Higher-order NLS Research Articles

Higher-order NLS with anisotropic dispersion and modulation spaces: a global existence and scattering result

Abstract In this paper we transfer a small data global existence and scattering result by Wang and Hudzik to the more general case of modulation spaces $M_{p, q}^s(\mathbb{R}^d)$ where q = 1 and $s \geq 0$ or $q \in (1, \infty]$ and $s \gt \frac{d}{q^{\prime}}$ and to the non-linear Schrödinger equation with higher-order anisotropic dispersion.

New abundant exact solutions for Kundu equation

In this paper, the higher order NLS equation with cubic-quintic nonlinear terms is studied, new abundant solitary solutions with traveling-wave envelope of this equation are obtained with the aid of a generalized auxiliary equation method and complex envelope non-traveling transform approach.

A note on the Painlevé test for nonlinear variable-coefficient PDEs

For a generalized nonlinear PDEs with variable coefficients, it is not Painlevé integrable unless the variable coefficients satisfy certain constraint conditions. In this note a generalized algorithm is proposed for the Painlevé test of nonlinear variable-coefficient PDEs. For the three steps of Painlevé test, i.e. leading order analysis, resonance determination and verification of resonant conditions, the analysis of parametric constraints is similar to those of nonlinear PDEs with constant coefficients given in my previous work. The main difference lies in the coefficients of Laurent series should have proper dependence according to the types of variable coefficients. By this generalized algorithm, several important nonlinear variable-coefficient PDEs, including KdV equation, mKdV equation, KP equation, NLS equation and higher-order NLS equation are studied and, in addition to rederiving all known P-integrable conditions, some new P-integrable models are obtained with the assistance of Maple.

Completely integrable nonlinear Schrödinger type equations on moving space curves

Using the Lamb formalism, we show that some completely integrable homogeneous and inhomogeneous nonlinear Schr\"odinger (NLS) type equations such as derivative NLS, extended NLS, higher-order NLS, inhomogeneous NLS, circularly and radially symmetric NLS, and generalized inhomogeneous radially symmetric NLS equations can be related to certain types of moving helical space curves.

A class of exact, periodic solutions of nonlinear envelope equations

A class of periodic solutions of nonlinear envelope equations, e.g., the nonlinear Schrödinger equation (NLS), is expressed in terms of rational functions of elliptic functions. The Hirota bilinear transformation and theta functions are used to extend and generalize this class of solutions first reported for NLS earlier in the literature. In particular a higher order NLS and the Davey–Stewartson (DS) equations are treated. Doubly periodic standing waves solutions are obtained for both the DSI and DSII equations. A symbolic manipulation software is used to confirm the validity of the solutions independently.

Prolongation structures of a higher-order nonlinear Schrodinger equation

A higher-order Schrodinger equation containing parameters, which is used to describe pulse propagation in optical fibres, is shown to admit an infinite-dimensional prolongation structure for exactly four combinations of the parameters, besides the classical NLS equation. For each of these cases, the structure of the resulting prolongation algebra is determined explicitly. For the first three cases the prolongation algebra is essentially a sub-algebra of A1(1), the fourth case turns out to be sub-algebra of the twisted Kac-Moody algebra A2(2). Using vector-field representations, related systems of differential equations for the (pseudo-) potential functions are given for each of the cases. The cases found here correspond exactly to the derived NLS equations I and II, the Hirota equation and the equation recently considered by Sasa and Satsuma (1991). The result of this paper strongly indicates that the considered higher-order NLS equation is completely integrable for precisely these four cases.