Multiple Permanent‐wave Trains in Nonlinear Systems

Abstract

Multiple permanent-wave trains in nonlinear systems are constructed by the asymptotic tailmatching method. Under some general assumptions, simple criteria for the construction are presented. Applications to fourth-order systems and coupled nonlinear Schrödinger equations are discussed.

Classification of the solitary waves in coupled nonlinear Schrödinger equations

“…In addition, there are abundant asymmetric solitary waves exemplified by the solutions (2.9). Studies on such solitary waves can be found in [7].…”

“…We have found that they often dominate the long time solution evolution [5,6]. These waves and their stability have been investigated by Mesentsev and Turitsyn [8], Kaup et Yang [7] among others. The work in [g-10] showed that there exists a family of symmetric, single-humped and stable solitary waves with an arbitrary polarization.…”

“…Based on some analytical results and numerical evidence, we conjectured that only the family of symmetric single-humped solitary waves is stable. In [7], we studied multi-hump permanent waves in general nonlinear systems. Under some assumptions, we developed simple criteria for constructing such permanent waves.…”

A tail-matching method for the linear stability of multi-vector-soliton bound states

“…Below, we use a simplified version of [18]'s method to construct multi-vector-soliton bound states in the coupled NLS equations [i.e., (2.4) and (2.5)], and reproduce the spacing formula (2.10). There are two reasons for our doing this: (i) to highlight the key ideas in the tail-matching method for the construction of multi-pulse bound states; (ii) to motivate a similar tail-matching idea for the linear-stability analysis of multi-pulse bound states (see Sec.…”

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

“…In certain physical situations, when there are two wavetrains moving with nearly the same group velocities, their interactions are then governed by the coupled NLS equations [34,39]. For example, the coupled NLS systems appear in the study of interactions of waves with different polarizations [8], the description of nonlinear modulations of two monochromatic waves [30], the interaction of Bloch-wave packets in a periodic system [35], the evolution of two orthogonal pulse envelopes in birefringent optical fiber [29], the evolution of two surface wave packets in deep water [34], to name a few.…”