N-soliton solutions and perturbation theory for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

Abstract

We present a simple approach for finding N -soliton solution and the corresponding Jost solutions of the derivative nonlinear Scrödinger equation with nonvanishing boundary conditions. Soliton perturbation theory based on the inverse scattering transform method is developed. As an application of the present theory we consider the action of the diffusive-type perturbation on a single bright/dark soliton.

Freak waves in laboratory and space plasmas

“…However, its applicability to studying the intermediate behaviour of solutions is restricted to the initial conditions for which the corresponding scattering problem can be solved analytically. To our knowledge, the only non-trivial exact solutions to the DNLS equation obtained so far are different kinds of the N-soliton solutions (see, e.,g., [48][49][50][51]). Since, at present, it is not clear if the scattering problem for the DNLS equation for the initial condition (17) can be solved analytically, the nonlinear evolution of this perturbation was studied numerically [43].…”

The rogue wave and breather solution of the Gerdjikov-Ivanov equation

“…Each element of the matrix of n-fold Darboux transformation of this system is expressed by a ratio of (n + 1) × (n + 1) determinant and n × n determinant of eigenfunctions, which implies the determinant representation of q [n] and r [n] generated from known solution q and r. By choosing some special eigenvalues and eigenfunctions according to the reduction conditions q [n] = −(r [n] ) * , the determinant representation of q [n] provides some new solutions of the GI equation. As examples, the breather solutions and rogue wave of the GI is given explicitly by two-fold DT from a periodic "seed" with a constant amplitude.1 regarded as an extension of the NLS when certain higher-order nonlinear effects are taken into account.Comparing with the intensive studies on the DNLSI [4,[7][8][9][10][11][12][13][14][15][16][17][18][19][20] from the point of view of physics and mathematics, there are only few works on the GI including soliton constructed by Darboux transformation(DT) [21], Hamiltonian structures [22], algebro-geometric solutions [23], hierarchy of the GI equation from an extended version of Drinfel'd-Sokolov formulation [24], Wronskian type solution [25] without using affine Lie groups. In order to show more possible physical relevance of the GI equation, and inspired by the importance of breather(BA) solution and rogue wave(RW) of the NLS [26-33], so we shall find these two types of solutions for the GI equation by DT.…”

High‐Order Solutions and Generalized Darboux Transformations of Derivative Nonlinear Schrödinger Equations

“…In this work, we extend the DT by the limit technique, so that it may be iterated at the same spectral parameter. The modified transformation is referred as gDT.The inverse scattering method was used to study DNLS with vanishing background (VBC) and non-vanishing background (NVBC) [3,[20][21][22][23]. The N -bright soliton formula for DNLS was established by Nakamura and Chen by the Hirota bilinear method [24] and the DT for DNLS was constructed by Imai [25] and Steudel [26] (see also [17]).…”