From breather solutions to lump solutions:A construction method for the Zakharov equation

Periodic solutions of the Zakharov equation are investigated.By performing the limit operationλ_(2l-1)→λ_(1)on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation,an order-n breather-positon solution is first obtained from a plane wave seed.It is then proven that an order-n lump solution can be further constructed by taking the limitλ_(1)→λ_(0)on the breather-positon solution,because the unique eigenvalueλ_(0)associated with the Lax pair eigenfunctionΨ(λ_(0))=0 corresponds to the limit of the infinite-periodic solutions.A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.

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.

Lump solution and interaction solutions to the fourth-order extended(2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper,we explore the exact solutions to the fourth-order extended(2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation.Based on Hirota bilinear method,lump solution,periodic cross-kink solutions and bright-dark soliton solutions were investigated.By calculating and solving,the peak and trough of lump solution are obtained,and the maximum and minimum points of each are solved.The three-dimensional plots and density plots of periodic cross-kink solution and bright-dark soliton solution are drawn and the dynamics of solutions under different parameters are observed.

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.

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.

A Pair of Resonance Stripe Solitons and Lump Solutions to a Reduced(3+1)-Dimensional Nonlinear Evolution Equation

With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.

Lump solutions to a generalized Hietarinta-type equation via symbolic computation

Lump solutions are one of important solutions to partial differential equations,both linear and nonlinear.This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms.The key is a Hirota bilinear form.Lump solutions are constructed via symbolic computations with Maple,and specific reductions of the resulting lump solutions are made.Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented,together with three-dimensional plots and density plots of the lump solutions.

General M-lumps, T-breathers, and hybrid solutions to (2+1)-dimensional generalized KDKK equation

A special transformation is introduced and thereby leads to the N-soliton solution of the(2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt(KDKK) equation.Then,by employing the long wave limit and imposing complex conjugate constraints to the related solitons,various localized interaction solutions are constructed,including the general M-lumps,T-breathers,and hybrid wave solutions.Dynamical behaviors of these solutions are investigated analytically and graphically.The solutions obtained are very helpful in studying the interaction phenomena of nonlinear localized waves.Therefore,we hope these results can provide some theoretical guidance to the experts in oceanography,atmospheric science,and weather forecasting.

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.

Dynamical analysis of diversity lump solutions to the(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure equation

The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equation by the Hirota bilinear method and test function method.With the help of Maple,we draw three-dimensional plots of the lump solution and lump-type solutions,and by observing the plots,we analyze the dynamic behavior of the(2+1)-dimensional dissipative AKNS equation.We find that the interaction solutions come in a variety of interesting forms.

Lump solutions and interaction solutions for(2+1)-dimensional KPI equation

The lump solutions and interaction solutions are mainly investigated for the(2+1)-dimensional KPI equation.According to relations of the undetermined parameters of the test functions,the N-soliton solutions are showed by computations of the Maple using the Hirota bilinear form for(2+1)-dimensional KPI equation.One type of the lump solutions for(2+1)-dimensional KPI equation has been deduced by the limit method of the N-soliton solutions.In addition,the interaction solutions between the lump and N-soliton solutions of it are studied by the undetermined interaction functions.The sufficient conditions for the existence of the interaction solutions are obtained.Furthermore,the new breather solutions for the(2+1)-dimensional KPI equation are considered by the homoclinic test method via new test functions including more parameters than common test functions.

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.

Integrability Tests and Some New Soliton Solutions of an Extended Potential Boiti-Leon-Manna-Pempinelli Equation

This paper is devoted to the study of a (2 + 1)-dimensional extended Potential Boiti-Leon-Manna-Pempinelli equation. Firstly, By means of the standard Weiss Tabor Carnevale approach and Kruskal’s simplification, we prove the painlevé non integrability of the equation. Secondly, A new breather solution and lump type solution are obtained based on the parameter limit method and Hirota’s bilinear method. Besides, some interaction behavior between lump type solution and N-soliton solutions (N is any positive integer) are studied. We construct the existence theorem of the interaction solution and give the process of calculation and proof. We also give a concrete example to illustrate the effectiveness of the theorem, and some spatial structure figures are displayed to reflect the evolutionary behavior of the interaction solutions with the change of soliton number N and time t.

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.

Lump and new interaction solutions to the (3+1)-dimensional nonlinear evolution equation

In this paper,a new(3+1)-dimensional nonlinear evolution equation is introduced,through the generalized bilinear operators based on prime number p=3.By Maple symbolic calculation,one-,two-lump,and breather-type periodic soliton solutions are obtained,where the condition of positiveness and analyticity of the lump solution are considered.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and breather-type periodic soliton are derived,by combining multi-exponential function or trigonometric sine and cosine functions with a quadratic one.In addition,new interaction solutions between a lump,periodic-solitary waves,and one-,two-or even three-kink solitons are constructed by using the ansatz technique.Finally,the characteristics of these various solutions are exhibited and illustrated graphically.

New lump, lump-kink, breather waves and other interaction solutions to the (3+1)-dimensional soliton equation

This study investigates the (3+1)-dimensional soliton equation via the Hirota bilinear approach and symbolic computations. We successfully construct some new lump, lump-kink, breather wave, lump periodic, and some other new interaction solutions. All the reported solutions are verified by inserting them into the original equation with the help of the Wolfram Mathematica package. The solution’s visual characteristics are graphically represented in order to shed more light on the results obtained. The findings obtained are useful in understanding the basic nonlinear fluid dynamic scenarios as well as the dynamics of computational physics and engineering sciences in the related nonlinear higher dimensional wave fields.

Multiple-order line rogue wave,lump and its interaction,periodic,and cross-kink solutions for the generalized CHKP equation

The multiple-order line rogue wave solutions method is emp loyed for searching the multiple soliton solutions for the generalized(2+1)-dimensional Camassa-HolmKadomtsev-Petviashvili(CHKP)equation,which contains first-order,second-order,and third-order waves solutions.At the critical point,the second-order derivative and Hessian matrix for only one point will be investigated and the lump solution has one minimum value.For the case,the lump solution will be shown the bright-dark lump structure and for another case can be present the dark lump structure-two small peaks and one deep hole.Also,the interaction of lump with periodic waves and the interaction between lump and soliton can be obtained by introducing the Hirota forms.In the meanwhile,the cross-kink wave and periodic wave solutions can be gained by the Hirota operator.The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.We alternative offer that the determining method is general,impressive,outspoken,and powerful and can be exerted to create exact solutions of various kinds of nonlinear models originated in mathematical physics and engineering.

Dynamical rational solutions and their interaction phenomena for an extended nonlinear equation

In this paper,we analyze the extended Bogoyavlenskii-Kadomtsev-Petviashvili(eBKP)equation utilizing the condensed Hirota's approach.In accordance with a logarithmic derivative transform,we produce solutions for single,double,and triple M-lump waves.Additionally,we investigate the interaction solutions of a single M-lump with a single soliton,a single M-lump with a double soliton,and a double M-lump with a single soliton.Furthermore,we create sophisticated single,double,and triple complex soliton wave solutions.The extended Bogoyavlenskii-Kadomtsev-Petviashvili equation describes nonlinear wave phenomena in fluid mechanics,plasma,and shallow water theory.By selecting appropriate values for the related free parameters we also create three-dimensional surfaces and associated counter plots to simulate the dynamical characteristics of the solutions offered.

Dynamics of mixed lump-soliton for an extended(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equation

A new(2+1)-dimensional higher-order extended asymmetric Nizhnik-Novikov-Veselov(eANNV)equation is proposed by introducing the additional bilinear terms to the usual ANNV equation.Based on the independent transformation,the bilinear form of the eANNV equation is constructed.The lump wave is guaranteed by introducing a positive constant term in the quadratic function.Meanwhile,different class solutions of the eANNV equation are obtained by mixing the quadratic function with the exponential functions.For the interaction between the lump wave and one-soliton,the energy of the lump wave and one-soliton can transfer to each other at different times.The interaction between a lump and two-soliton can be obtained only by eliminating the sixth-order bilinear term.The dynamics of these solutions are illustrated by selecting the specific parameters in three-dimensional,contour and density plots.

Lump,its interaction phenomena and conservation laws to a nonlinear mathematical model

We solve the Ostrovsky equation in the absence of the rotation effect using the Hirota bilinear method and symbolic calculation.Some unique interaction phenomena have been obtained between lump so-lution,breather wave,periodic wave,kink soliton,and two-wave solutions.All the obtained solutions are validated by putting them into the original problem using the Wolfram Mathematica 12.The physical characteristics of the solutions have been visually represented to shed additional light on the acquired re-sults.Furthermore,using the novel conservation theory,the conserved vectors of the governing equation have been generated.The discovered results are helpful in understanding particular physical phenomena in fluid dynamics as well as the dynamics of nonlinear higher dimensional wave fields in computational physics and ocean engineering and related disciplines.