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.     

mKdV-sine-Gordon方程丰富的相互作用解

基于mKdV-sine-Gordon方程的Wronsk解的形式和结构,提出了Wronsk形式展开法,通过这一方法求得了该方程的丰富的相互作用解,该方法的主要特征是不要求Wronsk行列式元素满足线性偏微分方程组。     

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.     

High-order breather,M-kink lump and semi-rational solutions of potential Kadomtsev-Petviashvili equation

N-kink soliton and high-order synchronized breather solutions for potential Kadomtsev-Petviashvili equation are derived by means of the Hirota bilinear method,and the limit process of high-order synchronized breathers are shown.Furthermore,M-lump solutions are also presented by taking the long wave limit.Additionally,a family of semi-rational solutions with elastic collision are generated by taking a long-wave limit of only a part of exponential functions,their interaction behaviors are shown by three-dimensional plots and contour plots.     

Investigation on the resonance phenomena of multi-solitons for the （3＋1）-dimensional Kadomtsev-Petviashvili equation

In this paper, we theoretically investigate the four-soliton interaction and their resonance phenomena of the (3+1)-dimensional Kadomtsev--Petviashvili (KP) equation. We find that the maximum amplitude of the resonantly created soliton can be 16 times that of one of the four equi-amplitude initial interacting solitons. We also find that the maximum amplitude can only be 4 times the initial soliton amplitude when the resonance phenomena does not take place. The case of four solitons with different amplitudes also has been studied analytically. The results indicate that the resonance phenomena still exists in this case. Numerical results confirm the theoretical predictions.     

Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids

In this article, we consider the(3+1)-dimensional generalized Kadomtsev–Petviashvili(GKP)equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the(oscillating-) W-and M-shaped waves, rational W-shaped waves, multi-peak solitary waves,(quasi-) Bell-shaped and W-shaped waves and(quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.     

Lump and lump-soliton interaction solutions for an integrable variable coefficient Kadomtsev-Petviashvili equation

For a variable coefficient Kadomtsev-Petviashvili (KP) equation the Lax pair as well as conjugate Lax pair are derived from the Painleve analysis.The N-fold binary Darboux transformation is presented in a compact form.As an application,the multi-lump,higher-order lump and general lump-soliton interaction solutions for the variable coefficient KP equation are obtained.Typical lump structures with amplitudes exponentially decaying to zero as the time tends to infinity and interactions between one lump and one soliton are shown.     

Lump and interaction solutions to the (3+1)-dimensional Burgers equation

The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.     

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 rogue wave and solitary solutions for the generalized BK equation via Hirota bilinear and SIVP schemes arising in fluid mechanics

The multiple lump solutions method is employed for the purpose of obtaining multiple soliton solutions for the generalized Bogoyavlensky-Konopelchenko(BK) equation. The solutions obtained contain first-order, second-order, and third-order wave solutions. At the critical point,the second-order derivative and Hessian matrix for only one point is investigated, and the lump solution has one maximum value. He's semi-inverse variational principle(SIVP) is also used for the generalized BK equation. Three major cases are studied, based on two different ansatzes using the SIVP. The physical phenomena of the multiple soliton solutions thus obtained are then analyzed and demonstrated in the figures below, using a selection of suitable parameter values.This method should prove extremely useful for further studies of attractive physical phenomena in the fields of heat transfer, fluid dynamics, etc.     

Soliton molecules and mixed solutions of the(2+1)-dimensional bidirectional Sawada-Kotera equation

In this paper, we investigate some new interesting solution structures of the(2+1)-dimensional bidirectional Sawada–Kotera(bSK) equation. We obtain soliton molecules by introducing velocity resonance. On the basis of soliton molecules, asymmetric solitons are obtained by changing the distance between two solitons of molecules. Based on the N-soliton solutions,several novel types of mixed solutions are generated, which include the mixed breather-soliton molecule solution by the module resonance of the wave number and partial velocity resonance,the mixed lump-soliton molecule solution obtained by partial velocity resonance and partial long wave limits, and the mixed solutions composed of soliton molecules(asymmetric solitons), lump waves, and breather waves. Some plots are presented to clearly illustrate the dynamic features of these solutions.     

Interaction phenomena between lump and solitary wave of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

In this paper, a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation is studied in liquid with gas bubbles. Based on the Hirota's bilinear form and symbolic computation, lump and interaction solutions between lump and solitary wave are obtained,which include a periodic-shape lump solution, a parabolic-shape lump solution, a cubic-shape lump solution, interaction solutions between lump and one solitary wave, and between lump and two solitary waves. The spatial structures called the bright lump wave and the bright-dark lump wave are discussed. Interaction behaviors of two bright-dark lump waves and a periodic-shape bright lump wave are also presented. Their interactions are shown in some 3D plots.     

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.     

Two new integrable Kadomtsev–Petviashvili equations with time-dependent coefficients: multiple real and complex soliton solutions

ABSTRACT

In this work, we develop two new integrable Kadomtsev–Petviashvili (KP) equations with time-dependent coefficients. The integrability property of each equation is explicitly demonstrated exhibiting the Painlevé test to confirm its integrability. Moreover, each equation admits multiple real and multiple complex soliton solutions. We introduce complex forms of the simplified Hirota's method to derive multiple complex soliton solutions. These two model equations are likely to be of applicative relevance, because it may be considered an application of a large class of nonlinear KP equations.     

New interaction solutions and nonlocal symmetry of an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form

The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method. The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables. The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly. Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.     

Exact solutions of the generalized Bretherton equation

The generalized Bretherton equation is studied. The Bäcklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.     

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.     

The D'Alembert type waves and the soliton molecules in a (2+1)-dimensional Kadomtsev-Petviashvili with its hierarchy equation

For a one (2+1)-dimensional combined Kadomtsev-Petviashvili with its hierarchy equation, the missing D'Alembert type solution is derived first through the traveling wave transformation which contains several special kink molecule structures. Further, after introducing the Bäcklund transformation and an auxiliary variable, the N-soliton solution which contains some soliton molecules for this equation, is presented through its Hirota bilinear form. The concrete molecules including line solitons, breathers and a lump as well as several interactions of their hybrid are shown with the aid of special conditions and parameters. All these dynamical features are demonstrated through the different figures.