A direct theory for the perturbed unstable nonlinear Schrödinger equation

Abstract

A direct perturbation theory for the unstable nonlinear Schrödinger equation with perturbations is developed. The linearized operator is derived and the squared Jost functions are shown to be its eigenfunctions. Then the equation of linearized operator is transformed into an equivalent 4ϫ4 matrix form with first order derivative in t and the eigenfunctions into a four-component row. Adjoint functions and the inner product are defined. Orthogonality relations of these functions are derived and the expansion of the unity in terms of the four-component eigenfunctions is implied. The effect of damping is discussed as an example.

Perturbations of Soliton Solutions to the Unstable Nonlinear Schrödinger and Sine‐Gordon Equations

“…The principal purpose of this paper is to reconsider the perturbed UNS equation via a direct perturbation expansion and phase‐averaged “conservation balances” following the methods described by, for example, Kodama and Ablowitz [16] and Grimshaw [17, 18], respectively. Our results from both the direct perturbation expansion and conservation balance approaches, of course, completely agree with each other, but contradict Huang et al [13]. Moreover, in the context of the dissipative problem, we are able to explicitly determine the first‐order perturbation field.…”

“…Indeed, this is the case for the dissipative perturbation example explicitly solved for later in this section. The results of this specific example can be directly compared with those in [13]. As we then show, the results so obtained do not agree with those in Huang et al [13].…”

“…It is also possible to obtain and using, in our opinion, the more physically intuitive approach of phase averaged conservation relations like that described by Grimshaw [17,18]. The demonstration that and are obtained using this second approach gives confidence in the results since the transport equations derived here for the perturbed UNS equation do not agree with those in [13].…”

“…To provide an explicit example that can be compared directly with the results of Huang et al [13] and with the known results for the perturbed NLS equation [14–16], we consider the dissipative perturbation in , where γ(ɛ t ) > 0 is a prescribed O coefficient function. Substitution of into and yields, respectively, which have the solutions where μ 0 =μ(0) and U 0 = U (0), respectively.…”

“…Huang et al [13] have presented a theory for the adiabatic deformation of the solitary wave solution to the UNS equation based on the inverse scattering formalism similar to that developed for integrable equations by Kaup and Newell [14] and catalogued for numerous other soliton models by Kivshar and Malomed [15]. The principal purpose of this paper is to reconsider the perturbed UNS equation via a direct perturbation expansion and phase‐averaged “conservation balances” following the methods described by, for example, Kodama and Ablowitz [16] and Grimshaw [17, 18], respectively.…”