Stability of Traveling Wave Solutions of Nonlinear Dispersive Equations of NLS Type

We present a rigorous modulational stability theory for periodic traveling wave solutions to equations of nonlinear Schrödinger type. For Hamiltonian dispersive equations with a non-singular symplectic form and d conserved quantities (in addition to the Hamiltonian), one expects that generically ℒ , the linearization around a periodic traveling wave, will have a particular Jordan structure. The kernel (ℒ) and the first generalized kernel (ℒ^2)/ (ℒ) are expected to be d dimensional, with no higher generalized kernels. The breakup of this Jordan block under perturbations arising from a change in boundary conditions dictates the modulational stability or instability of the underlying periodic traveling wave. This general picture is worked out in detail for equations of nonlinear Schrödinger (NLS) type. We give explicit genericity conditions that guarantee that the Jordan form is the generic one: these take the form of non-vanishing determinants of certain matrices whose entries can be expressed in terms of a finite number of moments of the traveling wave solution. Assuming that these genericity conditions are met we give a normal form for the small eigenvalues that result from the break-up of the generalized kernel, in the form of the eigenvalues of a quadratic matrix pencil. We compare these results to direct numerical simulation for the cubic and quintic focusing and defocusing NLS equations subject to both longitudinal and transverse perturbations. The stability of traveling waves of the cubic NLS subject to longitudinal perturbations has been previously studied using the integrability and our results agree with those in the literature. All of the remaining cases are new.