Interactions and Asymptotics of Dispersive Shock Waves -- Korteweg-de Vries Equation

Abstract

The long-time asymptotic solution of the Korteweg-de Vries equation for general, step-like initial data is analyzed. Each sub-step in well-separated, multi-step data forms its own single dispersive shock wave (DSW); at intermediate times these DSWs interact and develop multiphase dynamics. Using the inverse scattering transform and matched-asymptotic analysis it is shown that the DSWs merge to form a single-phase DSW, which is the `largest' one possible for the boundary data. This is similar to interacting viscous shock waves (VSW) that are modeled with Burgers' equation, where only the single, largest-possible VSW remains after a long time.

5 pages, 3 figures

References in corpus (4)

• On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
• Shocks in nonlocal media
• Observation of a gradient catastrophe generating solitons
• Formation of Quantum Shock Waves by Merging and Splitting Bose-Einstein Condensates

Cited by in corpus (8)

• Dispersive shock waves and modulation theory
• On the nonlinear dynamics of the traveling-wave solutions of the Serre system
• Oblique spatial dispersive shock waves in nonlinear Schrödinger flows
• Inverse scattering theory for Schrödinger operators with steplike potentials
• On the form of dispersive shock waves of the Korteweg-de Vries equation
• Whitham modulation equations and application to small dispersion asymptotics and long time asymptotics of nonlinear dispersive equations
• Numerical renormalization group algorithms for self-similar solutions of partial differential equations
• Formation of compression waves with multiscale asymptotics in the Burgers and KdV models