广义拟线性双曲型方程的对称势及其守恒定律

基于Lie群方法,研究广义拟线性双曲型方程的对称势和不变解.为了得到显式的不变解,关注物理上有趣的有对称势的情况.然后,利用局部的Lagrange函数逼近,在3种物理上引起注意的情况下,得到该方程的守恒定律.     

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u _t =(g(x)u _x ) _x +h(x)u ^m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).     

耦合KdV方程组的对称,精确解和守恒律

通过利用修正的CK直接方法建立了耦合KdV方程组的对称群理论.利用对称群理论和耦合KdV方程组的旧解得到了它们的新的精确解.基于上述理论和耦合KdV方程组的共轭方程组的理论,得到了耦合KdV方程组的守恒律.