Soliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamics

Abstract

The propagation of localized solitons in the presence of large-scale waves is a fundamental problem, both physically and mathematically, with applications in fluid dynamics, nonlinear optics and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg-de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A comprehensive review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton-rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite-gap description are used to describe soliton-rarefaction wave and soliton-dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the Inverse Scattering Transform. For transmitted solitons, far-field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave-mean field interaction

References in corpus (31)

• Dispersive shock waves and modulation theory
• On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
• Resolution of a shock in hyperbolic systems modified by weak dispersion
• Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
• The thermodynamic limit of the Whitham equations
• Soliton gas in integrable dispersive hydrodynamics
• Solitonic dispersive hydrodynamics: theory and observation
• Long-Time Asymptotics for the Korteweg-de Vries Equation with Steplike Initial Data
• Rarefaction Waves of the Korteweg-de Vries Equation via Nonlinear Steepest Descent
• Hydrodynamic Optical Soliton Tunneling
• Observation of dispersive shock waves, solitons, and their interactions in viscous fluid conduits
• Quantitative Analysis of Shock Wave Dynamics in a Fluid of Light
• On the long time asymptotic behavior of the modified Korteweg de Vries equation with step-like initial data
• Dissipative shock waves generated by a quantum-mechanical piston
• Soliton Generation and Multiple Phases in Dispersive Shock and Rarefaction Wave Interaction
• Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves
• KdV breathers on a cnoidal wave background
• Observation of Self-Cavitating Envelope Dispersive Shock Waves in Yttrium Iron Garnet Thin Films
• Dynamic soliton-mean flow interaction with nonconvex flux
• Soliton trapping, transmission and wake in modulationally unstable media
• Collisionally inhomogeneous Bose-Einstein condensates with a linear interaction gradient
• Motion of dark solitons in a non-uniform flow of Bose-Einstein condensate
• Undular Bores Generated by Fracture
• Solitons, the Korteweg-de Vries equation with step boundary values and pseudo-embedded eigenvalues
• Oblique interactions between solitons and mean flows in the Kadomtsev-Petviashvili equation
• Whitham equations and phase shifts for the Korteweg-deVries equation
• Propagation of wave packets along intensive simple waves
• Inverse scattering theory for Schrödinger operators with steplike potentials
• Nonlinear interactions between solitons and dispersive shocks in focusing media
• On the form of dispersive shock waves of the Korteweg-de Vries equation
• Shock Waves in a Superfluid with Higher-Order Dispersion

Cited by in corpus (5)

• KdV breathers on a cnoidal wave background
• Propagation of generalized Korteweg-de Vries solitons along large-scale waves
• Propagation of dark solitons of DNLS equation along a large-scale background
• Solitary wave-mean flow interaction in strongly nonlineardispersive shallow water waves
• Genus two KdV soliton gases and their long-time asymptotics