Lump and Stripe Soliton Solutions to the Generalized Nizhnik-Novikov-Veselov Equation

With the aid of the truncated Painlevé expansion, a set of rational solutions of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov (GNNV) equation with the quadratic function which contains one lump soliton is derived. By combining this quadratic function and an exponential function, the fusion and fission phenomena occur between one lump soliton and a stripe soliton which are two kinds of typical local excitations. Furthermore, by adding a corresponding inverse exponential function to the above function, we can derive the solution with interaction between one lump soliton and a pair of stripe solitons. The dynamical behaviors of such local solutions are depicted by choosing some appropriate parameters.     

Lump solitons,rogue wave solutions and lump-stripe interaction phenomena to an extended (3+1)-dimensional KP equation

By using a direct method, we construct the Hirota bilinear form for an extended (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Based on this bilinearization, the lump solitons and rogue wave solutions are investigated. Constraint conditions for the wave propagation and velocity for lump solitons are found and verified by figures. Also the lump-stripe interaction was investigated to show that the lump solitons will be swallowed by the stripe soliton. Finally, the dynamic behaviour for the obtained lump solution, rogue wave and lump-stripe soliton interaction by suitable special parameters is shown graphically.     

Lump,lumpoff and predictable rogue wave solutions to a dimensionally reduced Hirota bilinear equation

We study a simplified(3+1)-dimensional model equation and construct a lump solution for the special case of z=y using the Hirota bilinear method.Then,a more general form of lump solution is constructed,which contains more arbitrary autocephalous parameters.In addition,a lumpoff solution is also derived based on the general lump solutions and a stripe soliton.Furthermore,we figure out instanton/rogue wave solutions via introducing two stripe solitons.Finally,one can better illustrate these propagation phenomena of these solutions by analyzing images.     

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.     

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.     

Novel localized wave solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations.     

Analysis on lump,lumpoff and rogue waves with predictability to the (2 + 1)-dimensional B-type Kadomtsev–Petviashvili equation

In this work, we investigate the ($2+1$)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation, which can be used to describe weakly dispersive waves propagating in the quasi media and fluid mechanics. We construct the more general lump solutions, localized in all directions in space, with more arbitrary autocephalous parameters. By considering a stripe soliton generated completely by lump solution, a lumpoff solution is presented. Its lump part is cut by soliton part before or after a specific time, with a specific divergence relationship. Furthermore, combining a pair of stripe solitons, we obtain the special rogue waves when lump solution is cut by double solitons. Our results show that the emerging time and place of the rogue waves can be caught through tracking the moving path of lump solution, and confirming when and where it happens a collision with the visible soliton. Finally, some graphic analysis are discussed to understand the propagation phenomena of these 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.     

Solitons molecules,lump and interaction solutions to a (2+1)-dimensional Sharma–Tasso–Olver–Burgers equation

Different resonance constraints enrich the behavior of soliton solutions. The soliton molecules, which are the bound states of solitons, can be set off by the velocity resonance. The lump waves, which are localized in all directions in space, are theoretically regarded as a limit form of soliton in some ways. In this paper, a (2+1)-dimensional Sharma–Tasso–Olver–Burgers (STOB) equation is investigated. Soliton (kink) molecule, half periodic kink(HPK) and HPK molecule are studied. Then the lump solution is obtained and the interactions between lump and kink molecule are discussed. The kink molecule-lump solutions exhibit a fusion phenomenon and a rogue (instanton) phenomenon, respectively.     

Rogue waves of a (3+1)-dimensional BKP equation

We investigate certain rogue waves of a (3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method. We obtain semi-rational solutions in the determinant form, which contain two special interactions: (i) one lump develops from a kink soliton and then fuses into the other kink one; (ii) a line rogue wave arises from the segment between two kink solitons and then disappears quickly. We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time, which performs like a rogue wave. Furthermore, the higher-order semi-rational solutions describing the interaction between two lumps (one line rogue wave) and three kink solitons are presented.     

Lump and rogue wave solutions for the Broer-Kaup-Kupershmidt system

This paper retrieves lump solution for (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) system by the aid of Hirota bilinear method (HBM). We also obtain rogue wave solutions formed by the interaction of lump solution and a pair of stripe solitons. The dynamics of these solutions are figured out graphically by selecting suitable values to parameters.     

Lump,lumpoff and predictable rogue wave solutions to the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation

This paper mainly uses Hirota bilinear form to investigate the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. We obtain the general lump solutions and discuss its positiveness, the propagation path, amplitude and position at any time. Based on the general lump solutions, lumpoff solutions which a combination of lump solitons and stripe solitons, are also triumphantly acquired. Similarly, according to the general lump solutions, we are also consider a particular rogue wave by introducing a pair of stripe solitons, and research its predictability which include the time of the rogue wave appearance, position at time, propagation path and the maximum value of wave height. Finally, some figures are given to explain the movement mechanism of these solutions.     

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.     

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.     

Interaction Solutions for Kadomtsev-Petviashvili Equation with Variable Coefficients *

Based on the Hirota's bilinear form and symbolic computation, the Kadomtsev-Petviashvili equation with variable coefficients is investigated. The lump solutions and interaction solutions between lump solution and a pair of resonance stripe solitons are presented. Their dynamical behaviors are described by some three-dimensional plots and corresponding contour plots.     

Soliton Molecules,Asymmetric Solitons and Hybrid Solutions for(2+1)-Dimensional Fifth-Order Kd V Equation

Soliton molecules were first discovered in optical systems and are currently a hot topic of research. We obtain soliton molecules of the(2+1)-dimensional fifth-order Kd V system under a new resonance condition called velocity resonance in theory. On the basis of soliton molecules, asymmetric solitons can be obtained by selecting appropriate parameters. Based on the N-soliton solution, we obtain hybrid solutions consisting of soliton molecules,lump waves and breather waves by partial velocity resonance and partial long wave limits. Soliton molecules,and some types of special soliton resonance solutions, are stable under the meaning that the interactions among soliton molecules are elastic. Both soliton molecules and asymmetric solitons obtained may be observed in fluid systems because the fifth-order Kd V equation describes the ion-acoustic waves in plasmas, shallow water waves in channels and oceans.     

Space-Curved Resonant Line Solitons in a Generalized(2 + 1)-Dimensional Fifth-Order KdV System

On the basis of N-soliton solutions, space-curved resonant line solitons are derived via a new constraint proposed here, for a generalized(2 + 1)-dimensional fifth-order KdV system. The dynamic properties of these new resonant line solitons are studied in detail. We then discuss the interaction between a resonance line soliton and a lump wave in greater detail. Our results highlight the distinctions between the generalized(2+1)-dimensional fifth-order KdV system and the classical type.     

Resonance Y-type soliton solutions and some new types of hybrid solutions in the(2+1)-dimensional Sawada–Kotera equation

In this paper, based on N-soliton solutions, we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in (2+1)-dimensional integrable systems. Then, we take the (2+1)-dimensional Sawada–Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint. Next, by the long wave limit method, velocity resonance and module resonance, we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves, breather waves, high-order lump waves respectively. Finally, we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.     

Localized Spatial Soliton Excitations in (2+1)-Dimensional NonlinearSchrödinger Equation with Variable Nonlinearity and an ExternalPotential

We report on the localized spatial soliton excitations in the multidimensional nonlinear Schrödinger equation with radially variable nonlinearity coefficient and an external potential. By using Hirota's binary differential operators, we determine a variety of external potentials and nonlinearity coefficients that can support nonlinear localized solutions of different but desired forms. For some specific external potentials and nonlinearity coefficients, we discuss features of the corresponding (2+1)-dimensional multisolitonic solutions, including ring solitons, lump solitons, and soliton clusters.