The short pulse equation by a Riemann–Hilbert approach

The short pulse equation by a Riemann–Hilbert approach We develop a Riemann–Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation \begin{aligned} u_{xt}=u+\tfrac{1}{6}(u^3)_{xx} \end{aligned} u x t = u + 1 6 ( u 3 ) x x with zero boundary conditions (as $$|x|\rightarrow \infty$$ | x | → ∞ ). This approach is directly applied to a Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the longtime behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the longtime behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Letters in Mathematical Physics Springer Journals

The short pulse equation by a Riemann–Hilbert approach

, Volume 107 (7) – Feb 22, 2017
29 pages

/lp/springer_journal/the-short-pulse-equation-by-a-riemann-hilbert-approach-V0Z0jNs46M
Publisher
Springer Netherlands
Subject
Physics; Theoretical, Mathematical and Computational Physics; Complex Systems; Geometry; Group Theory and Generalizations
ISSN
0377-9017
eISSN
1573-0530
D.O.I.
10.1007/s11005-017-0945-z
Publisher site
See Article on Publisher Site

Abstract

We develop a Riemann–Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation \begin{aligned} u_{xt}=u+\tfrac{1}{6}(u^3)_{xx} \end{aligned} u x t = u + 1 6 ( u 3 ) x x with zero boundary conditions (as $$|x|\rightarrow \infty$$ | x | → ∞ ). This approach is directly applied to a Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the longtime behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the longtime behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.

Journal

Letters in Mathematical PhysicsSpringer Journals

Published: Feb 22, 2017

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