TY - JOUR

T1 - Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

AU - Ablowitz, Mark J.

AU - Cole, Justin T.

AU - El, Gennady A.

AU - Hoefer, Mark A.

AU - Luo, Xu‐Dan

N1 - Funding Information: The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for sup-port and hospitality during the program Dispersive Hydrodynamics when work on this paper was undertaken.This work was supported by EPSRC Grant Number EP/R014604/1. MJA was partially supported by NSF under Grant DMS-2005343. XL was partially supported by NSFC under Grant 12101590. MAH was partially supported by NSF under Grant DMS-1816934. GAE was partially supported by EPSRC under Grant EP/W032759/1.

PY - 2023/10/1

Y1 - 2023/10/1

N2 - The mathematical description of localized solitons in the presence of large-scale waves is a fundamental problem in nonlinear science, 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 topical 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.

AB - The mathematical description of localized solitons in the presence of large-scale waves is a fundamental problem in nonlinear science, 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 topical 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.

KW - Whitham theory

KW - dispersive shock wave

KW - inverse scattering transform

KW - rarefaction wave

KW - soliton

UR - http://www.scopus.com/inward/record.url?scp=85156154964&partnerID=8YFLogxK

U2 - 10.1111/sapm.12615

DO - 10.1111/sapm.12615

M3 - Review article

SN - 0022-2526

VL - 151

SP - 795

EP - 856

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

IS - 3

ER -