J. Hornick and D.E. Pelinovsky

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

Abstract:
We consider families of solitary waves in the Korteweg-de Vries (KdV) equation coupled with the linear Schrodinger (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 soliton 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.

Keywords:
Korteweg-de Vries equation; linear Schrodinger equation; solitary waves: existence and bifurcations; Morse index; orbital stability; spectral stability.