Existence and stability of ground-state solutions of a Schrödinger—KdV system

Abstract

We consider the coupled Schrödinger–Korteweg–de Vries system which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δi, ci, we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.

Existence of bound states for the coupled Schrödinger–KdV system with cubic nonlinearity

“…The global well-posedness of the Cauchy problem for the I.V.P. associated to (1) was solved recently by A. Corcho and F. Linares [5] in the energy space…”

On fractional Schrödinger systems of Choquard type

“…The advantage of utilizing this technique in our context is that this not only gives the existence of stationary solutions but also addresses the important stability issue of associated standing waves, since the energy and the power(s) involved in variational problems are conservation laws for the flow of associated NLS-type evolution equations (see [6,1] for the illustration of the method). To study the two-parameter problem (1.10), we follow the techniques developed in a series of papers [2,3,4] where the concentration compactness principle was used to study solitary waves for coupled Schrödinger and KdV systems. In order to establish relative compactness of energy minimizing sequences (and hence, existence and stability of minimizers) in the spirit of concentration compactness technique, one require to check certain strict inequalities involving the infimum of the minimization problem.…”

Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities

“…Now, for each each n, choose a number x n such that f (1) n (x) and f (2) n (x) = f (2) n (x + x n ) have disjoint support, and g (1) n (x) andg (2) n (x) = g (2) n (x + x n ) have disjoint support. Define…”

“…Lemma 2.2 guarantees that there exist numbers δ 1 > 0 and δ 2 > 0 such that for all sufficiently large n, (g (1) n ) x ≥ δ 1 and (g (2) n ) x ≥ δ 2 . Let δ = min(δ 1 , δ 2 ) > 0.…”

“…The Cazenave and Lions method has since been adapted by different authors to prove existence and stability results of a variety of nonlinear dispersive equations (see, for example, [2,3,9,14,31,33]). …”