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.     

Wronskian Form Solutions for a Variable Coefficient Kadomtsev–Petviashvili Equation

Starting from a simple transformation, and with the aid of symbolic computation, we establish the relation- ship between the solution of a generalized variable coefficient Kadomtsev-Petviashvili （vKP） equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover, we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.     

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.     

Novel Interaction Solutions to Kadomtsev-Petviashvili Equation

In this paper, based on the forms and structures of Wronskian solutions to soliton equations, a Wronskian form expansion method is presented to find a new class of interaction solutions to the Kadomtsev-Petviashvili equation. One characteristic of the method is that Wronskian entries do not satisfy linear partial differential equation.     

Effect of Polynomial Expressed Distribution Dust Particles for Unmagnetized Two-Ion-Temperature Dusty Plasma

Linear dispersion relation for linear wave and a Kadomtsev-Petviashvili （KP） equation for nonlinear wave are given for the unmagnetized two-ion-temperature cold dusty plasma with many different dust grain species, The numerical results of variations of linear dispersion with respect to the different dust size distribution are given, Moreover, how the amplitude, width, and propagation velocity of solitary wave vary vs different dust size distribution is also studied numerically in this paper.     

Similarity Reductions of （2＋1）-Dimensional Multi-component Broer-Kaup System

Painleve property of the （2＋1）-dimensional multi-component Broer-Kaup （BK） system is considered by using the standard Weiss Kruskal approaches. Applying the Clarkson and Kruskal （CK） direct method to the （2＋1）- dimensional multi-component BK system, some types of similarity reductions are obtained. By solving the reductions, one can get the solutions of the （2＋1）-dimensional multi-component BK system.     

Approximate symmetry reduction for perturbed nonlinear SchrSdinger equation

We investigate the one-dimensional nonlinear SchrSdinger equation with a perturbation of polynomial type. The approximate symmetries and approximate symmetry reduction equations are obtained with the approximate symmetry perturbation theory.     

Solitons for the cubic-quintic nonlinear Schr顜僤inger equation with varying coefficients

In this paper,by means of similarity transfomations,we obtain explicit solutions to the cubic-quintic nonlinear Schr顜僤inger equation with varying coefficients,which involve four free functions of space.Four types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.     

Symmetry Analysis and Similarity Solutions of a Resonant Davey-Stewartson System

Lie point symmetry analysis is performed to a recently proposed （2＋1）-dimensional nonlinear system, the resonant Davey-Stewartson equation. Some similarity solutions of the RDS equation are thus obtained.     

Similarity Reductions of Nonisospectral BKP Equation

Basing on the direct method developed by Clarkson and Kruskal, the nonisospectral BKP equation can be reduced to three types of （1＋1）-dimensional variable coefficients partial differential equations （PDEs）. Furthermore, on the basis of the idea of the symmetry group direct method by Lou et al., three types of reduction PDEs are all reduced to the related constant coefficients PDEs by some transformations.     

Similarity Reductions and Exact Solutions of a Coupled Korteweg de Vries System

For a special coupled Korteweg de Vries （KdV） system, its similarity solutions and reduction equations are obtained by the Clarkson and Kruskal＇s direct method. In addition, its new explicit soliton solutions and traveling wave solutions are found by the deformation and mapping method.     

Some Similarity Reduction Solutions to Two-Dimensional Incompressible Navier-Stokes Equation

We investigate the symmetry reduction for the two-dimensional incompressible Navier Stokes equation in conventional stream function form through Lie symmetry method and construct some similarity reduction solutions. Two special cases in [D.K. Ludlow, P.A. Clarkson, and A.P. Bassom, Stud. Appl. Math. 103 （1999） 183] and a theorem in [S.Y. Lou, M. Jia, X.Y. Tang, and F. Huang, Phys. Rev. E 75 （2007） 056318] are retrieved.     

Similarity Reductions of Nearly Concentric KdV Equation

Basing on the direct method developed by Clarkson and Kruskal, the nearly concentric Korteweg-de Vries （ncKdV） equation can be reduced to three types of （1＋1）-dimensional variable coefficients partial differential equations （PDEs） and three types of variable coefficients ordinary differential equation. Furthermore, three types of （1＋1）-dimensional variable coefficients PDEs are all reduced to constant coefficients PDEs by some transformations.     

New Exact Solutions and Evolution of Solitons in Background Waves for （2＋1）-Dimensional Modified Dispersive Water-Wave System

With the help of the conditional similarity reduction method, some new exact solutions of the （2＋1）- dimensional modified dispersive water-wave system （MDWW） are obtained. Based on the derived solution, we investigate the evolution of solitons in the background waves.     

Asymptotics for Large Time of Global Solutions to the Generalized Kadomtsev–Petviashvili Equation

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP where † = 1 or † = m 1. When „ = 2 and † = m 1, (KP) is known as the KPI equation, while „ = 2, † = + 1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case „ = 3, † = m 1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if „ S 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates: ||u(t)||_￥ ￡ C (1 + |t|)^-1 (log(2+|t|))^k, ||u_x(t)||_￥ ￡ C (1 + |t|)^-1 \|u(t)\|_\infty \le C (1 + |t|)^{-1} (\log (2+|t|))^{\kappa}, \|u_x(t)\|_\infty \le C (1 + |t|)^{-1} for all t ] R, where s = 1 if „ = 3 and s = 0 if „ S 4. We also find the large time asymptotics for the solution.     

Decomposition of the generalized KP, cKP and mKP and their exact solutions

The generalized (2+1)-dimensional KP, cKP and mKP are decomposed into the known (1+1)-dimensional soliton equations. Then, we show that the (1+1)-dimensional soliton equations give rise to the explicit soliton solutions of the generalized KP, cKP and mKP.     

Clarkson-Kruskal Direct Similarity Approach for Differential-Difference Equations

In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.     

Clarkson-Kruskal Direct Similarity Approach for Differential-Difference Equations

In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.     

Symmetries of the Classical Integrable Systems and 2-Dimensional Quantum Gravity: a ‘Map’

Abstract

We draw attention to the connections recently established by others between the classical integrable KdV and KP hierarchies in 1 + 1 and 2 + 1 dimensions respectively and the matrix models which relate to the partition functions of 2-dimensional (1 + 1 dimensional) quantum gravity. The symmetries of the classical KP hierarchy in 2 + 1 dimensions are fundamental to this connection.     

Relativistic Magnetosonic Solitary Wave in Magnetized Multi-Ion Plasma

In the presence of an applied uniform magnetic field Bo, the properties of 2-dimensional （2D） magnetosonic solitary waves of relativistic amplitude in the plasma containing electron, light ions He^＋, and heavy ions O＋ are presented. In the weakly relativistic limit, a Kadomtsev Petviashvili （KP） equation is derived by reductive perturbation method. We give the N-soliton solution of the KP equation and find dromion solutions of a potential of the physical field. The interaction law of the dromions is obtained, which shows there is no exchange of energy, momentum, and angular momentum before and after interaction of the dromions except for phase shifts.