The Inverse Scattering Transform for the Defocusing Nonlinear Schrödinger Equations with Nonzero Boundary Conditions. Issue 1 (13th November 2012)

• Record Type: Journal Article
• Authors: Demontis, F.; Prinari, B.; van der, C.; Vitale, F.
• Is Part Of: Studies in applied mathematics. Volume 131:Issue 1(2013)
• Page Start: 1
• Page End: 40
• Publication Date: 2012-11-13
• Subjects: Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/j.1467-9590.2012.00572.x
• Languages: English
• ISSNs: 0022-2526

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg25tchsxg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm572:equation:sapm572-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mo>±</mml:mo></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mo>±</mml:mo></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:math> as <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg25tchstt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm572:equation:sapm572-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> is presented. The direct problem is shown to be well posed for potentials <italic>q</italic> such that <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg25tchss8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm572:equation:sapm572-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mi>x</mml:mi><mml:mn>02010</mml:mn><mml:mo>;</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mo>±</mml:mo></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>, </mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann‐Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.</p> </abstract>