Three types of generalized Kadomtsev－Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized (2+1)-dimensional Kadomtsev--Petviashvili (KP) equations are derived from the baroclinic potential vorticity (BPV) equation, including the modified KP (mKP) equation, standard KP equation and cylindrical KP (cKP) equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.     

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.     

Hierarchies of multi-component mKP equations and theirs integrable couplings

First, a new multi-component modified Kadomtsev-Petviashvill (mKP) spectral problem is constructed by k-constraint imposed on a general pseudo-differential operator. Then, two hierarchies of multi-component mKP equations are derived, including positive non-isospectral mKP hierarchy and negative non-isospectral mKP hierarchy. Moreover, new integrable couplings of the resulting mKP soliton hierarchies are constructed by enlarging the associated matrix spectral problem.     

Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.     

Similarity Reductions and Similarity Solutions of the （3＋1）-Dimensional Kadomtsev-Petviashvili Equation

Employing the compatibility method, we obtain the symmetries of the （3＋1）-dimensional Kadomtsev Petviashvili （KP） equation. Four types of similarity reductions of the KP equation are obtained by solving the corresponding characteristic equations associated with symmetry equations. In addition, a lot of similarity solutions to the KP equation are obtadned.     

A Bilinear Backlund Transformation and Explicit Solutions for a （3＋1）-Dimensional Soliton Equation

Considering the bilinear form of a （3＋1）-dimensional soliton equation, we obtain a bilinear Backlund transformation for the equation. As an application, soliton solution and stationary rational solution for the （3＋1）- dimensional soliton equation are presented.     

New doubly periodic and multiple soliton solutions of the generalized （3＋l）-dimensional KP equation with variable coefficients

A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig－Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.     

Effects of dust size distribution on nonlinear waves in a dusty plasma

For two-dimensional unmagnetized dusty plasmas with many different dust grain species, a Kadomtsev--Petviashvili (KP) equation, a modified KP (mKP) equation and a coupled KP(cKP) equation for small, but finite amplitude dust-acoustic solitary waves are obtained for different physical conditions respectively. The influence of an arbitrary dust size distribution described by a polynomial expressed function on the properties of dust-acoustic solitary waves is investigated numerically. How dust size distribution affects the sign and the magnitude of nonlinear coefficient A(D) of KP (mKP) equation is also discussed in detail. It is noted that whether a compressive or a rarefactive solitary wave exists depends on the dust size distribution in some dusty plasmas.     

A new extended KP hierarchy

A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It provides a general way to construct soliton equations with sources and their Lax representations.     

Periodic Wave Solution to the （3＋1）-Dimensional Boussinesq Equation

One- and two-periodic wave solutions for （3＋l）-dimensional Boussinesq equation are presented by means of Hirota＇s bilinear method and the Riemann theta function. The soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.     

(4,3) and (5,2) Nonlinear Evolution Equations of (2+1)-Dimensional KP Hierarchy

The Kadometsev-Petviashvili (KP) equation is generalized to a class of equations of the (2+1)-dimensional KP hierarchy. The (4,3) and (5,2) integrable evolution equations with high nonlinearity are given in detail.     

Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations

In this Letter, the Exp-function method is used to seek generalized solitonary solutions of Riccati equation. Based on the Riccati equation and its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional Broer-Kaup-Kupershmidt equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.     

Infinitely-many conservation laws for two (2+1)-dimensional nonlinear evolution equations in fluids

In this paper, a method that can be used to construct the infinitely-many conservation laws with the Lax pair is generalized from the (1+1)-dimensional nonlinear evolution equations (NLEEs) to the (2+1)-dimensional ones. Besides, we apply that method to the Kadomtsev–Petviashvili (KP) and Davey–Stewartson equations in fluids, and respectively obtain their infinitely-many conservation laws with symbolic computation. Based on that method, we can also construct the infinitely-many conservation laws for other multidimensional NLEEs possessing the Lax pairs, including the cylindrical KP, modified KP and (2+1)-dimensional Gardner equations, in fluids, plasmas, optical fibres and Bose–Einstein condensates.     

The study of soliton fission and fusion in (2+1)-dimensional nonlinear system

By means of a variable separation approach and an extended homogeneous balance method, a general variable separation excitation of a (2+1)-dimensional nonlinear system is derived. Based on the derived solution with arbitrary functions, we reveal soliton fission and fusion phenomena in the (2+1)-dimensional soliton system.     

Explicit Solutions of (2+1)-Dimensional Canonical Generalized KP,KdV, and(2+1)-Dimensional Burgers Equations with Variable Coefficients

In this paper, using the generalized G'/G-expansion method and the auxiliary differential equation method, we discuss the (2+1)-dimensional canonical generalized KP (CGKP), KdV, and (2+1)-dimensional Burgers equations with variable coefficients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions.     

A new mapping method and its applications to nonlinear partial differential equations

In this Letter, a new mapping method is proposed for constructing more exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2+1)-dimensional Konopelchenko-Dubrovsky equation and the (2+1)-dimensional KdV equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained.     

Quasi-periodic and non-periodic waves in (2+1) dimensions

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. By means of the extended mapping approach, new exact quasi-periodic and non-periodic solutions for the (2+1)-dimensional nonlinear systems are displayed both analytically and graphically.     

Higher-Dimensional Integrable Systems Induced by Motions of Curves in Affine Geometries

We discuss the motions of curves by introducing an extra spatial variable or equivalently, moving surfaces in arffine geometries. It is shown that the 2 ＋1-dimensional breaking soliton equation and a 2 ＋ 1-dimensional nonlinear evolution equation regarded as a generalization to the 1 ＋ 1-dimensional KdV equation arise from such motions.     

Symmetry Reduction and Exact Solutions of the (3+1)-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the (3+1)-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the (3+1)-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the (3+1)-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the (3+1)-dimensional breaking soliton equation.