Bright and dark multi-solitons in Hamiltonian systems

Abstract

We consider the existence and spectral stability of multi-pulse solitary wave solutions in two Hamiltonian systems, a nonlinear Schrödinger equation which incorporates both fourth and second-order dispersion terms (NLS4) and the fifth-order Korteweg-De Vries equation (KdV5). For NLS4, we first show that a discrete family of bright multi-pulse solutions exists, which is characterized by the distances between consecutive copies of the the primary solitary wave. We then reduce the spectral stability problem to computing the determinant of a matrix which is, to leading order, block diagonal. Under additional assumptions, which can be verified numerically and are sufficient to prove orbital stability of the primary solitary wave, we show that all bright multi-solitons are spectrally unstable. We then look at a similar problem for KdV5 on a periodic domain, and show that brief instability bubbles form when eigenvalues collide on the imaginary axis. Finally, we return to NLS4, this time to the dark soliton regime. Since dark solitons decay to a nonzero background, we impose either Neumann or periodic boundary conditions. Numerical results suggest that, in contrast to bright multi-solitons, dark multi-solitons can be spectrally neutrally stable. In addition, eigenvalue collisions on the imaginary axis produce similar instability bubbles to those found in KdV5. Results of numerical timestepping experiments are shown for all systems, and these are interpreted using the spectral computations.

Ross Parker

RTG postdoctoral fellow / visiting assistant professor

I am a postdoctoral fellow and visiting assistant professor in the department of mathematics at Southern Methodist University.