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.     

The direct method for multisolitons and two-hump solitons in the Hirota-Satsuma system

The coupled Korteweg-de Vries (cKdV) system, originally proposed by Hirota and Satsuma [1], modeling the interaction of two long waves evolving with different dispersion relations, is considered. Multisoliton solutions are explicitly found in the Hirota-Satsuma system by means of an extension of the direct method due to Hirota [2]. Such multisolitons are found to be a complete family of interacting classical solitons consisting of the trivial ones corresponding to the decoupled N-soliton solution of the KdV equation and the ones coupled to the M-cKdV solitons, here simply referred to as the classical (N, M)-soliton solutions which are N+M interacting solitons indeed. Explicit analytical solutions, expressed in terms of hyperbolic functions, are found for the (0, 1), (1, 1) and (0, 2) cases while the process to generate the general (N, M)-soliton solutions is systematically described in detail. In the particular scenario of the (1, 1)-soliton, a novel two-hump one-soliton solution is obtained when a negative prestressing is prescribed for one of the waves in the merging of the 1-KdV and 1-cKdV solitons. Special attention is emphasized to the phase shift for the (1, 1) and (0, 2) interaction cases to asymptotically get a particular remarkable long range interaction phenomenon. Finally, some numerical evolutions are also considered to illustrate the interaction dynamics of the multisolitons and the new two-hump soliton in the different situations considered.     

Invariant interaction solutions for a supersymmetric mKdV equation

The bosonic supersymmetric modified KdV (BSmKdV) system is obtained by the bosonization approach. The nonlocal symmetry for the BSmKdV equation is obtained by the truncated Painlevé method. By introducing multiple new fields, the finite symmetry transformation for the BSmKdV equation is derived by applying Lie’s first principle to the prolonged systems. The similarity reductions related to the nonlocal symmetry are studied. The interaction solutions among the solitons and other complicated waves, including Painlevé II waves and periodic cnoidal waves, are presented through the reduction theorems. The concrete soliton-cnoidal interaction solutions are illustrated in detail by using the mapping and deformation method.     

Soliton solutions for a generalized fifth-order KdV equation with t-dependent coefficients

We consider a generalized fifth-order KdV equation with time-dependent coefficients exhibiting higher-degree nonlinear terms. This nonlinear evolution equation describes the interaction between a water wave and a floating ice cover and gravity-capillary waves. By means of the subsidiary ordinary differential equation method, some new exact soliton solutions are derived. Among these solutions, we can find the well known bright and dark solitons with sech and tanh function shapes, and other soliton-like solutions. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous KdV system supporting high-order dispersive and nonlinear effects.     

单离子各向异性影响下的一维铁磁链中的孤子

利用行波解法，研究了考虑交换各向异性和单离子各向异性的一维铁磁链，得出椭圆函数波解和孤子解,并讨论了单离子各向异性对椭圆函数波和孤子的影响.研究表明：单离子各向异性对一维铁磁链中椭圆函数波和孤子的激发及其稳定性有显著影响.单离子各向异性不利于易轴铁磁链中椭圆函数波和孤子解的激发，但有利于它们的稳定；而在易平面铁磁链中，单离子各向异性使椭圆函数波和孤子激发变得容易，但不利于它们的稳定.此外，还讨论了单离子各向异性对一维各向同性铁磁链中的孤子激发具有的影响. 关键词： 各向异性 单离子各向异性 孤子 铁磁链     

Excitation of low-amplitude solitons in systems with decay instability

A nonlinear equation for the interaction of a pair of opposite-traveling waves with a normally incident decay mode has been derived. It is shown that solutions of the equation are low-amplitude solitons of a new type.     

New Exact Jacobi Elliptic Function Solutions of Three—Dimensional Nonlinear Helmholtz Equation in a Nonlinear Kerr—Type Medium

In this letter the three-dimensional nonlinear Helmholtz equation is investigated.which describes electromagnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic function solutions are obtained,by using our extended Jacobian elliptic function expansion method.When the modulus m-→1 or 0,the corresponding solitary waves including bright solitons,dark solitons and new line solitons and singly periodic solutions can be also found.     

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and(2+1)-dimensional breaking soliton equations via a generalized exponential rational function(GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons,double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons,elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.     

Multiple-mode wave solutions in Vakhnenko equation

A new type of wave solutions, called as multiple-mode waves, which can be expressed in the superposition forms of more than two types of single-mode waves of Vakhnenko equation have been investigated in this Letter. A new general method for obtaining the multiple-mode waves is proposed, based on which four cases of the possible forms of wave solutions with two-mode have been derived. The explicit expressions of the two-mode waves as well as the existence conditions have been presented, which may be the nonlinear combinations between periodic waves, solitons, compactons, etc., with different wave speeds, respectively. It is pointed out that more complicated multiple-mode waves with more than three single-mode waves can be derived accordingly, which can be used to reveal the evolution of interactions between different types of waves, especially between various solitons.     

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the combination of the third- and fifth-order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are close to solitons of the modified Korteweg-de Vries equation. However, the height of large-amplitude solutions has a limit approaching which solitary waves widen and acquire a table like shape similar to soluitons of the Gardner equation. Numerical calculations confirm that the collision of solitons of the derived equation is inelastic. Inelasticity is the most pronounced in the interaction of unipolar pulses. The direction of the shift of the phase of the higher-amplitude soliton owing to the interaction of solitons of different polarities depends on the amplitudes of the pulses.     

Bäcklund transformation,infinitely-many conservation laws,solitary and periodic waves of an extended (3 + 1)-dimensional Jimbo–Miwa equation with time-dependent coefficients

In this paper, an extended (3+1)-dimensional Jimbo–Miwa equation with time-dependent coefficients is investigated, which comes from the second member of the Kadomtsev–Petviashvili hierarchy and is shown to be conditionally integrable. Bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived via the binary Bell polynomials and symbolic computation. With the help of the bilinear form, one-, two- and three-soliton solutions are obtained via the Hirota method, one-periodic wave solutions are constructed via the Riemann theta function. Additionally, propagation and interaction of the solitons are investigated analytically and graphically, from which we find that the interaction between the solitons is elastic and the time-dependent coefficients can affect the soliton velocities, but the soliton amplitudes remain unchanged. One-periodic waves approach the one-solitary waves with the amplitudes vanishing and can be viewed as a superposition of the overlapping solitary waves, placed one period apart.     

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.     

Interaction Solutions for (1+1)-Dimensional Higher-Order Broer-Kaup System

The (1+1)-dimensional higher-order Broer-Kaup (HBK) system is studied by consistent tanh expansion (CTE) method in this paper. It is proved that the HBK system is CTE solvable, and some exact interaction solutions among different nonlinear excitations such as solitons, rational waves, periodic waves, corresponding images are explicitly given.     

Interaction Behaviours Between Solitons and Cnoidal Periodic Waves for (2+1)-Dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

The consistent tanh expansion (CTE) method is employed to the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explicitly obtained. Concretely, we discuss a special kind of interaction solution in the form of tanh functions and Jacobian elliptic functions in both analytical and graphical ways. The results show that the profiles of the soliton-cnoidal periodic wave interaction solutions can be designed by choosing different values of wave parameters.     

Solitons and periodic waves for a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics

Under investigation in this paper is a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics. Soliton and one-periodic-wave solutions are obtained via the Hirota bilinear method and Hirota–Riemann method. Magnitude and velocity of the one soliton are derived. Graphs are presented to discuss the solitons and one-periodic waves: the coefficients in the equation can determine the velocity components of the one soliton, but cannot alter the soliton magnitude; the interaction between the two solitons is elastic; the coefficients in the equation can influence the periods and velocities of the periodic waves. Relation between the one-soliton solution and one-periodic wave solution is investigated.     

Nonlocalization of Nonlocal Symmetry and Symmetry Reductions of the Burgers Equation

Symmetry reduction method is one of the best ways to find exact solutions. In this paper, we study the possibility of symmetry reductions of the well known Burgers equation including the nonlocal symmetry. The related new group invariant solutions are obtained. Especially, the interactions among solitons, Airy waves, and Kummer waves are explicitly given.     

Nonlocalization of Nonlocal Symmetry and Symmetry Reductions of the Burgers Equation

Symmetry reduction method is one of the best ways to find exact solutions. In this paper, we study the possibility of symmetry reductions of the well known Burgers equation including the nonlocal symmetry. The related new group invariant solutions are obtained. Especially, the interactions among solitons, Airy waves, and Kummer waves are explicitly given.     

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.     

Temporal solitary waves due to optical rectification and the electro-optic effect

Temporal solitary waves that are due to the mutual interaction of optical rectification and the electro-optic effect in the presence of a second-order nonlinearity are studied. It is found that a two-parameter family of solitons exists. Analytical solutions are found for special cases. Numerical soliton solutions of the system of equations include single- and multiple-hump solitary waves.