Localized wave solutions and interactions of the (2+1)-dimensional Hirota-Satsuma-Ito equation

This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.

Abundant invariant solutions of extended(3+1)-dimensional KP-Boussinesq equation

Lie group analysis method is applied to the extended(3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation and the corresponding similarity reduction equations are obtained with various infinitesimal generators.By selecting suitable arbitrary functions in the similarity reduction solutions,we obtain abundant invariant solutions,including the trigonometric solution,the kink-lump interaction solution,the interaction solution between lump wave and triangular periodic wave,the two-kink solution,the lump solution,the interaction between a lump and two-kink and the periodic lump solution in different planes.These exact solutions are also given graphically to show the detailed structures of this high dimensional integrable system.

Dynamics of Nonlinear Waves in(2+1)-Dimensional Extended Boiti-Leon-Manna-Pempinelli Equation

Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamical characteristics of these solutions were displayed through graphical,particularly revealing fusion and ssion phenomena in the interaction of lump and the one-stripe soliton.

Variable Separation for(1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

A New Class of Periodic Solutions to （2＋1）-Dimensional KdV Equations

We investigate a new class of periodic solutions to (2+1)-dimensional KdV equations, by both the linear superposition approach and the mapping deformation method. These new periodic solutions are suitable combinations of the periodic solutions to the (2+1)-dimensional KdV equations obtained by means of the Jacobian elliptic function method, but they possess different periods and velocities.

A New (2+1)-Dimensional KdV Equation and Its Localized Structures

A new （2＋1）-dimensional KdV equation is constructed by using Lax pair generating technique. Exact solutions of the new equation are studied by means of the singular manifold method. Bgcklund transformation in terms of the singular manifold is obtained. And localized structures are also investigated.

A novel(2+1)-dimensional integrable KdV equation with peculiar solution structures

The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation.A novel(2+1)-dimensional KdV extension,the cKP3-4 equation,is obtained by combining the third member(KP3,the usual KP equation)and the fourth member(KP4)of the KP hierarchy.The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair.The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable.Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations.For instance,the soliton molecules and the missing D'Alembert type solutions(the arbitrary travelling waves moving in one direction with a fixed model dependent velocity)including periodic kink molecules,periodic kink-antikink molecules,few-cycle solitons,and envelope solitons exist for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.

Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method

The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.

Multisoliton Solutions of the (2+1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2+1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2+1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.

A Series of Exact Solutions for a New （2＋1）-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new (2+1)-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.

The Periodic Solitary Wave Solutions for the (2 + 1)-Dimensional Fifth-Order KdV Equation

The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.

Painlevé property, local and nonlocal symmetries, and symmetry reductions for a (2+1)-dimensional integrable KdV equation

The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.

Exact Solutions for (2 + 1)-Dimensional KdV-Calogero-Bogoyavlenkskii-Schiff Equation via Symbolic Computation

This paper constructs exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation with the help of symbolic computation. By means of the truncated Painlev expansion, the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear equation, through the trilinear-linear equation, we can obtain the explicit representation of exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation. We have depicted the profiles of the exact solutions by presenting their three-dimensional plots and the corresponding density plots.

New Periodic Wave Solutions and Their Interaction for （2＋1）-dimensional KdV Equation

A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contains two arbitrary functions） got by means of multilinear variable separation approach for （2＋1）-dimensional KdV equation. Limiting cases are considered and some localized excitations are derived, such as dromion, multidromions, dromion-antidromion, multidromions-antidromions, and so on. Some solutions of the dromion-antidromion and multidromions-antidromions are periodic in one direction but localized in the other direction. The interaction properties of these solutions, which are numerically studied, reveal that some of them are nonelastic and some are completely elastic. Furthermore, these results are visualized.

New Compacton-Like and Solitary Pattern-Like Solutions of （2＋1）-Dimensional Generalization of Modified KdV Equation

Recently some (1+1)-dimensional nonlinear wave equations with linearly dispersive terms were shown to possess compacton-like and solitary pattern-like solutions. In this paper, with the aid of Maple, new solutions of (2+1)-dimensional generalization of mKd V equation, which is of only linearly dispersive terms, are investigated using three new transformations. As a consequence, it is shown that this (2+ 1)-dimensional equation also possesses new compacton-like solutions and solitary pattern-like solutions.

The (3+1)-dimensional generalized mKdV-ZK equation for ion-acoustic waves in quantum plasmas as well as its non-resonant multiwave solution

The quantum hydrodynamic model for ion-acoustic waves in plasmas is studied.First,we design a new disturbance expansion to describe the ion fluid velocity and electric field potential.It should be emphasized that the piecewise function perturbation form is new with great difference from the previous perturbation.Then,based on the piecewise function perturbation,a(3+1)-dimensional generalized modified Korteweg–de Vries Zakharov–Kuznetsov(mKdV-ZK)equation is derived for the first time,which is an extended form of the classical mKdV equation and the ZK equation.The(3+1)-dimensional generalized time-space fractional mKdV-ZK equation is constructed using the semi-inverse method and the fractional variational principle.Obviously,it is more accurate to depict some complex plasma processes and phenomena.Further,the conservation laws of the generalized time-space fractional mKdV-ZK equation are discussed.Finally,using the multi-exponential function method,the non-resonant multiwave solutions are constructed,and the characteristics of ion-acoustic waves are well described.

Symmetries of(2+1)-Dimensional KdV-Ito Equation in the Bilinear Form^1

A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary functions of time t constitute an infinnite-dirmensional Lin algebra which contains two types of the Virasoro subalgebra.

Exact Traveling Wave Solutions for the (1 + 1)-Dimensional Compound KdVB Equation via the Novel (G'/G)-Expansion Method

In this work, while applying a new and novel (G'/G)-expansion version technique, we identify four families of the traveling wave solutions to the (1 + 1)-dimensional compound KdVB equation. The exact solutions are derived, in terms of hyperbolic, trigonometric and rational functions, involving various parameters. When the parameters are tuned to special values, both solitary, and periodic wave models are distinguished. State of the art symbolic algebra graphical representations and dynamical interpretations of the obtained solutions physics are provided and discussed. This in turn ends up revealing salient solutions features and demonstrating the used method efficiency.

Error estimates of H^1-Galerkin mixed finite element method for Schrdinger equation

An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.