Rogue Waves and Lump Solitons of the(3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves

In this paper, the(3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order,higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.     

New Doubly Periodic Waves of the （2＋1）-Dimensional Double Sine-Gordon Equation

New exact solutions of the （2 ＋1）-dimensional double sine-Gordon equation are studied by introducing the modified mapping relations between the cubic nonlinear Klein-Gordon system and double sine-Gordon equation. Two arbitrary functions are included into the Jacobi elliptic function solutions. New doubly periodic wave solutions are obtained and displayed graphically by proper selections of the arbitrary functions.     

Solitons and Waves in （2＋l）-Dimensional Dispersive Long-Wave Equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of an improved projective Riccati equation approach, the paper obtains several types of exact solutions to the （2＋l）-dimenslonal dispersive long-wave equation, including multiple-soliton solutions, periodic soliton solutions, and Weierstrass function solutions. From these solutions, apart from several multisoliton excitations, we derive some novel features of wave structures by introducing some types of lower-dimensional patterns.     

Symmetry Reduction of (2+1)-Dimensional Lax–Kadomtsev–Petviashvili Equation

The Lax–Kadomtsev–Petviashvili equation is derived from the Lax fifth order equation, which is an important mathematical model in fluid physics and quantum field theory. Symmetry reductions of the Lax–Kadomtsev–Petviashvili equation are studied by the means of the Clarkson–Kruskal direct method and the corresponding reduction equations are solved directly with arbitrary constants and functions.     

Abundant Traveling Wave Structures of (1+1)-Dimensional Sawada–Kotera Equation: Few Cycle Solitons and Soliton Molecules

Traveling wave solutions have been well studied for various nonlinear systems.However, for high order nonlinear physical models, there still exist various open problems.Here, travelling wave solutions to the well-known fifth-order nonlinear physical model, the Sawada–Kotera equation, are revisited.Abundant travelling wave structures including soliton molecules, soliton lattice, kink-antikink molecules, peak-plateau soliton molecules, few-cycle-pulse solitons, double-peaked and triple-peaked soli...     

Spatio-Temporal Deformation of Kink-Breather to the (2+1)-Dimensional Potential Boiti–Leon–Manna–Pempinelli Equation

In the paper, the rational breather soliton and kink solitary wave solution of the (2+1)-dimensional PBLMP equation are obtained by adopting Hirota bilinear method and selecting different test functions. Furthermore, it has been found that the fusion and degeneration of the kink solitary wave occur when interaction between the rational breather soliton and the kink solitary wave happens. These phenomena are very helpful in researching soliton dynamical complexity in the higher dimensional systems.     

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.     

Spatiotemporal Rogue Waves for the Variable-Coefficient (3+1)-Dimensional Nonlinear Schrdinger Equation

With the help of the similarity transformation connected the variable-coefficient (3+1)-dimensional nonlinear Schrdinger equation with the standard nonlinear Schrdinger equation, we firstly obtain first-order and second-order rogue wave solutions. Then, we investigate the controllable behaviors of these rogue waves in the hyperbolic dispersion decreasing profile. Our results indicate that the integral relation between the accumulated time T and the real time t is the basis to realize the control and manipulation of propagation behaviors of rogue waves, such as sustainment and restraint. We can modulate the value T 0 to achieve the sustained and restrained spatiotemporal rogue waves. Moreover, the controllability for position of sustainment and restraint for spatiotemporal rogue waves can also be realized by setting different values of X 0 .     

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.     

Periodic Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and （3＋ 1）-Dimensional Kadomtsev-Petviashvili Equation

For describing various complex nonlinear phenomena in the realistic world, the higher-dimensional nonlinear evolution equations appear more attractive in many fields of physical and engineering sciences. In this paper, by virtue of the Hirota bilinear method and Riemann theta functions, the periodic wave solutions for the （2＋1）-dimensional Boussinesq equation and （3＋1）-dimensional Kadomtsev Petviashvili （KP） equation are obtained. Furthermore, it is shown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.     

(2+1)-Dimensional Analytical Solutions of the Combining Cubic-Quintic Nonlinear Schrdinger Equation

We investigate analytical solutions of the(2+1)-dimensional combining cubic-quintic nonlinear Schrdinger(CQNLS) equation by the classical Lie group symmetry method.We not only obtain the Lie-point symmetries and some(1+1)-dimensional partial differential systems,but also derive bright solitons,dark solitons,kink or anti-kink solutions and the localized instanton solution.     

Vortex and Gaussian Solitons of (3+1)-Dimensional Spatially Modulated Cubic-Quintic-Septimal Nonlinear Schrdinger Equation with External Potential

Based on the variable separation principle and the similarity transformation, vortex soliton solution of a (3+1)-dimensional cubic-quintic-septimal nonlinear Schr¨odinger equation with spatially modulated nonlinearity under the external potential are obtained in the spatially modulated cubic-quintic-septimal nonlinear media. If the topological charge m = 0 and m ≠ 0, Gaussian solitons and vortex solitons can be constructed respectively. The shapes of vortex soliton possess similar structures when the value of l-m is same. Moreover, all phases of vortex solitons exist m-jump with the change of every jump as 2π/m, m-jumps, and thus totally realize the azimuthal change of 2π around their cores.     

Exact Periodic Wave Solution of Extended(2+1)-Dimensional Shallow Water Wave Equation with Generalized D_-operators

With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.     

Noncommutative Integrability of the Klein–Gordon and Dirac Equations in (2+1)-Dimensional Spacetime

Russian Physics Journal - Noncommutative integration of the Klein–Gordon and Dirac relativistic wave equations in (2+1)-dimensional Minkowski space is considered. It is shown that for all...     

Cross Soliton-Like Waves for the （2＋1）-Dimensional Breaking Soliton Equation

We construct a two-soliton-like solution for the （2＋1）-dimensionai breaking soliton equation. The obtained solution contains two arbitrary functions and hence can model various cross soliton-like waves including the two-solitary waves. We show the evolution of some special cross soliton-like waves diagrammatically.     

Darboux Transformations,Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schrdinger Equation

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.     

Annihilation Solitons and Chaotic Solitons for the （2＋1）-Dimensional Breaking Soliton System

By means of an improved mapping method and a variable separation method, a series of variable separation solutions including solitary wave solutions, periodic wave solutions and rational function solutions） to the （2＋1）-dimensional breaking soliton system is derived. Based on the derived solitary wave excitation, we obtain some special annihilation solitons and chaotic solitons in this short note.     

Periodic Homoclinic Wave of （1＋1）-Dimensional Long-Short Wave Equation

The exact periodic homoclinic wave of （1＋1）D long-short wave equation is obtained using an extended homoclinic test technique. This result shows complexity and variety of dynamical behaviour for a （1＋1）-dimensional long-short wave equation.     

New Exact Solutions and Interactions Between Two Solitary Waves for （3＋1）-Dimensional Jimbo-Miwa System

By means of an extended mapping approach and a linear variable separation approach, a new family of exact solutions of the （3＋1）-dimensional Jimbo-Miwa system is derived. Based on the derived solitary wave solution, we obtain some special localized excitations and study the interactions between two solitary waves of the system.