Periodic Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and （3＋ 1）-Dimensional Kadomtsev-Petviashvili Equation

For describing various complex nonlinear phenomena in the realistic world, the higher-dimensional nonlinear evolution equations appear more attractive in many fields of physical and engineering sciences. In this paper, by virtue of the Hirota bilinear method and Riemann theta functions, the periodic wave solutions for the （2＋1）-dimensional Boussinesq equation and （3＋1）-dimensional Kadomtsev Petviashvili （KP） equation are obtained. Furthermore, it is shown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.     

Soliton Solutions and Interaction Between a Line Soliton and a y-Periodic Soliton in Generalized （2W1）-Dimensional Nizhnik-Novikov-Veselov Equation

Using the variable separation approach, many types of exact solutions of the generalized (2 1)-dimensional Nizhnik-Novikov-Veselov equation are derived. One of the exact solutions of this model is analyzed to study the interaction between a line soliton and a y-periodic soliton.     

Abundant Multisoliton Structure of （3＋1）-Dimensional Breaking Soliton Equation

Using the extended homogeneous balance method, we obtained abundant exact solution structures of the (3 1)-dimensional breaking soliton equation. By means of the leading order term analysis, the nonlinear transformations of the (3 1)-dimensional breaking soliton equation are given first, and then some special types of single solitary wavesolutions and the multisoliton solutions are constructed.     

Doubly Periodic Propagating Wave for （2d-1）-Dimensional Breaking Soliton Equation

Using the variable separation approach, we obtain a general exact solution with arbitrary variable separation functions for the （2＋1）-dimensional breaking soliton system. By introducing Jacobi elliptic functions in the seed solution, two families of doubly periodic propagating wave patterns are derived. We investigate these periodic wave solutions with different modulus m selections, many important and interesting properties are revealed. The interaction of Jabcobi elliptic function waves are graphically considered and found to be nonelastic.     

Wronskian Form of N-Soliton Solution for the （2＋1）-Dimensional Breaking Soliton Equation

The Wronskian form of N-soliton solution for the （2-kl）-dimensional breaking soliton equation is obtained by resorting to the Hirota direct method.     

Exact Periodic-Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and（3＋1）-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

New Multi-soliton Solutions for the （2＋1）-Dimensional Kadomtsev-Petviashvili Equation

In this paper, an explicit N-fold Darboux transformation with multi-parameters for both a （1＋1）- dimensional Broer-Kaup （BK） equation and a （1＋1）-dimensional high-order Broer-Kaup equation is constructed with the help of a gauge transformation of their spectral problems. By using the Darboux transformation and new basic solutions of the spectral problems, 2N-soliton solutions of the BK equation, the high-order BK equation, and the Kadomtsev-Petviashvili （KP） equation are obtained.     

New Families of Soliton and Periodic Solutions of Bose-Einstein Condensation in Linear Magnetic Field and Time-Dependent Laser Field

By using a new generally projective Riccati equation method and with the help of symbolic computation,we consider a nonlinear Gross-Pitaevskii equation with weak bias magnetic and time-dependent laser fields. As a result,some new soliton solutions, rational function solution, and periodic solutions are obtained.     

Soliton and Periodic Travelling Wave Solutions for Quadratic Nonlinear Media

Many sets of the soliton and periodic travelling wave solutions for the quadratic χ^（2） nonlinear system are obtained by the Backlund transformation and the trial method. The property of the propagation for some travelling waves is investigated.     

New Multiple Soliton-like and Periodic Solutions for （2＋l）-Dimensional Canonical Generalized KP Equation with Variable Coefficients

In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the （2＋1）-dimensional OGKP equation with variable coetffcients.     

Periodic Soliton Solutions as Imbricate Series of Rational Solitons: Solutions to the Kadomtsev-Petviashvili Equation with Positive Dispersion

Abstract

An inclined periodic soliton solution can be expressed as imbricate series of rational soliton solutions. A convenient form of the imbrication is given by using the bilinear form. A lattice soliton solution which propagaties in any direction can be also constructed by doubly imbricating rational solitons.     

Soliton Asymptotics of Rear Part of Non-Localized Solutions of the Kadomtsev-Petviashvili Equation

Abstract

We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for x → ?∞ and study their large time asymptotic behavior. We prove that such solutions eject (for t → ∞) a train of curved asymptotic solitons which move behind the basic wave packet.     

Decay Mode Solutions for Kadomtsev-Petviashvili Equation

The decay mode solutions for the Kadomtsev-Petviashvili (KP) equation are derived by Hirota method (direct method). The decay mode solution is a new set of analytical solutions with Airy function.     

Breather-Type Periodic Soliton Solutions for (1+1)-Dimensional Sinh-Poisson Equation

In this paper,by using bilinear form and extended homoclinic test approach,we obtain new breather-type periodic soliton solutions of the (1+1)-dimensional Sinh-Poisson equation.These results demonstrate that the nonlinear evolution equation has rich dynamical behavior even if it is (1+1)-dimensional.     

Frobenius Manifold for the Dispersionless Kadomtsev-Petviashvili Equation

We consider a Frobenius structure associated with the dispersionless Kadomtsev – Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic energy with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.     

Exact Solutions for a Nonisospectral and Variable-Coefficient Kadomtsev-Petviashvili Equation

The bilinear form for a nonisospectral and variable-coefficient Kadomtsev-Petviashvili equation is obtained and some exact soliton solutions are derived by the Hirota method and Wronskian technique. We also derive the bilinear Backlund transformation from its Lax pairs and find solutions with the help of the obtained bilinear Bgcklund transformation.     

Interaction Between Line Soliton and Algebraic Soliton for Asymmetric Nizhnik-Novikov-Veselov Equation

Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for （2＋1）-dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton： 1） the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2） the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.     

Nonlocal Symmetries and Interaction Solutions for Potential Kadomtsev-Petviashvili Equation

The nonlocal symmetry for the potential Kadomtsev-Petviashvili(pKP)equation is derived by the truncated Painleve analysis.The nonlocal symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variable.Thanks to localization process,the finite symmetry transformations related with the nonlocal symmetry are obtained by solving the prolonged systems.The inelastic interactions among the multiple-front waves of the pKP equation are generated from the finite symmetry transformations.Based on the consistent tanh expansion method,a nonauto-B(a|¨)cklund transformation(BT)theorem of the pKP equation is constructed.We can get many new types of interaction solutions because of the existence of an arbitrary function in the nonauto-BT theorem.Some special interaction solutions are investigated both in analytical and graphical ways.