Dynamics and interaction scenarios of localized wave structures in the Kadomtsev–Petviashvili-based system

In this letter, we investigate the dynamics and various interaction scenarios of localized wave structures in the Kadomtsev–Petviashvili (KP)-based system. By using a combination of the Hirota’s bilinear method and the KP hierarchy reduction method, new families of determinant semi-rational solutions of the KP-based system are derived, including lump solitons and rogue-wave solitons. The generic interaction scenarios between distinct types of localized wave solutions are investigated. Our detailed study reveals different types of interaction phenomena: fusion of lumps and line solitons into line solitons, fission of line solitons into lumps and line solitons, a mixture of fission and fusion processes of lumps and line solitons, and the inelastic collision of line rogue waves and line solitons.     

Forced motion of breathers and domain boundaries against the background of nonlinear magnetization wave

Exact solutions of Landau–Lifshitz equation for a ferromagnet with an easy-axis anisotropy, which describe interaction of nonlinear precession wave of large amplitude with soliton-like objects, such as breathers, solitary domains and domain boundaries, are found by the “dressing” method. The change of the internal structure and physical parameters of solitons due to interaction with magnetization wave is analyzed. It is shown, that both solitary domains and domain walls move toward the wave. The conditions for the destruction of solitons by the nonlinear magnetization wave are obtained. An infinite series of integrals of motion, that stabilize the solitons on the background of magnetization wave, is found.     

On the Multi-soliton Solutions of Some Nonlinear Evolution Equations

We use the Hirota's technique to find the multi-soliton solutions of some nonlinear evolution equations. The interaction of the solitons is studied.     

Multisoliton Solutions of the Matrix KdV Equation

We consider multisoliton solutions of the matrix KdV equation. We obtain the formulas for changing phases and amplitudes during the interaction of two solitons and prove that no multiparticle effects appear during the multisoliton interaction. We find the conditions ensuring the symmetry of the corresponding solutions of the matrix KdV equation if they are constructed by the matrix Darboux transformation applied to the Schrödinger operator with zero potential.     

A Method for Solving the Hopf Equation and Generalization of Sobolev–Schwartz Distributions

We introduce distributions that are functionals with values in a nonarchimedian field of Laurent series. These objects naturally generalize Sobolev–Schwartz distributions, and we use them to find generalized solutions of the Hopf equation in the form of infinitely narrow solitons and shock waves. We propose a method for computation of the profile of an infinitely narrow soliton and of a shock wave described by the Hopf equation.     

Solitary wave families of a generalized microstructure PDE

Wave propagation in a generalized microstructure PDE, under the Mindlin relations, is considered. Limited analytic results exist for the occurrence of one family of solitary wave solutions of these equations. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work are also mentioned.     

Steady bifurcation and solitons in relativistic laser plasmas interaction

In this paper, steady bifurcation and solitons in relativistic laser plasmas interaction are investigated. At first, a new coupled equation for wake wave and the circularly polarized transversal electromagnetic wave is derived. It is a Hamiltonian system with two degrees of freedom. Then, a steady bifurcation analysis based on the coexistence of three different equilibrium states is given. Finally, a condition for predicting the existence of solitons is obtained in terms of the bifurcation control parameter and Hamiltonian function value. The soliton solutions are found numerically. It is shown that the solitons can exist in appropriate regime of vector potential frequency.     

Dynamics of an integrable Kadomtsev–Petviashvili-based system

Kadomtsev–Petviashvili (KP)-type equations are seen in fluid mechanics, plasma physics, and gas dynamics. Hereby we consider an integrable KP-based system. With the Hirota method, symbolic computation and truncated Painlevé expansion, we obtain bright one- and two-soliton solutions. Figures are plotted to help us understand the dynamics of regular and resonant interactions, and we find that the regular interaction of solitons is completely elastic. Based on the asymptotic and graphical behavior of the two-soliton solutions, we analyze two kinds of resonance between the solitons, both of which are non-completely elastic. A triple structure, a periodic resonant structure in the procedure of interactions and a high wave hump in the vicinity of the crossing point, can be observed. Through the linear stability analysis, instability condition for the soliton solutions can be given, which might be useful, e.g., for the ship traffic on the surface of water.     

Soliton solutions of a BBM(m, n) equation with generalized evolution

We consider a BBM(m, n) equation which is a generalization of the celebrated Benjamin-Bona-Mahony equation with generalized evolution term. By using two solitary wave ansatze in terms of sech^p(x) and tanh^p(x) functions, we find exact analytical bright and dark soliton solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of existence of solitons are presented. Note that, it is always useful and desirable to construct exact analytical solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方 法,即改进的广义射影Ricccati方程方法,求解(2 1)维色散长波方程,得到该方程的新的 更一般形式的行波解,包括扭状孤波解,钟状解,孤子解和周期解．并对部分新形式孤波解画 图示意．     

Soliton Solutions for a Generalized Inhomogeneous Variable-Coefficient Hirota Equation with Symbolic Computation

Under investigation in this paper is a generalized inhomogeneous variable- coefficient Hirota equation. Through the Hirota bilinear method and symbolic computation, the bilinear form and analytic one-, two- and N-soliton solutions for such an equation are obtained, respectively. Properties of those solitons in the inhomogeneous media are discussed analytically. We get the soliton with the property that the larger the amplitude is, the narrower and slower the pulse is. Dynamics of that soliton can be regarded as a repulsion of the soliton by the external potential barrier. During the interaction of two solitons, we observe that the larger the value of the coefficient β in the equation is, the larger the distance of the two solitons is.     

On the shapes of two-dimensional soliton perturbations in simple lattices

The Toda lattice and the discrete Korteweg-de Vries equation generalized to two dimensions are studied numerically. The interactions are assumed to be identical in both directions. It is shown that the equations have solutions in the form of plane linear and localized solitons. In contrast to equations integrable by the inverse scattering method, the parameters of solitons change in the course of their interaction and additional wave structures are formed. The basic types of solutions characterizing these processes are presented.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方法,即改进的广义射影Ricccati方程方法, 求解(2+1)维色散长波方程, 得到该方程的新的更一般形式的行波解, 包括扭状孤波解, 钟状解,孤子解和周期解. 并对部分新形式孤波解画图示意.     

一类一维阵列的孤波特征

该文研究了一类由二自由度可积哈密顿系统构成的一维阵列的行波解,发现在长波极限下,问题可约化为分析哈密顿系统在扰动下的同异宿轨道的情形．当无扰系统具有共振时,利用能量──相方法,得到该系统存在同、异宿到不动点和周期轨的充分条件,在该条件下相应地一维阵列存在一组具有孤波特征的行波,同时给出了一个Ｎ脉冲孤立子波的例子．     

Symmetries for a family of Boussinesq equations with nonlinear dispersion

In this paper, we make a full analysis of a family of Boussinesq equations which include nonlinear dispersion by using the classical Lie method of infinitesimals. We consider travelling wave reductions and we present some explicit solutions: solitons and compactons.For this family, we derive nonclassical and potential symmetries. We prove that the nonclassical method applied to these equations leads to new symmetries, which cannot be obtained by Lie classical method. We write the equations in a conserved form and we obtain a new class of nonlocal symmetries. We also obtain some Type-II hidden symmetries of a Boussinesq equation.     

Different physical structures of solutions for a generalized Boussinesq wave equation

A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.     

Linear Waves and Nonlinear Wave Interactions in a Bounded Three‐Layer Fluid System

We investigate possible linear waves and nonlinear wave interactions in a bounded three‐layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode‐one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode‐two solitary waves. Mathematically, owing to a small loss of energy to dispersive tails during the interaction, the waves are not solitons. However, this energy loss is extremely minute, and because the dispersively coupled waves in the system exhibit the three types of Lax KdV interactions, we conclude that for all intents and purposes the solitary waves exhibit soliton behavior.     

Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation

It had been found that some nonlinear wave equations have the so-called “W/M”-shape-peaks solitons. What is the dynamical behavior of these solutions? To answer this question, all traveling wave solutions in the parameter space are investigated for a integrable water wave equation from a dynamical systems theoretical point of view. Exact explicit parametric representations of all solitary wave solutions are given.