Traveling waves of nonlinear Schr\"{o}dinger equation including higher order dispersions

The solitary wave solution and periodic solutions expressed in terms of elliptic Jacobi's functions are obtained for the nonlinear Schr\"{o}dinger equation governing the propagation of pulses in optical fibers including the effects of second, third and fourth order dispersion. The approach is based on the reduction of the generalized nonlinear Schr\"{o}dinger equation to an ordinary nonlinear differential equation. The periodic solutions obtained form one-parameter family which depend on an integration constant $p$. The solitary wave solution with ${\rm sech}^2$ shape is the limiting case of this family with $p=0$. The solutions obtained describe also a train of soliton-like pulses with ${\rm sech}^2$ shape. It is shown that the bounded solutions arise only for special domains of integration constant.Comment: We consider in this paper also the case with negative parameter $\gamma$ (defocusing nonlinearity

The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painleve test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation

We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the "energy" parameter $E$. We show that as $|E| \to \infty$, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when $| E |$ is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied

On pattern structures of the N-soliton solution of the discrete KP equation over a finite field

The existence and properties of coherent pattern in the multisoliton solutions of the dKP equation over a finite field is investigated. To that end, starting with an algebro-geometric construction over a finite field, we derive a "travelling wave" formula for $N$-soliton solutions in a finite field. However, despite it having a form similar to its analogue in the complex field case, the finite field solutions produce patterns essentially different from those of classical interacting solitons.Comment: 12 pages, 3 figure