Exact solutions for a higher-order nonlinear Schrödinger equation

Abstract

We performed a systematic analysis of exact solutions for the higher-order nonlinear Schrodinger equation iP~+Prr=a, g~t(t~'+azg~P~+ia, (P~P~')r+(a4+ia, )P(~g~')r that describes wave propagation in nonlinear dispersive media. The method consists of the determination of all transformations that reduce the equation to ordinary differential equations that are solved whenever possible.All obtained solutions fall into one of the following categories: "bright" or "dark" solitary waves, solitonic waves, regular and singular periodic waves, shock waves, accelerating waves, and selfsimilar solutions. They are expressed in terms of simple functions except for few cases given in terms of the less-known Painleve transcendents.

Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations

“…its solutions can be written down in closed form. Such conditions have been obtained by FlorjanCzyk and Gagnon [21] who also discuss the connection with Painlev6 theory (see Cariello and Tabor [21]). We will obtain similar though less general results by direct inspection of the ODE's.…”

“…It turns out that the unperturbed system (1.5) leads to an integrable [21] dynamical system (1.3) whose orbits can be calculated analytically. In particular, we find a two-parameter family of pulses indexed by v and to, whose existence is connected to the Hamiltonian property of the dynamical system (1.3) resulting from eq.…”

“…(2.1) has a continuous symmetry [21], i.e. it is invariant under a one-parameter family of transformations A ~ J~(A)=A~.…”

“…These can be obtained from a set of three ordinary differential equations (ODE's) in the variable ~ = x -vt. Specifically, we set [19][20][21] A(x,t) =e-i~ 't,4(x-vt), (1.2a) .…”

On the polynomial solution of the first Painlevé equation

“…The corresponding equations are non-linear second-order ordinary differential equations, used by physicists and mathematicians since their discovery to describe a growing variety of systems. Some examples involve the description of the asymptotic behavior of non-linear equations [2], statistical mechanics [3], correlation functions of the XY model [4], bidimensional ising model [5], superconductivity [6], Bose-Einstein condensation [6], stimulated Raman dispersion [7], quantum gravity and quantum field theory [8], aleatory matrix models [9], topologic field theory (e.g., the so-called WittenDijkgraaf-Verlinde-Verlinde equations) [10], general relativity [11], solutions of Einstein axialsymmetric equations [11], negative curvature surfaces [12], plasma physics [6], Hele-Shaw problems [13] and non-linear optics [14]. During the last years, more and more researchers are interested in these equations and they have found interesting analytic, geometric, and algebraic properties.…”