Bifurcations of solitary waves in a coupled system of long and short waves

TL;DR

This paper analyzes bifurcations of solitary waves in a coupled KdV and Schrödinger system, identifying a sequence of solutions including ground and excited states, and connecting them with known exact solutions.

Contribution

It characterizes a sequence of bifurcations of solitary waves in a coupled KdV–Schrödinger system, linking them to known solutions and excited states.

Findings

First bifurcation yields the KdV soliton with the Schrödinger ground state.

Subsequent bifurcations involve excited states of the Schrödinger equation.

The first two bifurcations are connected to exact solutions used in literature.

Abstract

We consider families of solitary waves in the Korteweg--de Vries (KdV) equation coupled with the linear Schr\"{o}dinger (LS) equation. This model has been used to describe interactions between long and short waves. To characterize families of solitary waves, we consider a sequence of local (pitchfork) bifurcations of the uncoupled KdV solitons. The first member of the sequence is the KdV soliton coupled with the ground state of the LS equation, which is proven to be the constrained minimizer of energy for fixed mass and momentum. The other members of the sequence are the KdV solitons coupled with the excited states of the LS equation. We connect the first two bifurcations with the exact solutions of the KdV--LS system frequently used in the literature.