{2}^{2}\omega _{m}^{4}+2\gamma P\beta _{2}\omega _{m}^{2}}}.}" loading="lazy"></dd></dl>

This dispersion relation is vitally dependent on the sign of the term within the square root, as if positive, the wavenumber will be <a href="Real_number" title="Real number">real</a>, corresponding to mere <a href="Oscillation" title="Oscillation">oscillations</a> around the unperturbed solution, whilst if negative, the wavenumber will become <a href="Imaginary_number" title="Imaginary number">imaginary</a>, corresponding to exponential growth and thus instability. Therefore, instability will occur when

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{2}^{2}\omega _{m}^{2}+2\gamma P\beta _{2}<0,}</annotation>

</semantics>

</math><img src="./4b445795b9a5dbd8f904449f25ba5f6033ab0dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.8ex; height:3.176ex;" alt="{\displaystyle \beta _{2}^{2}\omega _{m}^{2}+2\gamma P\beta _{2}<0,}" loading="lazy">   that is for   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}}.}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}}.}</annotation>

</semantics>

</math><img src="./76f775ab97739c4495d93b1636051d89de1e2deb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.681ex; height:5.843ex;" alt="{\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}}.}" loading="lazy"></dd></dl>

This condition describes the requirement for anomalous dispersion (such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \beta _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \beta _{2}}</annotation>

</semantics>

</math><img src="./1f15e61fd88984b8045789cba12de729bab6a692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.632ex; height:2.676ex;" alt="{\displaystyle \gamma \beta _{2}}" loading="lazy"> is negative). The gain spectrum can be described by defining a gain parameter as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\equiv 2|\Im \{k_{m}\}|,}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\equiv 2|\Im \{k_{m}\}|,}</annotation>

</semantics>

</math><img src="./b5c7428e8957bdb895bc111951baa0f6974dab2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.816ex; height:2.843ex;" alt="{\displaystyle g\equiv 2|\Im \{k_{m}\}|,}" loading="lazy"> so that the power of a perturbing signal grows with distance as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\,e^{gz}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\,e^{gz}.}</annotation>

</semantics>

</math><img src="./8592f177d93891a7577885c4cf497a1c4ff98e29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.654ex; height:2.343ex;" alt="{\displaystyle P\,e^{gz}.}" loading="lazy"> The gain is therefore given by

<mrow>

<mtr>

<mtd>

<msqrt>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g={\begin{cases}2{\sqrt {-\beta _{2}^{2}\omega _{m}^{4}-2\gamma P\beta _{2}\omega _{m}^{2}}},&{\text{for }}\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}},\\[2ex]0,&{\text{for }}\displaystyle \omega _{m}^{2}\geq -2{\frac {\gamma P}{\beta _{2}}},\end{cases}}}</annotation>

</semantics>

</math><img src="./a84ce71136981fd3f1fd145162d9ca35220365d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:50.356ex; height:13.843ex;" alt="{\displaystyle g={\begin{cases}2{\sqrt {-\beta _{2}^{2}\omega _{m}^{4}-2\gamma P\beta _{2}\omega _{m}^{2}}},&{\text{for }}\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}},\\[2ex]0,&{\text{for }}\displaystyle \omega _{m}^{2}\geq -2{\frac {\gamma P}{\beta _{2}}},\end{cases}}}" loading="lazy"></dd></dl>

where as noted above, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{m}}</annotation>

</semantics>

</math><img src="./238a329c04974a4de95ddcfc91409a2269aca64c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.121ex; height:2.009ex;" alt="{\displaystyle \omega _{m}}" loading="lazy"> is the difference between the frequency of the perturbation and the frequency of the initial light. The growth rate is maximum for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}=-\gamma P/\beta _{2}.}">

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ^{2}=-\gamma P/\beta _{2}.}</annotation>

</semantics>

</math><img src="./6feed5565e611f6acbc51f5709e8643205133bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.594ex; height:3.176ex;" alt="{\displaystyle \omega ^{2}=-\gamma P/\beta _{2}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Modulation_instability_in_soft_systems">Modulation instability in soft systems</h2></div>

Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a> Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index.<a href="#cite_note-13">[13]</a> Modulation instability of spatially and temporally incoherent light is possible owing to the non-instantaneous response of photoreactive systems, which consequently responds to the time-average intensity of light, in which the femto-second fluctuations cancel out.<a href="#cite_note-14">[14]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBenjaminFeir1967" class="citation journal cs1"><a href="T._Brooke_Benjamin" class="mw-redirect" title="T. Brooke Benjamin">Benjamin, T. Brooke</a>; Feir, J.E. (1967). "The disintegration of wave trains on deep water. Part 1. Theory". Journal of Fluid Mechanics. 27 (3): 417–430. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967JFM....27..417B">1967JFM....27..417B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS002211206700045X">10.1017/S002211206700045X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121996479">121996479</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFBenjamin1967" class="citation journal cs1"><a href="T._Brooke_Benjamin" class="mw-redirect" title="T. Brooke Benjamin">Benjamin, T.B.</a> (1967). "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems". Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 299 (1456): 59–76. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967RSPSA.299...59B">1967RSPSA.299...59B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1967.0123">10.1098/rspa.1967.0123</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121661209">121661209</a>.</cite> Concluded with a discussion by <a href="Klaus_Hasselmann" title="Klaus Hasselmann">Klaus Hasselmann</a>.

</li>

<li id="cite_note-agrawal-3">^ <a href="#cite_ref-agrawal_3-0">a</a> <a href="#cite_ref-agrawal_3-1">b</a> <a href="#cite_ref-agrawal_3-2">c</a> <a href="#cite_ref-agrawal_3-3">d</a> <cite id="CITEREFAgrawal1995" class="citation book cs1">Agrawal, Govind P. (1995). Nonlinear fiber optics (2nd ed.). San Diego (California): Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-045142-5</bdi>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFYuenLake1980" class="citation journal cs1">Yuen, H.C.; Lake, B.M. (1980). "Instabilities of waves on deep water". Annual Review of Fluid Mechanics. 12: 303–334. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980AnRFM..12..303Y">1980AnRFM..12..303Y</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.fl.12.010180.001511">10.1146/annurev.fl.12.010180.001511</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFPiliptetskiiRustamov1965" class="citation journal cs1">Piliptetskii, N. F.; Rustamov, A. R. (31 May 1965). <a rel="nofollow" class="external text" href="http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml">"Observation of Self-focusing of Light in Liquids"</a>. JETP Letters. 2 (2): 55–56.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBespalovTalanov1966" class="citation journal cs1">Bespalov, V. I.; Talanov, V. I. (15 June 1966). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200731112029/http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml">"Filamentary Structure of Light Beams in Nonlinear Liquids"</a>. ZhETF Pisma Redaktsiiu. 3 (11): 471–476. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966ZhPmR...3..471B">1966ZhPmR...3..471B</a>. Archived from <a rel="nofollow" class="external text" href="http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml">the original</a> on 31 July 2020. Retrieved 17 February 2021.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFJanssen2003" class="citation journal cs1">Janssen, Peter A.E.M. (2003). <a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0485%282003%2933%3C863%3ANFIAFW%3E2.0.CO%3B2">"Nonlinear four-wave interactions and freak waves"</a>. Journal of Physical Oceanography. 33 (4): 863–884. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003JPO....33..863J">2003JPO....33..863J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0485%282003%2933%3C863%3ANFIAFW%3E2.0.CO%3B2">10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFDystheKrogstadMüller2008" class="citation journal cs1">Dysthe, Kristian; Krogstad, Harald E.; Müller, Peter (2008). "Oceanic rogue waves". Annual Review of Fluid Mechanics. 40 (1): 287–310. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008AnRFM..40..287D">2008AnRFM..40..287D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.fluid.40.111406.102203">10.1146/annurev.fluid.40.111406.102203</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFBurgessShimmellSaravanamuttu2007" class="citation journal cs1">Burgess, Ian B.; Shimmell, Whitney E.; Saravanamuttu, Kalaichelvi (2007-04-01). "Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium". Journal of the American Chemical Society. 129 (15): 4738–4746. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007JAChS.129.4738B">2007JAChS.129.4738B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fja068967b">10.1021/ja068967b</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-7863">0002-7863</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17378567">17378567</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFBaskerBrookSaravanamuttu2015" class="citation journal cs1">Basker, Dinesh K.; Brook, Michael A.; Saravanamuttu, Kalaichelvi (2015). "Spontaneous Emergence of Nonlinear Light Waves and Self-Inscribed Waveguide Microstructure during the Cationic Polymerization of Epoxides". The Journal of Physical Chemistry C. 119 (35): 20606–20617. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facs.jpcc.5b07117">10.1021/acs.jpcc.5b07117</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFBiriaMalleyKahanHosein2016" class="citation journal cs1">Biria, Saeid; Malley, Philip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-03-03). "Tunable Nonlinear Optical Pattern Formation and Microstructure in Cross-Linking Acrylate Systems during Free-Radical Polymerization". The Journal of Physical Chemistry C. 120 (8): 4517–4528. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facs.jpcc.5b11377">10.1021/acs.jpcc.5b11377</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1932-7447">1932-7447</a>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFBiriaMalleyKahanHosein2016" class="citation journal cs1">Biria, Saeid; Malley, Phillip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-11-15). "Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring". ACS Macro Letters. 5 (11): 1237–1241. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facsmacrolett.6b00659">10.1021/acsmacrolett.6b00659</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35614732">35614732</a>.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFKewitschYariv1996" class="citation journal cs1">Kewitsch, Anthony S.; Yariv, Amnon (1996-01-01). <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/2845/1/KEWol96.pdf">"Self-focusing and self-trapping of optical beams upon photopolymerization"</a> (PDF). Optics Letters. 21 (1): 24–6. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996OptL...21...24K">1996OptL...21...24K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2Fol.21.000024">10.1364/ol.21.000024</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1539-4794">1539-4794</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19865292">19865292</a>.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://www.springer.com/us/book/9783540416531">Spatial Solitons | Stefano Trillo | Springer</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFZakharovOstrovsky2009" class="citation journal cs1"><a href="Vladimir_E._Zakharov" title="Vladimir E. Zakharov">Zakharov, V.E.</a>; Ostrovsky, L.A. (2009). <a rel="nofollow" class="external text" href="http://www.math.umass.edu/~kevrekid/math697/sdarticle_ZO.pdf">"Modulation instability: The beginning"</a> (PDF). Physica D: Nonlinear Phenomena. 238 (5): 540–548. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009PhyD..238..540Z">2009PhyD..238..540Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physd.2008.12.002">10.1016/j.physd.2008.12.002</a>.</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Physical_oceanography593" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Physical_oceanography593" style="font-size:114%;margin:0 4em"><a href="Physical_oceanography" title="Physical oceanography">Physical oceanography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Wind_wave" title="Wind wave">Waves</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Airy_wave_theory" title="Airy wave theory">Airy wave theory</a></li>

<li><a href="Ballantine_scale" title="Ballantine scale">Ballantine scale</a></li>

<li><a href="Boussinesq_approximation_(water_waves)" title="Boussinesq approximation (water waves)">Boussinesq approximation</a></li>

<li><a href="Breaking_wave" title="Breaking wave">Breaking wave</a></li>

<li><a href="Clapotis" title="Clapotis">Clapotis</a></li>

<li><a href="Cnoidal_wave" title="Cnoidal wave">Cnoidal wave</a></li>

<li><a href="Cross_sea" title="Cross sea">Cross sea</a></li>

<li><a href="Dispersion_(water_waves)" title="Dispersion (water waves)">Dispersion</a></li>

<li><a href="Equatorial_wave" title="Equatorial wave">Equatorial waves</a></li>

<li><a href="Gravity_wave" title="Gravity wave">Gravity wave</a></li>

<li><a href="Green's_law" title="Green's law">Green's law</a></li>

<li><a href="Infragravity_wave" title="Infragravity wave">Infragravity wave</a></li>

<li><a href="Internal_wave" title="Internal wave">Internal wave</a></li>

<li><a href="Iribarren_number" title="Iribarren number">Iribarren number</a></li>

<li><a href="Kelvin_wave" title="Kelvin wave">Kelvin wave</a></li>

<li><a href="Kinematic_wave" title="Kinematic wave">Kinematic wave</a></li>

<li><a href="Longshore_drift" title="Longshore drift">Longshore drift</a></li>

<li><a href="Luke's_variational_principle" title="Luke's variational principle">Luke's variational principle</a></li>

<li><a href="Mild-slope_equation" title="Mild-slope equation">Mild-slope equation</a></li>

<li><a href="Radiation_stress" title="Radiation stress">Radiation stress</a></li>

<li><a href="Rogue_wave" title="Rogue wave">Rogue wave</a>

<ul><li><a href="Draupner_wave" title="Draupner wave">Draupner wave</a></li></ul></li>

<li><a href="Rossby_wave" title="Rossby wave">Rossby wave</a></li>

<li><a href="Rossby-gravity_waves" title="Rossby-gravity waves">Rossby-gravity waves</a></li>

<li><a href="Sea_state" title="Sea state">Sea state</a></li>

<li><a href="Seiche" title="Seiche">Seiche</a></li>

<li><a href="Significant_wave_height" title="Significant wave height">Significant wave height</a></li>

<li><a href="Soliton" title="Soliton">Soliton</a></li>

<li><a href="Stokes_drift" title="Stokes drift">Stokes drift</a></li>

<li><a href="Stokes_problem" title="Stokes problem">Stokes problem</a></li>

<li><a href="Stokes_wave" title="Stokes wave">Stokes wave</a></li>

<li><a href="Swell_(ocean)" title="Swell (ocean)">Swell</a></li>

<li><a href="Trochoidal_wave" title="Trochoidal wave">Trochoidal wave</a></li>

<li><a href="Tsunami" title="Tsunami">Tsunami</a>

<ul><li><a href="Megatsunami" title="Megatsunami">megatsunami</a></li></ul></li>

<li><a href="Undertow_(water_waves)" title="Undertow (water waves)">Undertow</a></li>

<li><a href="Ursell_number" title="Ursell number">Ursell number</a></li>

<li><a href="Wave_action_(continuum_mechanics)" title="Wave action (continuum mechanics)">Wave action</a></li>

<li><a href="Wave_height" title="Wave height">Wave height</a></li>

<li><a href="Wave_nonlinearity" title="Wave nonlinearity">Wave nonlinearity</a></li>

<li><a href="Wave_power" title="Wave power">Wave power</a></li>

<li><a href="Wave_radar" title="Wave radar">Wave radar</a></li>

<li><a href="Wave_setup" title="Wave setup">Wave setup</a></li>

<li><a href="Wave_shoaling" title="Wave shoaling">Wave shoaling</a></li>

<li><a href="Wave_turbulence" title="Wave turbulence">Wave turbulence</a></li>

<li><a href="Wave%E2%80%93current_interaction" title="Wave–current interaction">Wave–current interaction</a></li>

<li><a href="Waves_and_shallow_water" title="Waves and shallow water">Waves and shallow water</a>

<ul><li><a href="One-dimensional_Saint-Venant_equations" class="mw-redirect" title="One-dimensional Saint-Venant equations">one-dimensional Saint-Venant equations</a></li>

<li><a href="Shallow_water_equations" title="Shallow water equations">shallow water equations</a></li></ul></li>

<li><a href="Wind_fetch" title="Wind fetch">Wind fetch</a></li>

<li><a href="Wind_setup" title="Wind setup">Wind setup</a></li>

<ul><li><a href="Wind_wave_model" title="Wind wave model">model</a></li></ul></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="10" style="width:1px;padding:0 0 0 2px"><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Ocean_current" title="Ocean current">Circulation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Atmospheric_circulation" title="Atmospheric circulation">Atmospheric circulation</a></li>

<li><a href="Baroclinity" title="Baroclinity">Baroclinity</a></li>

<li><a href="Boundary_current" title="Boundary current">Boundary current</a></li>

<li><a href="Coriolis_force" title="Coriolis force">Coriolis force</a></li>

<li><a href="Coriolis%E2%80%93Stokes_force" title="Coriolis–Stokes force">Coriolis–Stokes force</a></li>

<li><a href="Craik%E2%80%93Leibovich_vortex_force" title="Craik–Leibovich vortex force">Craik–Leibovich vortex force</a></li>

<li><a href="Downwelling" title="Downwelling">Downwelling</a></li>

<li><a href="Ekman_layer" class="mw-redirect" title="Ekman layer">Ekman layer</a></li>

<li><a href="Ekman_spiral" class="mw-redirect" title="Ekman spiral">Ekman spiral</a></li>

<li><a href="Ekman_transport" title="Ekman transport">Ekman transport</a></li>

<li><a href="El_Ni%C3%B1o%E2%80%93Southern_Oscillation" title="El Niño–Southern Oscillation">El Niño–Southern Oscillation</a></li>

<li><a href="General_circulation_model" title="General circulation model">General circulation model</a></li>

<li><a href="Geochemical_Ocean_Sections_Study" title="Geochemical Ocean Sections Study">Geochemical Ocean Sections Study</a></li>

<li><a href="Geostrophic_current" title="Geostrophic current">Geostrophic current</a></li>

<li><a href="Global_Ocean_Data_Analysis_Project" title="Global Ocean Data Analysis Project">Global Ocean Data Analysis Project</a></li>

<li><a href="Gulf_Stream" title="Gulf Stream">Gulf Stream</a></li>

<li><a href="Humboldt_Current" title="Humboldt Current">Humboldt Current</a></li>

<li><a href="Hydrothermal_circulation" title="Hydrothermal circulation">Hydrothermal circulation</a></li>

<li><a href="Langmuir_circulation" title="Langmuir circulation">Langmuir circulation</a></li>

<li><a href="Longshore_drift" title="Longshore drift">Longshore drift</a></li>

<li><a href="Loop_Current" title="Loop Current">Loop Current</a></li>

<li><a href="Modular_Ocean_Model" title="Modular Ocean Model">Modular Ocean Model</a></li>

<li><a href="Ocean_current" title="Ocean current">Ocean current</a></li>

<li><a href="Ocean_dynamical_thermostat" title="Ocean dynamical thermostat">Ocean dynamical thermostat</a></li>

<li><a href="Ocean_dynamics" title="Ocean dynamics">Ocean dynamics</a></li>

<li><a href="Ocean_gyre" title="Ocean gyre">Ocean gyre</a></li>

<li><a href="Overflow_(oceanography)" title="Overflow (oceanography)">Overflow</a></li>

<li><a href="Princeton_Ocean_Model" title="Princeton Ocean Model">Princeton Ocean Model</a></li>

<li><a href="Rip_current" title="Rip current">Rip current</a></li>

<li><a href="Subsurface_ocean_current" title="Subsurface ocean current">Subsurface ocean current</a></li>

<li><a href="Sverdrup_balance" title="Sverdrup balance">Sverdrup balance</a></li>

<li><a href="Thermohaline_circulation" title="Thermohaline circulation">Thermohaline circulation</a>

<ul><li><a href="Shutdown_of_thermohaline_circulation" class="mw-redirect" title="Shutdown of thermohaline circulation">shutdown</a></li></ul></li>

<li><a href="Upwelling" title="Upwelling">Upwelling</a></li>

<li><a href="Whirlpool" title="Whirlpool">Whirlpool</a></li>

<li><a href="Wind_generated_current" title="Wind generated current">Wind generated current</a></li>

<li><a href="World_Ocean_Circulation_Experiment" title="World Ocean Circulation Experiment">World Ocean Circulation Experiment</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Tide" title="Tide">Tides</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amphidromic_point" title="Amphidromic point">Amphidromic point</a></li>

<li><a href="Earth_tide" title="Earth tide">Earth tide</a></li>

<li><a href="Internal_tide" title="Internal tide">Internal tide</a></li>

<li><a href="Lunitidal_interval" title="Lunitidal interval">Lunitidal interval</a></li>

<li><a href="Perigean_spring_tide" title="Perigean spring tide">Perigean spring tide</a></li>

<li><a href="Rule_of_twelfths" title="Rule of twelfths">Rule of twelfths</a></li>

<li><a href="Slack_tide" title="Slack tide">Slack tide</a></li>

<li><a href="Theory_of_tides" title="Theory of tides">Theory of tides</a></li>

<li><a href="Tidal_bore" title="Tidal bore">Tidal bore</a></li>

<li><a href="Tidal_force" title="Tidal force">Tidal force</a></li>

<li><a href="Tidal_power" title="Tidal power">Tidal power</a></li>

<li><a href="Tidal_race" title="Tidal race">Tidal race</a></li>

<li><a href="Tidal_range" title="Tidal range">Tidal range</a></li>

<li><a href="Tidal_resonance" title="Tidal resonance">Tidal resonance</a></li>

<li><a href="Tide_gauge" title="Tide gauge">Tide gauge</a></li>

<li><a href="Tideline" title="Tideline">Tideline</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Landform" title="Landform">Landforms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abyssal_fan" title="Abyssal fan">Abyssal fan</a></li>

<li><a href="Abyssal_plain" title="Abyssal plain">Abyssal plain</a></li>

<li><a href="Atoll" title="Atoll">Atoll</a></li>

<li><a href="Bathymetric_chart" title="Bathymetric chart">Bathymetric chart</a></li>

<li><a href="Carbonate_platform" title="Carbonate platform">Carbonate platform</a></li>

<li><a href="Coastal_geography" title="Coastal geography">Coastal geography</a></li>

<li><a href="Continental_margin" title="Continental margin">Continental margin</a></li>

<li><a href="Continental_rise" title="Continental rise">Continental rise</a></li>

<li><a href="Continental_shelf" title="Continental shelf">Continental shelf</a></li>

<li><a href="Contourite" title="Contourite">Contourite</a></li>

<li><a href="Guyot" title="Guyot">Guyot</a></li>

<li><a href="Hydrography" title="Hydrography">Hydrography</a></li>

<li><a href="Knoll_(oceanography)" title="Knoll (oceanography)">Knoll</a></li>

<li><a href="Ocean_bank" title="Ocean bank">Ocean bank</a></li>

<li><a href="Oceanic_basin" title="Oceanic basin">Oceanic basin</a></li>

<li><a href="Oceanic_plateau" title="Oceanic plateau">Oceanic plateau</a></li>

<li><a href="Oceanic_trench" title="Oceanic trench">Oceanic trench</a></li>

<li><a href="Passive_margin" title="Passive margin">Passive margin</a></li>

<li><a href="Seabed" title="Seabed">Seabed</a></li>

<li><a href="Seamount" title="Seamount">Seamount</a></li>

<li><a href="Submarine_canyon" title="Submarine canyon">Submarine canyon</a></li>

<li><a href="Submarine_volcano" title="Submarine volcano">Submarine volcano</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Plate_tectonics" title="Plate tectonics">Plate tectonics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convergent_boundary" title="Convergent boundary">Convergent boundary</a></li>

<li><a href="Divergent_boundary" title="Divergent boundary">Divergent boundary</a></li>

<li><a href="Fracture_zone" title="Fracture zone">Fracture zone</a></li>

<li><a href="Hydrothermal_vent" title="Hydrothermal vent">Hydrothermal vent</a></li>

<li><a href="Marine_geology" title="Marine geology">Marine geology</a></li>

<li><a href="Mid-ocean_ridge" title="Mid-ocean ridge">Mid-ocean ridge</a></li>

<li><a href="Mohorovi%C4%8Di%C4%87_discontinuity" title="Mohorovičić discontinuity">Mohorovičić discontinuity</a></li>

<li><a href="Oceanic_crust" title="Oceanic crust">Oceanic crust</a></li>

<li><a href="Outer_trench_swell" title="Outer trench swell">Outer trench swell</a></li>

<li><a href="Ridge_push" title="Ridge push">Ridge push</a></li>

<li><a href="Seafloor_spreading" title="Seafloor spreading">Seafloor spreading</a></li>

<li><a href="Slab_suction" title="Slab suction">Slab suction</a></li>

<li><a href="Slab_window" title="Slab window">Slab window</a></li>

<li><a href="Subduction" title="Subduction">Subduction</a></li>

<li><a href="Transform_fault" title="Transform fault">Transform fault</a></li>

<li><a href="Vine%E2%80%93Matthews%E2%80%93Morley_hypothesis" title="Vine–Matthews–Morley hypothesis">Vine–Matthews–Morley hypothesis</a></li>

<li><a href="Volcanic_arc" title="Volcanic arc">Volcanic arc</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ocean zones</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Benthic_zone" title="Benthic zone">Benthic</a></li>

<li><a href="Deep_ocean_water" title="Deep ocean water">Deep ocean water</a></li>

<li><a href="Littoral_zone" title="Littoral zone">Littoral</a></li>

<li><a href="Mesopelagic_zone" title="Mesopelagic zone">Mesopelagic</a></li>

<li><a href="Oceanic_zone" title="Oceanic zone">Oceanic</a></li>

<li><a href="Pelagic_zone" title="Pelagic zone">Pelagic</a></li>

<li><a href="Photic_zone" title="Photic zone">Photic</a></li>

<li><a href="Swash" title="Swash">Swash</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Sea_level" title="Sea level">Sea level</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Deep-ocean_Assessment_and_Reporting_of_Tsunamis" title="Deep-ocean Assessment and Reporting of Tsunamis">Deep-ocean Assessment and Reporting of Tsunamis</a></li>

<li><a href="Global_Sea_Level_Observing_System" title="Global Sea Level Observing System">Global Sea Level Observing System</a></li>

<li><a href="North_West_Shelf_Operational_Oceanographic_System" title="North West Shelf Operational Oceanographic System">North West Shelf Operational Oceanographic System</a></li>

<li><a href="Sea-level_curve" title="Sea-level curve">Sea-level curve</a></li>

<li><a href="Sea_level_drop" title="Sea level drop">Sea level drop</a></li>

<li><a href="Sea_level_rise" title="Sea level rise">Sea level rise</a></li>

<li><a href="World_Geodetic_System" title="World Geodetic System">World Geodetic System</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Acoustical_oceanography" class="mw-redirect" title="Acoustical oceanography">Acoustics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Deep_scattering_layer" title="Deep scattering layer">Deep scattering layer</a></li>

<li><a href="Ocean_acoustic_tomography" title="Ocean acoustic tomography">Ocean acoustic tomography</a></li>

<li><a href="Sofar_bomb" title="Sofar bomb">Sofar bomb</a></li>

<li><a href="SOFAR_channel" title="SOFAR channel">SOFAR channel</a></li>

<li><a href="Underwater_acoustics" title="Underwater acoustics">Underwater acoustics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satellites</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Jason-1" title="Jason-1">Jason-1</a></li>

<li><a href="OSTM/Jason-2" title="OSTM/Jason-2">OSTM/Jason-2</a></li>

<li><a href="Jason-3" title="Jason-3">Jason-3</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ocean_acidification" title="Ocean acidification">Acidification</a></li>

<li><a href="Benthic_lander" title="Benthic lander">Benthic lander</a></li>

<li><a href="Color_of_water" title="Color of water">Color of water</a></li>

<li><a href="DSV_Alvin" title="DSV Alvin">DSV Alvin</a></li>

<li><a href="Marginal_sea" class="mw-redirect" title="Marginal sea">Marginal sea</a></li>

<li><a href="Marine_energy" title="Marine energy">Marine energy</a></li>

<li><a href="Marine_pollution" title="Marine pollution">Marine pollution</a></li>

<li><a href="Mooring_(oceanography)" title="Mooring (oceanography)">Mooring</a></li>

<li><a href="National_Oceanographic_Data_Center" title="National Oceanographic Data Center">National Oceanographic Data Center</a></li>

<li><a href="Ocean" title="Ocean">Ocean</a></li>

<li><a href="Ocean_exploration" title="Ocean exploration">Explorations</a></li>

<li><a href="Ocean_observations" title="Ocean observations">Observations</a></li>

<li><a href="Ocean_reanalysis" title="Ocean reanalysis">Reanalysis</a></li>

<li><a href="Ocean_surface_topography" title="Ocean surface topography">Ocean surface topography</a></li>

<li><a href="Ocean_temperature" title="Ocean temperature">Ocean temperature</a></li>

<li><a href="Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Ocean thermal energy conversion</a></li>

<li><a href="Oceanography" title="Oceanography">Oceanography</a>

<ul><li><a href="Outline_of_oceanography" title="Outline of oceanography">Outline of oceanography</a></li></ul></li>

<li><a href="Pelagic_sediment" title="Pelagic sediment">Pelagic sediment</a></li>

<li><a href="Sea_surface_microlayer" title="Sea surface microlayer">Sea surface microlayer</a></li>

<li><a href="Sea_surface_temperature" title="Sea surface temperature">Sea surface temperature</a></li>

<li><a href="Seawater" title="Seawater">Seawater</a></li>

<li><a href="Science_On_a_Sphere" title="Science On a Sphere">Science On a Sphere</a></li>

<li><a href="Ocean_stratification" title="Ocean stratification">Stratification</a></li>

<li><a href="Thermocline" title="Thermocline">Thermocline</a></li>

<li><a href="Underwater_glider" title="Underwater glider">Underwater glider</a></li>

<li><a href="Water_column" title="Water column">Water column</a></li>

<li><a href="World_Ocean_Atlas" title="World Ocean Atlas">World Ocean Atlas</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li> Category</li>

<li> <a href="https://commons.wikimedia.org/wiki/Category:Physical_oceanography" class="extiw external" title="commons:Category:Physical oceanography">Commons</a></li>

<li> <a href="Portal%3AOceans" title="Portal:Oceans">Oceans portal</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-21" href="https://en.wikipedia.org/wiki/?title=Modulational_instability&oldid=1296649136">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>