A general two-coupled nonlinear Schrodinger system is investigated with symbolic computation. The system is regarded to be a more general model than other coupled nonlinear Schrodinger systems since its coefficients of the self-phase modulation, cross-phase modulation, and four-wave mixing terms are arbitrary. Painleve-integrability associated with the system is examined by means of Painleve test. As a result, Painleve–Backlund transformation is constructed with truncating the Laurent series at the constant level term. In addition, a family of explicit solutions corresponding to the vacuum solutions are derived through the Painleve–Backlund transformation.

Painlevé-integrability and explicit solutions of the general two-coupled nonlinear Schrödinger system in the optical fiber communications

Korteweg–de Vries (KdV)-type equations describe certain nonlinear phenomena in fluids and plasmas. In this paper, three-coupled KdV equations corresponding to the Neumann system of the fourth-order eigenvalue problem is investigated. Through the dependent variable transformations, bilinear forms of such equations are obtained, from which the multi-soliton solutions are derived. Soliton propagation and interaction are analyzed: (1) Bell- and anti-bell-shaped solitons are found; (2) Among the soliton images, one depends on the sign of wave numbers k
 i
 ’s (i=1,2,3), while the others are independent of such a sign; (3) Interaction between two solitons and among three solitons are elastic, i.e., the amplitude and velocity of each soliton remain unvaried after the interaction except for the phase shift.

Multi-soliton solutions for the three-coupled KdV equations engendered by the Neumann system

In this paper, we are mainly concerned with the (2+1)-dimensional generalized Korteweg–de Vries equation in fluid dynamics. Based on the translation transformation and Hirota bilinear method, we study the excitations of nonlinear lump-type waves on a constant background. A remarkable feature of these lump-type waves is that under some parameter conditions, these lump-type wave solutions can be converted into some amusing nonlinear wave structures, including the W-shaped solitary wave, double-peak solitary wave, parallel solitary wave, multi-peak solitary wave and periodic wave solutions. These results do not have an analog in the standard Kadomtsev–Petviashvili equation. The transition condition between the lump-type wave and other nonlinear wave solutions is presented. The dynamical behaviors of these nonlinear wave solutions are investigated analytically and illustrated graphically. Furthermore, the existence conditions for these nonlinear wave solutions are exhibited explicitly. Our results further enrich the nonlinear wave theories for the (2+1)-dimensional generalized Korteweg–de Vries equation.

State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation

We derive analytical spatial soliton solutions of a (2 + 1)-dimensional nonlinear Schrodinger equation with power-law nonlinearity in $$\mathcal {PT}$$
 -symmetric potentials. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulations. Moreover, some dynamical characteristics of these solutions, such as the phase switch, the power, and the transverse power-flow density, are also examined.

Stable localized spatial solitons in \mathcal {PT}-symmetric potentials with power-law nonlinearity

With symbolic computation, this paper investigates some integrable properties of a two-dimensional generalization of the Korteweg-de Vries equation, i.e., the Bogoyavlensky–Konoplechenko model, which can govern the interaction of a Riemann wave propagating along the $y$-axis and a long wave propagating along the $x$-axis. Within the framework of Bell-polynomial manipulations, Bell-polynomial expressions are firstly given, which then are cast into bilinear forms. The $N$-soliton solutions in the form of an $N$th-order polynomial in the $N$ exponentials and in terms of the Wronskian determinant are, respectively, constructed with the Hirota bilinear method and Wronskian technique. Bilinear Backlund transformation is also derived with the achievement of a family of explicit solutions.

Integrability with symbolic computation on the Bogoyavlensky–Konoplechenko model: Bell-polynomial manipulation, bilinear representation, and Wronskian solution

The new (2+1)-dimensional generalized KdV equation which exists the bilinear form is mainly discussed. We prove that the equation does not admit the Painleve property even by taking the arbitrary constant a=0. However, this result is different from Radha and Lakshmanan’s work. In addition, based on Hirota bilinear method, periodic wave solutions in terms of Riemann theta function and rational solutions are derived, respectively. The asymptotic properties of the periodic wave solutions are analyzed in detail.

Exact solutions and Painlevé analysis of a new (2 + 1)-dimensional generalized KdV equation

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Exact solutions and Painlevé analysis of a new (2 + 1)-dimensional generalized KdV equation