与一个非局域对称有关的Kaup-Kupershmidt方程新的精确解

利用Kaup-Kupershmidt（KK）方程的一个非局域对称，可在两种不同的方法上找到方程新的精确解．首先，用标准的展开近似得到KK方程有限的Lie-B?cklund变换和单孤子解．其次，把一些局域对称与这个非局域对称组合起来，给出其群不变解，进而可求得新的孤子解 关键词：     

(3+1)维Jimbo-Miwa方程的精确解及局域激发

在符号计算软件 Maple 的帮助下,利用投射方程法和变量分离法,得到了(3+1)维 Jimbo-Miwa(JM)方程的新显式精确解. 根据得到的孤立波解,研究了 JM 方程新颖的局域激发. 关键词： 投射方程法 Jimbo-Miwa方程 精确解 局域激发     

广义Zakharov-Kuznetsov方程的对称及精确解

利用改进的直接方法给出了一类广义Zakharov-Kuznetsov方程ut auux bu2ux cuxxx duxyy=0新显式解与旧显式解之间的关系,并且得到了该方程的对称.这些对称推广了已有文献中应用Steinberg s相似方法获得的结果.利用广义Zakharov-Kuznetsov方程新旧显式解之间的关系,本文在已有显式解的基础上给出了方程新的显式解.这些解对于研究某些复杂的物理现象,以及验证数值求解法则的可行性有重要的意义.     

Boussinesq-Burgers方程新的精确解

基于改进的投影Riccati方程的解,提出一种新的构造非线性演化方程精确解的方法.通过这种方法,我们得导到了Boussinesq-Burgers方程各种类型的精确解,包括Jacobi和Weierstrass周期函数解.这种方法与数学软件Maple结合,简单易行,有助于探索其他非线性演化方程的精确解.     

非线性Klein-Gordon方程新的精确解

将行波变换替换为更一般的函数变换,推广了修正的Jacobi椭圆函数展开方法.给出了非线性 Klein-Gordon方程新的周期解.当模m→1或m→0时,这些解退化成相应的孤立波解、三 角函数解和奇异的行波解.对于某些非线性方程,在一定条件下一般变换退化为行波约化. 关键词： Jacobi椭圆函数 非线性发展方程 精确解     

Broer-Kau-Kupershmidt方程组的对称、约化和精确解

利用修正的CK直接方法得到了Broer-Kau-Kupershmidt (简写为BKK)方程组的对称、约化, 通过解约化方程得到了该方程组的一些精确解, 包括双曲函数解、 三角函数解、 有理函数解、 艾里函数解、 幂级数解和 孤立子解等. 关键词： 修正的CK直接方法 BKK方程组 对称、约化 精确解     

修正的BBM方程新的精确解

应用进一步修正的简单方程法对修正的 Benjamin -Bona -Mahoney (mBBM )方程进行求解,给出了mBBM方程新的精确类孤波解,取定某些参数值,便可得到精确孤波解.这种方法也可用于寻找其它常系数以及变系数非线性发展方程(组)的精确解,具有一定的普适性.     

非球谐振子势Schroedinger方程的精确解

将非球谐振子势V(r)=ar^2 br^4 cr^6径向波函数展开为指数函数与多项式函数的乘积，应用多项式函数的系数关系确定了体系的能级和波函数．结果表明，体系处于束缚态时，势参数a，b，c必须满足一定的约束条件．     

Liouville方程的八类精确解

利用Poisson括号的正则不变性求得了Liouville方程的八类精确解:1)“重力”势系统,2)谐振系统,3)正负平方幂函数势系统,4)双曲函数势系统,5)三角函数势系统,6)PschlTeller势系统,7)“引(斥)力”势系统和8)Kratzer势系统.得到了“化动量正则变换”的一般方法.在求解后两个系统Liouville方程的过程中还应用了Routh方法和Binet方法. 关键词：     

Nonlocal symmetry,optimal systems,and explicit solutions of the mKdV equation

The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.     

Nonlocal symmetry constraints and exact interaction solutions of the (2+1) dimensional modified generalized long dispersive wave equation

In this paper, nonlocal symmetry of the (2+1) dimensional modified generalized long dispersive wave system and its applications are investigated. The nonlocal symmetry related to the eigenfunctions in Lax pairs is derived, and infinitely many nonlocal symmetries are obtained. By introducing three potentials, the prolongation is found to localize the given nonlocal symmetry. Various finite-and infinite-dimensional integrable models are constructed by using the nonlocal symmetry constraint method. Moreover, applying the general Lie symmetry approach to the enlarged system, the finite symmetry transformation and similarity reductions are computed to give novel exact interaction solutions. In particular, the explicit soliton-cnoidal wave solution is obtained for the modified generalized long dispersive wave system, and it can be reduced to the two-dark-soliton solution in one special case.     

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N=2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.     

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

Localization of nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

The nonlocal symmetry of the Sawada–Kotera(SK) equation is constructed with the known Lax pair. By introducing suitable and simple auxiliary variables, the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further. On the other hand, the group invariant solutions of the SK equation are constructed with the classic Lie group method.In particular, by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.     

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.     

Residual Symmetry Reduction and Consistent Riccati Expansion of the Generalized Kaup-Kupershmidt Equation

The residual symmetry of the generalized Kaup-Kupershmidt(gKK) equation is obtained from the truncated Painlevé expansion and localized to a Lie point symmetry in a prolonged system. New symmetry reduction solutions of the prolonged system are given by using the standard Lie symmetry method. Furthermore, the g KK equation is proved to integrable in the sense of owning consistent Riccati expansion and some new B¨acklund transformations are given based on this property, from which interaction solutions between soliton and periodic waves are given.