Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation

Nonlocal symmetries related to the Bäcklund transformation (BT) for the modified KdV-sine-Gordon (mKdV-SG) equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations. Then through the parameter expansion procedure, an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived. Finally, several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.     

Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation

In nonlinear physics,it is very difficult to study interactions among different types of nonlinear waves.In this paper,the nonlocal symmetry related to the truncated Painleve′expansion of the(2+1)-dimensional Burgers equation is localized after introducing multiple new variables to extend the original equation into a new system.Then the corresponding group invariant solutions are found,from which interaction solutions among different types of nonlinear waves can be found.Furthermore,the Burgers equation is also studied by using the generalized tanh expansion method and a new Ba¨cklund transformation(BT)is obtained.From this BT,novel interactive solutions among different nonlinear excitations are found.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

1-Soliton solution of the generalized Zakharov-Kuznetsov equation with nonlinear dispersion and time-dependent coefficients

In this Letter, the 1-soliton solution of the Zakharov-Kuznetsov equation with power law nonlinearity and nonlinear dispersion along with time-dependent coefficients is obtained. There are two models for this kind of an equation that are studied. The constraint relation between these time-dependent coefficients is established for the solitons to exist. Subsequently, this equation is again analysed with generalized evolution. The solitary wave ansatz is used to carry out this investigation.     

A DIRECT PROCEDURE ON THE INTEGRABILITY OF NONISOSPECTRAL AND VARIABLE-COEFFICIENT MKdV EQUATION

An elementary and systematic method based on binary Bell polynomials is applied to nonisospectral and variable-coefficient MKdV (vcMKdV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite local conservation laws are obtained step by step, without too much clever guesswork.     

A second order numerical scheme for the improved Boussinesq equation

A finite-difference scheme arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation is used for the numerical solution of the improved Boussinesq equation (IBq). The resulting linear scheme, which is analyzed for local truncation error and stability, is tested numerically and conclusions with corresponding results known in the bibliography are derived.     

Bäcklund transformations,nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the $n$th Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type $N$-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.     

A simple-looking relative of the Novikov,Hirota-Satsuma and Sawada-Kotera equations

We study the simple-looking scalar integrable equation f_xxt 3( f_x f_t 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).     

First variation formula and conservation laws in several independent discrete variables

This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of R^p and for the discrete theory of partial differential operators that are common in the Calculus of Variations. A central result is the construction of a unique decomposition of certain partial difference operators into two components, one that is a vector bundle morphism and other one that leads to boundary terms. Application of this result to the differential of the discrete Lagrangian leads to unique discrete Euler and momentum forms not depending either on the choice of reference on the base lattice or on the choice of coordinates on the configuration manifold, and satisfying the corresponding discrete first variation formula. This formula leads to discrete Euler equations for critical points and to exact discrete conservation laws for infinitesimal symmetries of the Lagrangian density, with a clear physical interpretation.     

On the nonlocal symmetries of certain nonlinear oscillators and their general solution

In this Letter we establish the integrability of two nonlinear oscillators through group theoretical method. We utilize the algorithm given in [M.L. Gandarias, M.S. Bruzon, J. Nonlinear Math. Phys. 18 (2011) 123] and construct nonlocal symmetries for these two oscillators. From the knowledge of the latter we derive first integral and general solution for these two nonlinear nonpolynomial oscillator equations.     

Baicklund transformations for the Burgers equation via localization of residual symmetries

We obtain the non-local residual symmetry related to truncated Painlev~ expansion of Burgers equation. In order to localize the residual symmetry, we introduce new variables to prolong the original Burgers equation into a new system. By using Lie＇s first theorem, we obtain the finite transformation for the localized residual symmetry. More importantly, we also Iocalize the linear superposition of multiple residual symmetries to find the corresponding finite transformations. It is interesting to find that the n-th B~icklund transformation for Burgers equation can be expressed by determinants in a compact way.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

Conservation laws of the generalized short pulse equation

We show that the generalized short pulse equation is nonlinearly self-adjoint with differential substitution.Moreover,any adjoint symmetry is a differential substitution of nonlinear self-adjointness,and vice versa.Consequently,the general conservation law formula is constructed and new conservation laws for some special cases are found.     

An integrable coupled Alice–Bob modified Korteweg de-Vries system: Lax pairs,Bäcklund transformations,residual symmetries and exact solutions

ABSTRACT

A coupled Alice–Bob modified Korteweg de-Vries (mKdV) system is established from the mKdV equation in this paper, which is nonlocal and suitable to model two-place entangled events. The Lax integrability of the coupled Alice–Bob mKdV system is proved by demonstrating three types of Lax pairs. By means of the truncated Painlevé expansion, auto-Bäcklund transformation of the coupled Alice–Bob mKdV system and Bäcklund transformation between the coupled Alice–Bob mKdV system and the Schwarzian mKdV equation are demonstrated. Nonlocal residual symmetries of the coupled Alice–Bob mKdV system are researched. To obtain localized Lie point symmetries of residual symmetries, the coupled Alice–Bob mKdV system is extended to a system consisting six equations. Calculation on the prolonged system shows that it is invariant under the scaling transformations, space-time translations, phase translations and Galilean translations. One-parameter group transformation and one-parameter subgroup invariant solutions are obtained. The consistent Riccati expansion (CRE) solvability of the coupled Alice–Bob mKdV system is proved and some interaction structures between soliton–cnoidal waves are obtained by CRE. Moreover, Jacobi periodic wave solutions, solitary wave solutions and singular solutions are obtained by elliptic function expansion and exponential function expansion.     

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain B?cklund transformation associated with truncated Painlevéexpansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal B?cklund transformations of the derived conservation laws for the first time.     

Adiabatic perturbations for compactons under dissipation and numerically-induced dissipation

Compacton propagation under dissipation shows amplitude damping and the generation of tails. The numerical simulation of compactons by means of dissipative schemes also show the same behaviors. The truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis. For dissipative schemes, or when artificial dissipation is added, the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically-induced tails. In this paper, such methods are applied to the K(2,2)$K(2\text{,}2)$ Rosenau–Hyman equation, showing a very good agreement between perturbative and numerical results.     

Kaup-Kupershmidt方程的非局域对称及新的精确解

利用机械化算法得到了Kaup-Kupershmidt方程的非局域对称、约化,通过解约化方程得到了该方程的一些新的精确解.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.