Soliton molecules,T-breather molecules and some interaction solutions in the(2+1)-dimensional generalized KDKK equation

By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-breather molecules,T-breather–L-soliton molecules and some interaction solutions when N≤6.Dynamical behaviors of these solutions are discussed analytically and graphically.The method adopted can be effectively used to construct soliton molecules and T-breather molecules of other nonlinear evolution equations.The results obtained may be helpful for experts to study the related phenomenon in oceanography and atmospheric science.

Interaction solutions for the second extended(3+1)-dimensional Jimbo–Miwa equation

Based on the Hirota bilinear method,the second extended(3+1)-dimensional Jimbo–Miwa equation is established.By Maple symbolic calculation,lump and lump-kink soliton solutions are obtained.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions.Furthermore,periodiclump wave solution is derived via the ansatz including hyperbolic and trigonometric functions.Finally,3D plots,2D curves,density plots,and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.

Residual symmetry, interaction solutions, and conservation laws of the(2+1)-dimensional dispersive long-wave system

We explore the （2＋l）-dimensional dispersive long-wave （DLW） system. From the standard truncated Painleve expansion, the Baicklund transformation （BT） and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the （2＋l）-dimensional DLW system is consistent Riccati expansion （CRE） solvable. If the special form of （CRE）-consistent tanh-function expansion （CTE） is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.

Localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations

We focus on the localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations.Based on the Hirota bilinear method and the test function method,we construct the exact solutions to the extended equations including lump solutions,lump–kink solutions,and two other types of interaction solutions,by solving the underdetermined nonlinear system of algebraic equations for associated parameters.Finally,analysis and graphical simulation are presented to show the dynamical characteristics of our solutions and the interaction behaviors are revealed.

Rational solutions and interaction solutions for(2+1)-dimensional nonlocal Schrodinger equation

A chain of novel higher order rational solutions with some parameters and interaction solutions of a(2+1)-dimensional reverse space–time nonlocal Schrodinger(NLS)equation was derived by a generalized Darboux transformation(DT)which is derived by Taylor expansion and determinants.We obtained a series of higher-order rational solutions by one spectral parameter and we could get the periodic wave solution and three kinds of interaction solutions,singular breather and periodic wave interaction solution,singular breather and traveling wave interaction solution,bimodal breather and periodic wave interaction solution by two spectral parameters.We found a general formula for these solutions in the form of determinants.We also analyzed the complex wave structures of the dynamic behaviors and the effects of special parameters and presented exact solutions for the(2+1)-dimensional reverse space–time nonlocal NLS equation.

Novel Interaction Solutions to Kadomtsev-Petviashvili Equation

In this paper,based on the forms and structures of Wronskian solutions to soliton equations,a Wronskianform 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.

Painleve Integrability, Consistent Riccati Expansion Solvability and Interaction Solution for the Coupled mKdV-BLMP System

The integrability of the coupled, modified KdV equation and the potential Boiti-Leon-Manna-Pempinelli (mKdV-BLMP) system is investigated using the Painlevé analysis approach. It is shown that this coupled system possesses the Painlevé property in both the principal and secondary branches. Then, the consistent Riccati expansion (CRE) method is applied to the coupled mKdV-BLMP system. As a result, it is CRE solvable for the principal branch while non-CRE solvable for the secondary branch. Finally, starting from the last consistent differential equation in the CRE solvable case, soliton, multiple resonant soliton solutions and soliton-cnoidal wave interaction solutions are constructed explicitly.

Residual symmetries, consistent-Riccati-expansion integrability, and interaction solutions of a new(3+1)-dimensional generalized Kadomtsev–Petviashvili equation

With the aid of the Painlevé analysis, we obtain residual symmetries for a new(3+1)-dimensional generalized Kadomtsev–Petviashvili(gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the(3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.

New lump solutions and several interaction solutions and their dynamics of a generalized(3+1)-dimensional nonlinear differential equation

In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.

New Lump Solution and Their Interactions with N-Solitons for a Shallow Water Wave Equation

By employing the Hirota’s bilinear method and different test functions, the breather solutions of HSI equation with different structures are obtained based on symbolic calculation with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions. The interaction between lump solution and soliton solution is constructed in the form of lump solution, and the motion trajectory of lump is obtained. In addition, the theorem of lump solitons and N-solitons superposition is given and proved. The superposition formula of lump is derived from the theorem, and its spatial evolution behavior is given.

Interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation in incompressible fluid

This paper aims to search for the solutions of the(2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation.Lump solutions,breather solutions,mixed solutions with solitons,and lump-breather solutions can be obtained from the N-soliton solution formula by using the long-wave limit approach and the conjugate complex method.We use both specific circumstances and general higher-order forms of the hybrid solutions as examples.With the help of maple software,we create density and 3D graphs to summarize the dynamic properties of these solutions.Additionally,it is possible to observe how the solutions'trajectory,velocity,and shape vary over time.

Breather, lump, and interaction solutions to a nonlocal KP system

A new high-dimensional two-place Alice–Bob-Kadomtsev–Petviashvili(AB-KP)equation is proposed by applying the Alice–Bob-Bob–Alice principle and shifted-parity,delayed time reversal,charge conjugation(■)principle to the usual KP equation.Based on the dependent variable transformation,the bilinear form of the AB-KP system is constructed.Explicit trigonometric-hyperbolic,rational and rational-hyperbolic solutions are presented by taking advantage of the Hirota bilinear method.The obtained breather,lump,and interaction solutions enrich the solution structure of nonlocal nonlinear systems.The three-dimensional graphs of these nonlinear wave solutions are demonstrated by choosing the specific parameters.

Nonlinear Self-Adjointness, Conservation Laws and Soliton-Cnoidal Wave Interaction Solutions of(2+1)-Dimensional Modified Dispersive Water-Wave System

This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.

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

In this paper, the truncated Painleve 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 dimcult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus rn = 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.

CRE Solvability, Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation

This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.

Multiple lump molecules and interaction solutions of the Kadomtsev-Petviashvili Ⅰ equation

In this paper,a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation.The high-order multiple lump molecules consisting of M N-lump molecules are constructed by means of the Mth-order determinant and the non-homogeneous polynomial in the degree of 2N.The interaction solutions describing P line solitons radiating P of the M N-lump molecules are constructed.The dynamic behaviors of some specific solutions are analyzed through numerical simulation.All the results will enrich our understanding of the multiple lump waves of the Kadomtsev–Petviashvili I equation.

Localization of nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

The nonlocal symmetry of the Sawada–Kotera(SK)equation is constructed with the known Lax pair.By introducing suitable and simple auxiliary variables,the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further.On the other hand,the group invariant solutions of the SK equation are constructed with the classic Lie group method.In particular,by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.

Interaction Solutions for (1+1)-Dimensional Higher-Order Broer–Kaup System

The(1+1)-dimensional higher-order Broer–Kaup(HBK) system is studied by consistent tanh expansion(CTE) method in this paper. It is proved that the HBK system is CTE solvable, and some exact interaction solutions among different nonlinear excitations such as solitons, rational waves, periodic waves, corresponding images are explicitly given.

Breather molecules and localized interaction solutions in the(2+1)-dimensional BLMP equation

The(2+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP)equation is an important integrable model.In this paper,we obtain the breather molecule,the breather-soliton molecule and some localized interaction solutions to the BLMP equation.In particular,by employing a compound method consisting of the velocity resonance,partial module resonance and degeneration of the breather techniques,we derive some interesting hybrid solutions mixed by a breather-soliton molecule/breather molecule and a lump,as well as a bell-shaped soliton and lump.Due to the lack of the long wave limit,it is the first time using the compound degeneration method to construct the hybrid solutions involving a lump.The dynamical behaviors and mathematical features of the solutions are analyzed theoretically and graphically.The method introduced can be effectively used to study the wave solutions of other nonlinear partial differential equations.

Backlund transformations, consistent Riccati expansion solvability, and soliton-cnoidal interaction wave solutions of Kadomtsev-Petviashvili equation

The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlev6 expansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE) solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.