Interactions between Solitons and Cnoidal Periodic Waves of the Gardner Equation

The Gardner equation is one of the most important prototypic models in nonlinear physics. Many scholars pay much attention to the Gardner equation and various nonlinear excitations of the Gardner equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this work, with the help of the Riccati equation, the Gardner equation is solved by the consistent Riccati expansion. Furthermore, we obtain the soliton-cnoidal wave interaction solutions of the Gardner equation.     

Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation

In nonlinear physics,it is very difficult to study interactions among different types of nonlinear waves.In this paper,the nonlocal symmetry related to the truncated Painleve′expansion of the(2+1)-dimensional Burgers equation is localized after introducing multiple new variables to extend the original equation into a new system.Then the corresponding group invariant solutions are found,from which interaction solutions among different types of nonlinear waves can be found.Furthermore,the Burgers equation is also studied by using the generalized tanh expansion method and a new Ba¨cklund transformation(BT)is obtained.From this BT,novel interactive solutions among different nonlinear excitations are found.     

Invariance of Painlev property for some reduced (1+1)-dimensional equations

We study the Painleve′property of the (1+1)-dimensional equations arising from the symmetry reduction for the (2+1)-dimensional ones.Firstly,we derive the similarity reduction of the (2+1)-dimensional potential Calogero–Bogoyavlenskii–Schiff (CBS)equation and Konopelchenko–Dubrovsky (KD)equations with the optimal system of the admitted onedimensional subalgebras.Secondly,by analyzing the reduced CBS,KD,and Burgers equations with Painleve′test,respectively,we find both the Painleve′integrability,and the number and location of resonance points are invariant,if the similarity variables include all of the independent variables.     

Dynamic behaviors of the lump solutions and mixed solutions to a(2+1)-dimensional nonlinear model

In this paper, we propose a combined form of the bilinear Kadomtsev–Petviashvili equation and the bilinear extended(2+1)-dimensional shallow water wave equation, which is linked with a novel(2+1)-dimensional nonlinear model. This model might be applied to describe the evolution of nonlinear waves in the ocean. Under the effect of a novel combination of nonlinearity and dispersion terms, two cases of lump solutions to the(2+1)-dimensional nonlinear model are derived by searching for the quadratic...     

Nonlocal symmetry and exact solutions of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation

In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the(2+1)-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion(CRE) solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.     

Lump Solutions and Interaction Phenomenon for (2+1)-Dimensional Sawada–Kotera Equation

In this paper, a class of lump solutions to the (2+1)-dimensional Sawada–Kotera equation is studied by searching for positive quadratic function solutions to the associated bilinear equation. To guarantee rational localization and analyticity of the lumps, some sufficient and necessary conditions are presented on the parameters involved in the solutions. Then, a completely non-elastic interaction between a lump and a stripe of the(2+1)-dimensional Sawada–Kotera equation is obtained, which shows a lump solution is drowned or swallowed by a stripe soliton. Finally, 2-dimensional curves, 3-dimensional plots and density plots with particular choices of the involved parameters are presented to show the dynamic characteristics of the obtained lump and interaction solutions.     

Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation

The Boussinesq equation is one of important prototypic models in nonlinear physics.Various nonlinear excitations of the Boussinesq equation have been found by many methods.However,it is very difcult to find interaction solutions among diferent types of nonlinear excitations.In this peper,two equivalent very simple methods,the truncated Painlev′e analysis and the generalized tanh function expansion approaches,are developed to find interaction solutions between solitons and any other types of Boussinesq waves.     

Integrability and Solutions of the(2+1)-dimensional Hunter–Saxton Equation

In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations.Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation,a non-isospectral Lax-pair of the(2+1)-dimensional Hunter–Saxton equation is derived.In addition,exact singular solutions with a finite number of corners are obtained.Furthermore,the(2+1)-dimensional μ-Hunter–Saxton equation is presented,and its exact peaked traveling wave solutions are derived.     

Novel interaction phenomena of the(3+1)-dimensional Jimbo–Miwa equation

In this paper,we give the general interaction solution to the(3+1)-dimensional Jimbo–Miwa equation.The general interaction solution contains the classical interaction solution.As an example,by using the generalized bilinear method and symbolic computation by using Maple software,novel interaction solutions under certain constraints of the(3+1)-dimensional Jimbo–Miwa equation are obtained.Via three-dimensional plots,contour plots and density plots with the help of Maple,the physical characteristics and structures of these waves are described very well.These solutions greatly enrich the exact solutions to the(3+1)-dimensional Jimbo–Miwa equation found in the existing literature.     

Construction of Soliton-Cnoidal Wave Interaction Solution for the (2+1)-Dimensional Breaking Soliton Equation

In this paper, the truncated Painlev′e analysis and the consistent tanh expansion(CTE) method are developed for the(2+1)-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is difficult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus m = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.     

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding(2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation(BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing(2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the(2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the(2+1)-dimensional AKNS equation(also called the Davey-Stewartson hierarchy), a kind of(2+1)-dimensional Schr¨odinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new(2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the(2+1)-dimensional integrable coupling, which is further reduced to the standard(2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known(1+1)-dimensional AKNS hierarchy, the(1+1)-dimensional nonlinear Schr¨odinger equation are all special cases of the(2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the(2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.     

Oscillating multidromion excitations in higher-dimensional nonlinear lattice with intersite and external on-site potentials using symbolic computation

We show by an extensive method of quasi-discrete multiple-scale approximation that nonlinear multi-dimensional lattice waves subjected to intersite and external on-site potentials are found to be governed by (N +1)-dimensional nonlinear Schro¨dinger (NLS) equation. In particular, the resonant mode interaction of (2+1)-dimensional NLS equation has been identified and the theory allows the inclusion of transverse effect. We apply the exponential function method to the (2+1)-dimensional NLS equation and obtain the class of soliton solutions with a purely algebraic computational method. Notably, we discuss in detail the effects of the external on-site potentials on the explicit form of the soliton solution generated recursively. Under the action of the external on-site potentials, the model presents a rich variety of oscillating multidromion patterns propagating in the system.     

On the Quasi-Periodic Wave Solutions and Asymptotic Analysis to a(3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

In this paper, a(3+1)-dimensional generalized Kadomtsev–Petviashvili(GKP) equation is investigated,which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the generalized Bell's polynomials, we succinctly construct the Hirota's bilinear equation to the GKP equation. By virtue of multidimensional Riemann theta functions, a lucid and straightforward way is presented to explicitly construct multiperiodic Riemann theta function periodic waves(quasi-periodic waves) for the(3+1)-dimensional GKP equation. Interestingly,the one-periodic waves are well-known cnoidal waves, which are considered as one-dimensional models of periodic waves.The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional that they have two independent spatial periods in two independent horizontal directions. Finally, we analyze asymptotic behavior of the multiperiodic periodic waves, and rigorously present the relationships between the periodic waves and soliton solutions by a limiting procedure.     

(2+1)-DIMENSIONAL DERIVATIVE NONLINEAR SCHRODINGER EQUATION

A(2+1)-dimensional multi-component derivative nonlinear Schr?dinger (DNLS) equation is obtained from the symmetry constraint of the modified Kadomtsev-Petviashvili equation, The model is proved to be inte- grable under the meaning that it possesses the Paitdevé property and the infinitely many generalized symmetries which constitute a generalized W_∞ algebra, An integrable DNLS hierarchy is obtained from the flow equation of infinitely many symntetries of the DNLS equation.     

Higher-order rogue waves with controllable fission and asymmetry localized in a (3 + 1)-dimensional generalized Boussinesq equation

The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations.Such a nonlinear model considered in this paper as the concrete example is the(3+1)-dimensional generalized Boussinesq(gB) equation,and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters.It is indicated by the(3+1)-dimensional gB equation that the embedded param...     

SYMMETRIES AND DROMION SOLUTION OF A (2+1)-DIMENSIONAL NONLINEAR SOHR?DINGER EQUATION

The Painlevé property, infinitely many symmetries and exact solutions of a (2+1)-dimensional nonlinear Schr?dinger equation, which are obtained from the constraints of the Kadomtsev-Petviashvili equation, are studied in this paper. The Painlevé property is proved by the Weiss-Kruskal approach, the infinitely many symmetries are obtained by the formal series symmetry method and the dromion-like solution which is localized exponentially in all directions is obtained by a variable separation method.     

Higher Dimensional Camassa–Holm Equations

Utilizing some conservation laws of the(1+1)-dimensional Camassa–Holm(CH) equation and/or its reciprocal forms, some(n+1)-dimensional CH equations for n ≥ 1 are constructed by a modified deformation algorithm.The Lax integrability can be proven by applying the same deformation algorithm to the Lax pair of the(1+1)-dimensional CH equation. A novel type of peakon solution is implicitly given and expressed by the Lambert W function.     

Extended Jacobi elliptic function method and its applications to （2＋l）-dimensional dispersive long-wave equation

An extended Jacobi elliptic function method is proposed for constructing the exact double periodic solutions of nonlinear partial differential equations (PDEs) in a unified way. It is shown that these solutions exactly degenerate to the many types of soliton solutions in a limited condition. The Wu－Zhang equation (which describes the (2+1)-dimensional dispersive long wave) is investigated by this means and more formal double periodic solutions are obtained.     

A KdV-Type Wronskian Formulation to Generalized KP,BKP and Jimbo–Miwa Equations

The purpose of this paper is to introduce a class of generalized nonlinear evolution equations, which can be widely applied to describing a variety of phenomena in nonlinear physical science. A Kd V-type Wronskian formulation is constructed by employing the Wronskian conditions of the Kd V equation. Applications are made for the(3+1)-dimensional generalized KP, BKP and Jimbo–Miwa equations, thereby presenting their Wronskian sufficient conditions.An N-soliton solution in terms of Wronskian determinant is obtained. Under a dimensional reduction, our results yield Wronskian solutions of the Kd V equation.     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion(CTE) method is applied to the(2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution,and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlev′e truncated expansion method.And we investigate interactive properties of solitons and periodic waves.