New Soliton Solutions with Compact Support for a Family of Two-Parameter Regularized Long-Wave Boussinesq Equations

Searching for special solitary wave solutions with compact support is of important significance in soliton theory. In this paper, to understand the role of nonlinear dispersion in pattern formation, a family of the regularized longwave Boussincsq equations with fully nonlinear dispersion (simply called R(m, n) equations), utt + a( un )xx + b(um )xxtt = 0(a, b const.), is studied. New solitary wave solutions with compact support of R(m, n) equations are found. In addition we find another compacton solutions of the two special cases, R(2, 2) equation and R(3, 3) equation. It is found that the nonlinear dispersion term in a nonlinear evolution equation is not a necessary condition of that it possesses compacton solutions.     

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion

In this paper, similarity reductions of Boussinesq-like equations with nonlinear dispersion (simply called B(m,n) equations) u_tt=(uⁿ)_xx+(u^m)_xxxx, which is a generalized model of Boussinesq equation u_tt=(u²)_xx+u_xxxx and modified Bousinesq equation u_tt=(u³)_xx+u_xxxx, are considered by using the direct reduction method. As a result, several new types of similarity reductions are found. Based on the reduction equations and some simple transformations, we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of B(1,n) equations and B(m,m) equations, respectively.     

Abundant Symmetries and Exact Compacton—Like Structures in the Two—Parameter Family of the Estevez—Mansfield—Clarkson Equations

The two-parameter family of Estevez-Mansfield-Clarkson equations with fully nonlinear dispersion (called E(m,n) equations),(uz^m)zzτ γ(uz^nuτ)z uττ=0 which is a generalized model of the integrable Estevez-Mansfield-Clarkson equation uzzzτ γ(uzuzτ uzzuτ) uττ=0,is presented.Five types of symmetries of the E9m,n) equation are obtained by making use of the direct reduction method.Using these obtained reductions and some simple transformations,we obtain the solitary-like wave solutions of E(1,n) equation.In addition,we also find the compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions,they reemerge with the same coherent shape) of E(3,2) equation and E(m,m-1) for its potentials,say,uz,and compacton-like solutions of E(m,m-1) equations,respectively.Whether there exist compacton-like solutions of the other E(m,n) equation with m≠n 1 is still an open problem.     

一类非线性方程的compacton解及其移动compacton解

研究一类非线性方程,即广义Camassa-Holm方程C(n):ut+kux+β1u\{xxt\}+β2u\{n+1\}x+β3uxun\{xx\}+β4uun\{xxx\}=0.通过四种拟设得到丰富的精确解，特别是当k≠0时得到了com pacton解,当k=0时得到了移动compacton解.最后利用线 性化的方法得到了其他形式的广义Camassa-Holm方程的compacton解. 关键词： 广义Camassa-Holm方程 compacton解 移动compacton解     

New Symmetry Reductions,Dromions—Like and Compacton Solutions for a 2D BS（m,n） Equations Hierarchy with Fully Nonlinear Dispersion

We have found two types of important exact solutions,compacton solutions,which are solitary waves with the property that after colliding with their own kind,they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction,in the (1 1)D,(1 2)D and even (1 3)D models,and dromion solutions (exponentially decaying solutions in all direction) in many (1 2)D and (1 3)D models.In this paper,symmetry reductions in (1 2)D are considered for the break soliton-type equation with fully nonlinear dispersion (called BS(m,n) equation)ut b(u^m)xxy 4b(u^n δx^-1uy)x=0,which is a generalized model of (1 2)D break soliton equation ut buxxy 4buuy 4buxδx^-1uy=0,by using the extended direct reduction method.As a result,six types of symmetry reductions are obtained.Starting from the reduction equations and some simple transformations,we obtain the solitary wavke solutions of BS(1,n) equations,compacton solutions of BS(m,m-1) equations and the compacton-like solution of the potential form (called PBS(3,2)) ωxt b(ux^m)xxy 4b(ωx^nωy)x=0.In addition,we show that the variable ∫^x uy dx admits dromion solutions rather than the field u itself in BS(1,n) equation.     

New Compacton Solutions and Solitary Pattern Solutions for Modified Nonlinearly Dispersive inK（m,n, a,b） Equation

In this paper exact solutions of a new modified nonlinearly dispersive equation （simply called inK（m, n, a, b） Ua Ub equation）, u^m-1 ut ＋ α（ u^n）x ＋β（u^a（u^b）xx）x = 0, is investigated by using some direct algorithms. As a result, abundant new compacton solutions （solitons with the absence of infinite wings） and solitary pattern solutions （having infinite slopes or cusps） are obtained.     

Exact traveling wave solutions to nonlinear diffusion and wave equations

By the introduction of some ansatz equations, we have obtained several new classes of traveling (solitary) wave solutions to the nonlinear diffusion equation $$f_1 (u)u_t + f_2 (u)u_x + f_3 (u)u_{xx} + f_4 (u)u_x^2 = f_5 (u)$$ and the nonlinear wave equation $$f_1 (u)u_u + f_2 (u)u_t + f_3 (u)u_{xx} + f_4 (u)u_x + f_5 (u)u_x^2 + \cdots = f_6 (u)$$ Some applications of these solutions are discussed.     

Soliton-type solutions for two models in mathematical physics

In this paper, the generalised Klein-Gordon and Kadomtsov–Petviashvili Benjamin–Bona–Mahony equations with power law nonlinearity are investigated. Our study is based on reducing the form of both equations to a first-order ordinary differential equation having the travelling wave solutions. Subsequently, soliton-type solutions such as compacton and solitary pattern solutions are obtained analytically. Additionally, the peaked soliton has been derived where it exists under a specific restrictions. In addition to the soliton solutions, the mathematical method which is exploited in this work also creates a few amount of travelling wave solutions.     

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.     

New Compacton Solutions of Nonlinearly Dispersive R(m, n) Equations

In this paper, we establish exact solutions for the R(m,n) equations by using an sn-cn method. As a result, abundant new compactons, i.e. solitons with the absence of infinite wings, new type of Jacobi elliptic function, solitary wave and periodic wave solutions, of this equation are obtained with minimal calculations. The properties of the R(m,n) equations are shown in figures.     

Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

Dynamical system theory is applied to the integrable nonlinear wave equation $u_t±(u^3−u^2)x+(u^3)xxx=0$. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation corresponds to the case of wave speed $c$=0. In the case of $c^6$≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.     

一类广义Boussinesq方程和Boussinesq-Burgers方程新的显式精确解

利用一种推广的代数方法，求解了一类广义Boussinesq方程(B(m，n)方程)和Boussinesq-Burgers方程(B-B方程).获得了其多种形式的显式精确解，包括孤波解、三角函数解、有理函数解、Jacobi椭圆函数周期解和Weierstrass椭圆函数周期解，进一步丰富了这两类方程的解. 关键词： Boussinesq方程 Boussinesq-Burgers方程 推广的代数方法 显式精确解     

Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarie?s fractional derivative

In this Letter, the fractional variational iteration method using He?s polynomials is implemented to construct compacton solutions and solitary pattern solutions of nonlinear time-fractional dispersive KdV-type equations involving Jumarie?s modified Riemann-Liouville derivative. The method yields solutions in the forms of convergent series with easily calculable terms. The obtained results show that the considered method is quite effective, promising and convenient for solving fractional nonlinear dispersive equations. It is found that the time-fractional parameter significantly changes the soliton amplitude of the solitary waves.     

The projective Riccati equation expansion method and variable separation solutions for the nonlinear physical differential equation in physics

Using the projective Riccati equation expansion (PREE) method, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for two nonlinear physical models are obtained. Based on one of the variable separation solutions and by choosing appropriate functions, new types of interactions between the multi-valued and single-valued solitons, such as a peakon-like semi-foldon and a peakon, a compacton-like semi-foldon and a compacton, are investigated.     

Compactons versus solitons

We investigate whether the recently proposed PT-symmetric extensions of generalized Korteweg-de Vries equations admit genuine soliton solutions besides compacton solitary waves. For models which admit stable compactons having a width which is independent of their amplitude and those which possess unstable compacton solutions the Painlevé test fails, such that no soliton solutions can be found. The Painlevé test is passed for models allowing for compacton solutions whose width is determined by their amplitude. Consequently, these models admit soliton solutions in addition to compactons and are integrable.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation

Certain generalizations of one of the classical Boussinesq-type equations, $$u_{tt} = u_{xx} - (u^2 + u_{xx} )_{xx} $$ are considered. It is shown that the initial-value problem for this type of equation is always locally well posed. It is also determined that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds. These two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations. This contrasts with the results of blow-up obtained recently by Kalantarov and Ladyzhenskaya for the same type of equation, and casts additional light upon the results for the original version (*) of this class of equations obtained via inverse-scattering theory by Deift, Tomei and Trubowitz.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions are performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

Compact and non compact structures of the phi-four equation

In this paper, we establish exact special solutions of the phi-four equation. We use the reduction of order method and the He’s variational principle for this equation. So, we get general formulae for the compacton, solitary pattern, soliton, and periodic solutions.     

Bosonized Supersymmetric Sawada–Kotera Equations: Symmetries and Exact Solutions

The Bosonized Supersymmetric Sawada–Kotera(BSSK) system is constructed by applying bosonization method to a Supersymmetric Sawada–Kotera system in this paper. The symmetries on the BSSK equations are researched and the calculation shows that the BSSK equations are invariant under the scaling transformations, the space-time translations and Galilean boosts. The one-parameter invariant subgroups and the corresponding invariant solutions are researched for the BSSK equations. Four types of reduction equations and similarity solutions are proposed. Period Cnoidal wave solutions, dark solitary wave solutions and bright solitary wave solutions of the BSSK equations are demonstrated and some evolution curves of the exact solutions are figured out.