Lump and mixed rogue-soliton solutions to the 2+1 dimensional Ablowitz-Kaup-Newell-Segur equation

In this paper, the 2+1 dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation which obtained from the potential Boiti-Leon-Manna-Pempi nelli (pBLMP) equation, is introduced. Through the bilinear method and ansatz technique, the rational solutions consisting of rogue wave and lump soliton solutions are constructed, where we discuss the condition of guaranteeing the positiveness and analyticity of the lump solutions. The collection of a quadratic function with an exponential function describing rational-exponential solutions is proved, the interaction consisting of one lump and one soliton with fission and fusion phenomena. The second kind of interaction comprises the line rogue wave and soliton solution, which is inelastic. With the usage of the extended homoclinic test approach, the homoclinic breather-wave solution is derived. The characteristics of these various solutions are exhibited and illustrated graphically.     

Interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation for nonlinear waves

In this paper, we focus on the interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation. With symbolic computation, two types of interaction solutions including lump-kink and lump-soliton ones are derived through mixing two positive quadratic functions with an exponential function, or two positive quadratic functions with a hyperbolic cosine function in the bilinear equation. The completely non-elastic interaction between a lump and a stripe is presented, which shows the lump is drowned or shallowed by the stripe. The interaction between lump and soliton is also given, where the lump moves from one branch to the other branch of the soliton. These phenomena exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.     

Rogue wave and interaction phenomenon to (1+1)-dimensional Ito equation

A novel type of exact rogue wave is found for the (1+1)-dimensional Ito equation, which is generated by the interaction solution between an algebraic localized soliton (named “lump”) and an exponentially localized twin soliton. In addition, the interaction solution among triangular periodic wave and twin soliton is also proposed. Three special interaction phenomenons are displayed by some visual figures, respectively.     

Lump wave and hybrid solutions of a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.     

LUMP AND INTERACTION SOLUTIONS TO LINEAR (4+1)-DIMENSIONAL PDES

Taking a class of linear(4+1)-dimensional partial differential equations as examples, we would like to show that there exist lump solutions and interaction solutions in(4+1)-dimensions. We will compute abundant lump solutions and interaction solutions to the considered linear(4+1)-dimensional partial differential equations via symbolic computations,and plot three specific solutions with Maple plot tools, which supplements the existing literature on lump, rogue wave and breather solutions and their interaction solutions in soliton theory.     

Interaction among a lump,periodic waves,and kink solutions to the fractional generalized CBS‐BK equation

The Hirota bilinear method is prepared for searching the diverse soliton solutions for the fractional generalized Calogero‐Bogoyavlenskii‐Schiff‐Bogoyavlensky‐Konopelchenko (CBS‐BK) equation. Also, the Hirota bilinear method is used to finding the lump and interaction with two stripe soliton solutions. Interaction among lumps, periodic waves, and multi‐kink soliton solutions will be investigated. Also, the solitary wave, periodic wave, and cross‐kink wave solutions will be examined for the fractional gCBS‐BK equation. The graphs for various fractional order α are plotted to contain 3D plot, contour plot, density plot, and 2D plot. We construct the exact lump and interaction among other types solutions, by solving the under‐determined nonlinear system of algebraic equations for the associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions and the interaction behaviors are revealed. The existence conditions are employed to discuss the available got solutions.     

Lump Solutions, Lump-soliton Interaction Phenomena and Breather-soliton Solutions to a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli-like Equation

The (3+1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (BLMP-like equation) is introduced by the generalized bilinear operators $D_{p}$ associated with $p=3$. The lump solutions, lump-soliton interaction phenomena and breather-soliton solutions are discussed to the (3+1)-dimensional BLMP-like equation based on the generalized bilinear method with symbolic computation system Mathematica. In order to observe the behavior of those solutions, we fix the value of $z$, then give the 3D-graphs of some solutions at different times. We find a lump solution moved in oblique direction; a lump-soliton interaction phenomenon is appeared and disappeared along with the time. We also see a kink-breather soliton moved in oblique direction.     

(2+1)维AKNS方程的行波解研究(英文)

Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic solution. It also proves that the result is consistent with the soliton solution of simplify Hirota bilinear method by Wazwaz and illustrate the solution are right travelling wave solution.     

Chaos,solitons and fractals in hidden symmetry models

A spontaneous symmetry breaking (or hidden symmetry) model is reduced to a system nonlinear evolution equations integrable via an appropriate change of variables, by means of the asymptotic perturbation (AP) method, based on spatio-temporal rescaling and Fourier expansion. It is demonstrated the existence of coherent solutions as well as chaotic and fractal patterns, due to the possibility of selecting appropriately some arbitrary functions. Dromion, lump, breather, instanton and ring soliton solutions are derived and the interaction between these coherent solutions are completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Finally, one can construct lower dimensional chaotic patterns such as chaotic–chaotic patterns, periodic–chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution.     

一类非线性偏微分方程组的直接解法

根据Hopf-Cole变换法和试探函数法的基本思想,引入一个变换,并把它应用于求解(2+1)维破裂孤子方程组、(2+1)维Nizhnik-Novikov-Vesslov方程组和(2+1)维Broer-Kaup方程组,得到了这三个方程组的许多新的解析解,包括孤波解和奇异行波解.该方法也适用于其它方程组.     

Nonautonomous characteristics of lump solutions for a (2+1)-dimensional Korteweg–de Vries equation with variable coefficients

We study a $(2+1)$-dimensional Korteweg–de Vries (KdV) equation with variable coefficients. By virtue of Hirota method, we present three types of nonautonomous lump solutions including the bright, bright–dark and dark lump ones. By considering different types of dispersion coefficients, we investigate the characteristics of trajectories, velocities and displacements of nonautonomous bright lump wave, which are different from the case of its constant-coefficient counterpart. We finally demonstrate the periodic attraction and repulsion interaction between a lump wave and a soliton. Our results might provide some physical insights into the relevant fields in nonlinear science.     

Higher-Order Solitons in the N-Wave System

The soliton dressing matrices for the higher-order zeros of the Riemann–Hilbert problem for the N -wave system are considered. For the elementary higher-order zero, that is, whose algebraic multiplicity is arbitrary but the geometric multiplicity is 1, the general soliton dressing matrix is derived. The theory is applied to the study of higher-order soliton solutions in the three-wave interaction model. The simplest higher-order soliton solution is presented. In the generic case, this solution describes the breakup of a higher-order pumping wave into two higher-order elementary waves, and the reverse process. In non-generic cases, this solution could describe (i) the merger of a pumping sech wave and an elementary sech wave into two elementary waves (one sech and the other one higher order); (ii) the breakup of a higher-order pumping wave into two elementary sech waves and one pumping sech wave; and the reverse processes. This solution could also reproduce fundamental soliton solutions as a special case.     

Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics

In this paper, we devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic, and soliton solutions, (f) Jacobi, and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, combined KdV–MKdV, Camassa–Holm, Kaup–Kupershmidt, Jaulent–Miodek, (2+1)-dimensional dispersive long wave, new (2+1)-dimensional generalized Hirota, (2+1)-dimensional breaking soliton and double sine-Gordon equations. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.     

Fractional traveling wave solutions of the (2 + 1)-dimensional fractional complex Ginzburg–Landau equation via two methods

A (2 + 1)-dimensional fractional complex Ginzburg–Landau equation is solved via fractional Riccati method and fractional bifunction method, and exact traveling wave solutions including soliton solution and combined soliton solutions are constructed based on Mittag–Leffler function. A series of fractional orders is used to demonstrate the graphical representation and physical interpretation of the resulting solutions. The role of the fractional order is revealed.     

Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev-Petviashvili equation

In this paper, the one- and two-periodic wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation are presented by means of the Hirota’s bilinear method and the Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.     

关于求最优分批问题的全部解

<正> 在文章[2],[3],[4],[5]中已对任何N>n>1及0≤δ≤1/2(以下总假设上列不等式满足)研究了最优分批问题的满足[2]中定理5.1的条件组     

Consistent Riccati expansion for finding interaction solutions of (2+1)‐dimensional modified dispersive water‐wave system

A consistent Riccati expansion (CRE) is proposed to solve the (2+1)‐dimensional modified dispersive water‐wave (MDWW) system. It is proved that the MDWW system is CRE solvable. Furthermore, new exact interaction solutions, namely, soliton‐trigonometric waves, trigonometric waves‐soliton, soliton‐cosine periodic waves, and soliton‐cnoidal waves are explicitly derived.     

Solitons, chaos and fractals in the (2 + 1)-dimensional dispersive long wave equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of a projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional dispersive long wave (DLW) equation which include multiple soliton solution, periodic soliton solution and Weierstrass function solution. Subsequently, several multisolitons are derived and some novel features are revealed by introducing lower-dimensional patterns.     

A general bilinear form to generate different wave structures of solitons for a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation

In this paper, we investigate a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation (3D‐BMLP). By using bilinear forms under certain conditions, we obtain different wave structures for the 3D‐BMLP. Among these waves, lump waves, breather waves, mixed waves, and multi‐soliton wave solutions are constructed. The propagation and the dynamical behavior of the obtained solutions are discussed for different values of the free parameters.