The short pulse equation by a Riemann-Hilbert approach

2016/08/07 by de Monvel, Anne Boutet, Shepelsky, Dmitry, Zielinski, Lech
#35B40 #35Q15 #35Q51 #35Q53 (Primary) #37K15 #37K40 (Secondary) #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1608.02249

Abstract

We develop a Riemann-Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation u_xt=u+(1)/(6)(u³)_xx with zero boundary conditions (as |x|→∞). This approach is directly applied to the 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 long-time behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the long-time behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.

Related

• Long-time asymptotics for the short pulse equation
• The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach
• Long-time asymptotic behavior for the complex short pulse equation
• Soliton and Periodic solutions of the Short Pulse Model Equation
• Soliton resolution, asymptotic stability and Painlevé transcendents in the combined Wadati-Konno-Ichikawa and short-pulse equation