Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

Abstract

This text is a survey of recent results on traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. We present the existence, nonexistence and stability results and we describe the main ideas used in proofs.

Hamiltonian Dynamics of a Particle Interacting with a Wave Field

“…whence the bounds ( 22) and (23). We recall that the constant ε 0 appears in the hypotheses of Theorem 3.3 and must be sufficiently small.…”

“…The first term on the right-hand side is controlled by using (62). To bound the remaining term, we apply the first inequality in (54) to ∇W and then use (23):…”

“…For results on stability of traveling waves, see [18,8,9,26,3,16]. See also [23] for a recent review of results on traveling waves in nonlinear Schrödinger equations with nonzero conditions at infinity.…”