Stability criterion for multicomponent solitary waves

Abstract

We obtain the most general matrix criterion for stability and instability of multi-component solitary waves considering a system of N incoherently coupled nonlinear Schrödinger equations. Soliton stability is studied as a constrained variational problem which is reduced to finite-dimensional linear algebra. We prove that unstable (all real and positive) eigenvalues of the linear stability problem for multi-component solitary waves are connected with negative eigenvalues of the Hessian matrix, the latter is constructed for the energetic surface of N −component spatially localized stationary solutions.

Instabilities and bifurcations of nonlinear impurity modes

“…The opposite will be true for local maxima. The Hessian matrix here is given by ∂P/∂Λ, where P is the power or (squared) L 2 norm of the solution P = ||u|| 2 2 [31]. Recall that the L p norm is defined as ||u|| p = ( n |u n | p ) 1/p .…”

Stability of Spatial Optical Solitons

“…From Eqs. (16) and (19), it follows that Q(ν = 0) = L −1 1 µΦ|Φ . To calculate this value, we differentiate the equality L 0 Φ = 0 with respect to the propagation constant and obtain…”

“…Nevertheless, in some cases it is possible to extend the stability analysis to multi-parameter solitary waves. Below, we follow the recent original work by Pelinovsky and Kivshar [19] and demonstrate how the multi-parameter generalization of the Vakhitov-Kolokolov stability criterion can be derived for a system of N coupled NLS equations. Such a theory includes, as a limiting case, the bifurcation analysis near the marginal stability curve.…”

“…[19] can be extended, at least in principle, to other types of solitary waves, such as incoherent solitons in non-Kerr media, parametric solitary waves in χ (2) optical media, etc. In all such cases, the results on stability and instability of solitons can be readily obtained with a rigorous generalization of some of the previously known results of the multi-scale asymptotic theory.…”

“…Recently, a general matrix criterion for the stability and instability of multi-component solitary waves was derived [19] for a system of N incoherently coupled NLS equations. In this general approach, the soliton stability is studied as a constrained variational problem reduced to finite-dimensional linear algebra.…”