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<title>Riemann-Hilbert-Problem</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Riemann-Hilbert-Problem</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Riemann-Hilbert-Probleme (kurz RHP oder auch Riemann-Hilbert-Analysis) sind eine Klasse von mathematischen Problemstellungen, in denen eine <a href="Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertige Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> gesucht wird.

Die Problemstellung ist folgende: Gegeben sei eine <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientierte</a>, <a href="Glatte_Funktion" title="Glatte Funktion">glatte</a> <a href="Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma \in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50ae48d94330adee0ca98345e1a5aad7c4838250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.972ex; height:2.176ex;" alt="{\displaystyle \Gamma \in \mathbb {C} }" loading="lazy"> und eine Jump-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy">, um von einer Seite der Kurve auf die andere Seite zu gelangen. Das Ziel ist es nun, die darunterliegende Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> zu rekonstruieren, welche <a href="Analytische_Funktion" title="Analytische Funktion">analytisch</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \setminus \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \setminus \Gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb8209aae4cdb977713d071dd86bc5b47023e41c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.326ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \setminus \Gamma }" loading="lazy"> ist.

Die Probleme sind nach den deutschen Mathematikern <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> und <a href="David_Hilbert" title="David Hilbert">David Hilbert</a> benannt und haben mannigfaltige Anwendungen in der <a href="Mathematik" title="Mathematik">Mathematik</a> und <a href="Physik" title="Physik">Physik</a>, unter anderem trifft man sie in der Theorie der <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">nichtlinearen partiellen Differentialgleichungen</a> an.

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

Viele Problemstellungen lassen sich als Riemann-Hilbert-Probleme formulieren und sind Startpunkt für <a href="Asymptotische_Analyse" title="Asymptotische Analyse">asymptotische Analyse</a>. Klassische Riemann-Hilbert-Probleme sind das Lösen von Differentialgleichungen wie der <a href="Painlev%C3%A9-Gleichungen" title="Painlevé-Gleichungen">Painlevé-Gleichung vom Typ 2</a> (<a href="Airy-Funktion" title="Airy-Funktion">Airy-Funktion</a>) oder das Finden von <a href="Orthogonale_Polynome" title="Orthogonale Polynome">orthogonalen Polynome</a>, wie man sie in der Spektraltheorie von <a href="Zufallsmatrix" title="Zufallsmatrix">Zufallsmatrizen</a> benötigt.

Auch lässt sich das Finden von Lösungen für <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">nichtlineare partielle Differentialgleichungen</a> wie der <a href="Korteweg-de-Vries-Gleichung" title="Korteweg-de-Vries-Gleichung">KdV-Gleichung</a> als RHP formulieren.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Riemann-Hilbert-Problem">Riemann-Hilbert-Problem</h2></div>

Eine orientierte, glatte Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \subset \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma \subset \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fce8aadc4112fac06bf31597b053e3f2b666cd6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.229ex; height:2.176ex;" alt="{\displaystyle \Gamma \subset \mathbb {C} }" loading="lazy"> teilt die komplexe Ebene in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{+}\cup \Omega _{-}\cup \Gamma }">

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \Omega _{+}\cup \Omega _{-}\cup \Gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45416f9cc1bb383932c3386d76ac74058050c17b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.996ex; height:2.509ex;" alt="{\displaystyle \Omega _{+}\cup \Omega _{-}\cup \Gamma }" loading="lazy"> auf, wobei die positive Seite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{+}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aece5d68daf75b9115c8bda0d66c4a0c1d462888.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Omega _{+}}" loading="lazy"> links von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> liegt.

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}(z)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/534ec0d875cd99c09a4fdc30eaf7cb7657f3a3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.662ex; height:2.843ex;" alt="{\displaystyle M_{+}(z)}" loading="lazy"> respektive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{-}(z)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/196c9104f76c8d368acf1d9b3b4621c2fd98464c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.662ex; height:2.843ex;" alt="{\displaystyle M_{-}(z)}" loading="lazy"> bezeichnen wir die Limits von der positiven Seite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{+}\in \Omega _{+}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle x_{+}\in \Omega _{+}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d4f6dd5b7d1eaca33dd839cc8798d4c3e55af83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.87ex; height:2.509ex;" alt="{\displaystyle x_{+}\in \Omega _{+}}" loading="lazy"> resp. der negativen Seite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{-}\in \Omega _{-}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{-}\in \Omega _{-}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4105223e5d035763ce9497fcf2cd7595548aa2cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.87ex; height:2.509ex;" alt="{\displaystyle x_{-}\in \Omega _{-}}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \Gamma }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e004f6d4ebe37811f6370b11b353cb159800040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.382ex; height:2.176ex;" alt="{\displaystyle z\in \Gamma }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle M_{+}(z):=\lim \limits _{x_{+}\to z}M(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d71ea58f2061893c4fe0f8162988f27b1b52d316.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.931ex; height:4.176ex;" alt="{\displaystyle M_{+}(z):=\lim \limits _{x_{+}\to z}M(x)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle M_{-}(z):=\lim \limits _{x_{-}\to z}M(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/270ef9fe11d54a0360f02033845de360dda9140c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.931ex; height:4.176ex;" alt="{\displaystyle M_{-}(z):=\lim \limits _{x_{-}\to z}M(x)}" loading="lazy">,</dd></dl>

sofern diese existieren. Weiter sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> die Menge der Punkte, in denen sich die Kurve selbst überschneidet. Dann definiere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{u}:=\Gamma \setminus u}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{u}:=\Gamma \setminus u}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d0f2807990d29d0f483c1f4fae757f819605d2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.348ex; height:2.843ex;" alt="{\displaystyle \Gamma _{u}:=\Gamma \setminus u}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Formulierung">Formulierung</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \subset \mathbb {C} }">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma \subset \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fce8aadc4112fac06bf31597b053e3f2b666cd6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.229ex; height:2.176ex;" alt="{\displaystyle \Gamma \subset \mathbb {C} }" loading="lazy"> eine orientierte, glatte Kurve und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J:\Gamma _{u}\to \operatorname {GL} (n,\mathbb {C} )}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J:\Gamma _{u}\to \operatorname {GL} (n,\mathbb {C} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0588e923f38af355d9fa89ff4ef0a76f52d38d7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.841ex; height:2.843ex;" alt="{\displaystyle J:\Gamma _{u}\to \operatorname {GL} (n,\mathbb {C} )}" loading="lazy"> eine <a href="Glatte_Funktion" title="Glatte Funktion">glatte Funktion</a>.

Dann definiert das paar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Gamma ,J)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (\Gamma ,J)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbc1a1644df6caf667e8ee2f6108f28b644f409f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.767ex; height:2.843ex;" alt="{\displaystyle (\Gamma ,J)}" loading="lazy"> ein Riemann-Hilbert-Problem:

Gesucht wird eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M:\mathbb {C} \to \mathbb {C} ^{n\times n}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M:\mathbb {C} \to \mathbb {C} ^{n\times n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b354e932731c54cfb70d4dc758975f96331a5314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.833ex; height:2.343ex;" alt="{\displaystyle M:\mathbb {C} \to \mathbb {C} ^{n\times n}}" loading="lazy">, so dass

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> ist <a href="Analytische_Funktion" title="Analytische Funktion">analytisch</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \setminus \Gamma }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \setminus \Gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb8209aae4cdb977713d071dd86bc5b47023e41c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.326ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \setminus \Gamma }" loading="lazy">.</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{+}(z)=M_{-}(z)J(z),\quad z\in \Gamma _{u}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e93206bdf67ca1cb9f609e5f2e6c46ac9fadafc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.702ex; height:2.843ex;" alt="{\displaystyle M_{+}(z)=M_{-}(z)J(z),\quad z\in \Gamma _{u}}" loading="lazy">.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z)\to I_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/772dc73c30a1331d4bc8c33dda42ef3322a82b23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.195ex; height:2.843ex;" alt="{\displaystyle M(z)\to I_{n}}" loading="lazy"> wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\to \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/380ee74715fed4cd83f76bacac1add6aad2d58c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.026ex; height:1.843ex;" alt="{\displaystyle z\to \infty }" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Existenz_und_Eindeutigkeit_der_Lösung">Existenz und Eindeutigkeit der Lösung</h2></div>

Die Existenz einer Lösung zu einem RHP zu zeigen ist keine triviale Aufgabe und oft schwieriger als die Eindeutigkeit. Eine klassische Methode für ein RHP ist die <a href="Methode_des_steilsten_Anstiegs" class="mw-redirect" title="Methode des steilsten Anstiegs">Methode des steilsten Anstiegs</a> (<a href="Englische_Sprache" title="Englische Sprache">englisch</a> Method of steepest descent) von Deift und Zhou.<a href="#cite_note-2">[2]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Skalares_Hilbert-Riemann-Problem">Skalares Hilbert-Riemann-Problem</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =\mathbb {R} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma =\mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc9445f6e097bfca1e5148308ef9ed52451e5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.229ex; height:2.176ex;" alt="{\displaystyle \Gamma =\mathbb {R} }" loading="lazy">, orientiert in Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy">.<a href="#cite_note-3">[3]</a>

Da wir eine skalare Funktion suchen, können wir unter Anwendung des Logarithmus die Problemstellung etwas umschreiben

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log M_{+}(z)-\log M_{-}(z)=\log J(z),\quad z\in \Gamma _{u}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47ca89b2ceb3a6c66df7c6598004ba77a9c6d03e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.619ex; height:2.843ex;" alt="{\displaystyle \log M_{+}(z)-\log M_{-}(z)=\log J(z),\quad z\in \Gamma _{u}}" loading="lazy">,</dd></dl>

welches sich mit Sokhotski-Plemeljs Formel lösen lässt. Die Lösung hat folgende Form

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log M(z)={\frac {1}{2\pi i}}\int _{\mathbb {R} }{\frac {\log J(s)}{s-z}}\mathrm {d} s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10274f3407c052881fc635248d758ef983d1c476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.365ex; height:6.176ex;" alt="{\displaystyle \log M(z)={\frac {1}{2\pi i}}\int _{\mathbb {R} }{\frac {\log J(s)}{s-z}}\mathrm {d} s}" loading="lazy">,</dd></dl>

allerdings existiert dieses Integral nicht immer.

<div class="mw-heading mw-heading3"><h3 id="Nichtlineare_Schrödinger-Gleichung">Nichtlineare Schrödinger-Gleichung</h3></div>

Betrachte die nichtlineare <a href="Schr%C3%B6dinger-Gleichung" class="mw-redirect" title="Schrödinger-Gleichung">Schrödinger-Gleichung</a><a href="#cite_note-4">[4]</a>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}i\psi _{t}+\psi _{xx}-2|\psi |^{2}\psi =0,\\\psi (x,0)=\psi _{0}(x)\in {\mathcal {S}}(\mathbb {R} )\end{cases}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a1e7330f190f91d19053bc0538ea8888251742.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.387ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}i\psi _{t}+\psi _{xx}-2|\psi |^{2}\psi =0,\\\psi (x,0)=\psi _{0}(x)\in {\mathcal {S}}(\mathbb {R} )\end{cases}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0751119ebd490eacaceee66c787196ffae55c316.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.98ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} )}" loading="lazy"> den <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Raum</a> bezeichnet. Wir wählen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma =\mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc9445f6e097bfca1e5148308ef9ed52451e5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.229ex; height:2.176ex;" alt="{\displaystyle \Gamma =\mathbb {R} }" loading="lazy">, orientiert in Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy"> und weiter sei für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \Gamma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e004f6d4ebe37811f6370b11b353cb159800040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.382ex; height:2.176ex;" alt="{\displaystyle z\in \Gamma }" loading="lazy">

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{x,t}(z)={\begin{pmatrix}1-|r(z)|^{2}&-{\bar {r}}e^{-2i(2tz^{2}+xz)}\\r(z)e^{2i(2tz^{2}+xz)}&1\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7de8693ac7d7d59f4c2def2b09db010b687551df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.571ex; height:7.509ex;" alt="{\displaystyle J_{x,t}(z)={\begin{pmatrix}1-|r(z)|^{2}&-{\bar {r}}e^{-2i(2tz^{2}+xz)}\\r(z)e^{2i(2tz^{2}+xz)}&1\end{pmatrix}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(z)\in {\mathcal {S}}(\mathbb {R} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r(z)\in {\mathcal {S}}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0858fd862792aae93b263ab03e738c5f2e669c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.766ex; height:2.843ex;" alt="{\displaystyle r(z)\in {\mathcal {S}}(\mathbb {R} )}" loading="lazy"> den durch die <a href="Inverse_Streutransformation" title="Inverse Streutransformation">inverse Streutransformation</a> zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"> assoziierten Reflexionskoeffizient bezeichnet.

Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Gamma ,J_{x,t})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f569a5fc3487dc0aad11b39fa14f8ac647f20f46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.81ex; height:3.009ex;" alt="{\displaystyle (\Gamma ,J_{x,t})}" loading="lazy"> ein RHP.

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Thomas Bothner: <cite style="font-style:italic">On the origins of Riemann–Hilbert problems in mathematics</cite>. In: IOP Publishing (Hrsg.): <cite style="font-style:italic">Nonlinearity</cite>. 2021, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088/1361-6544%2Fabb543">10.1088/1361-6544/abb543</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Riemann-Hilbert-Problem&rft.atitle=On+the+origins+of+Riemann-Hilbert+problems+in+mathematics&rft.au=Thomas+Bothner&rft.btitle=Nonlinearity&rft.date=2021&rft.doi=10.1088%2F1361-6544%2Fabb543&rft.genre=book" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Percy Deift, Xin Zhou: <cite style="font-style:italic">A steepest descent method for oscillatory Riemann-Hilbert problems</cite>. In: American Mathematical Society (Hrsg.): <cite style="font-style:italic">Bulletin of the American Mathematical Society (N.S.)</cite>. Nr. 26, 1992, S. 119–124, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307/2946540">10.2307/2946540</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Riemann-Hilbert-Problem&rft.atitle=A+steepest+descent+method+for+oscillatory+Riemann-Hilbert+problems&rft.au=Percy+Deift%2C+Xin+Zhou&rft.date=1992&rft.doi=10.2307%2F2946540&rft.genre=journal&rft.issue=26&rft.jtitle=Bulletin+of+the+American+Mathematical+Society+%28N.S.%29&rft.pages=119-124" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Percy Deift: <cite style="font-style:italic">Orthogonal Polynomials and Random Matrices: A-Riemann Hilbert Approach</cite>. Hrsg.: American Mathematical Society. Rhode Island 2000, ISBN 978-0-8218-8344-0, S. 2.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Riemann-Hilbert-Problem&rft.au=Percy+Deift&rft.btitle=Orthogonal+Polynomials+and+Random+Matrices%3A+A-Riemann+Hilbert+Approach&rft.date=2000&rft.genre=book&rft.isbn=9780821883440&rft.pages=2&rft.place=Rhode+Island" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Percy Deift: <cite style="font-style:italic">Orthogonal Polynomials and Random Matrices: A-Riemann Hilbert Approach</cite>. Hrsg.: American Mathematical Society. Rhode Island 2000, ISBN 978-0-8218-8344-0, S. 12.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Riemann-Hilbert-Problem&rft.au=Percy+Deift&rft.btitle=Orthogonal+Polynomials+and+Random+Matrices%3A+A-Riemann+Hilbert+Approach&rft.date=2000&rft.genre=book&rft.isbn=9780821883440&rft.pages=12&rft.place=Rhode+Island" style="display:none">

</li>

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