On the formation of bound states by interacting nonlocal solitary waves

Abstract

Steady, multi-hump bound states are constructed asymptotically by matching the oscillatory and exponentially varying tail components of weakly nonlocal solitary waves. The asymptotic procedure is applied to the nonlinear Schrödinger (NLS) equation with a third-order-derivative dispersive term that governs the propagation of pulses near the zero-dispersion wavelength in optical fibers. Families of symmetric, locally confined bound states having two, three and four humps are found analytically for discrete values of the third-order dispersion, consistent with numerical solutions of the perturbed NLS equation. The Korteweg-de Vries equation with a fifth-order-derivative dispersive perturbation — a model equation for gravity-capillary waves on a shallow layer — is also considered, and both symmetric and asymmetric two-hump solution families with oscillatory tails are constructed.