Evolution theorem for a class of perturbed envelope soliton solutions

Abstract

Envelope soliton solutions of a class of generalized nonlinear Schrödinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift. For ∂η2N >0, the system is stable in the sense of Liapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relatons, and others.

Stability of the Bloch wall via the Bogomolnyi decomposition in elliptic coordinates

“…is J −1 H, with J the skew-symmetric matrix (16). The multiplication by a skew-symmetric matrix changes the spectral properties of an operator; for example, the continuous spectrum of H lies on the positive real axis whereas the continuous spectrum of J −1 H consists of pure imaginary λ.…”

Soliton solutions for quasilinear Schrödinger equations, I

“…Quasilinear equations of form (1) appear more naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of h, the superfluid film equation in plasma physics by Kurihara in [13] (cf. [14]) for h(s) = s. In the case h(s) = (1 + s) 1/2 , (1) models the self-channeling of a high-power ultra short laser in matter; see [4], [6], [8], [23] and the references in [5]. Equation (1) also appears in plasma physics and fluid mechanics [13], [14], [17], [19], [21], in the theory of Heisenberg ferromagnets and magnons [2], [12], [15], [22], [25], in dissipative quantum mechanics [10] and in condensed matter theory [18].…”

MULTIPLE SOLUTIONS FOR A CLASS OF QUASILINEAR SCHRÖDINGER SYSTEM IN ℝN

“…In the case l(s) = √ 1 + s, Eq. (1.2) models the self-channeling of a high-power ultra short laser in matter (see [10]). For more mathematical models in physics described by (1.2), see [14] and the references therein.…”