Rigorous justification of the Whitham modulation theory for equations of NLS type. Issue 2 (11th May 2021)

• Record Type: Journal Article
• Authors: Clarke, W.A.; Marangell, R.
• Is Part Of: Studies in applied mathematics. Volume 147:Issue 2(2021)
• Page Start: 577
• Page End: 621
• Publication Date: 2021-05-11
• Subjects: modulational stability -- nonlinear Schrödinger equation -- nonlinear waves -- partial differential equations -- Whitham modulation theory; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12390
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: We study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi‐linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken from Stability of Travelling wave solutions of Nonlinear Dispersive equations of NLS type, where Leisman et al . compute the spectrum of the linearized operator near the origin via an analysis of Jordan chains. We derive the modulation equations using Whitham's formal modulation theory, in particular the variational principle applied to an averaged Lagrangian. We use the genericity conditions assumed in the rigorous theory of Leismen et al . ​ to direct the homogenization of the modulation equations. As a result of the agreement between the equation for the characteristics and the normal form from the linear theory, we show that the hyperbolicity of the Whitham system is a necessary condition for modulational stability of the underlying wave.