变系数mKdV方程的椭圆函数周期解

Solutions of mKdV Equation with Variable Coefficients in Jacobi Elliptic and Periodic Functions

通过引入一个波变换,将变系数mKdV方程约化为常微分方程。假设方程的系数满足特定的约束条件,借助符号计算软件Mathematica和扩展的F-展开函数法,在拟设法、齐次平衡原理和Jacobi椭圆函数展开法的基础上,求得了精确解的浓缩公式。利用第一类椭圆方程中P,Q,R的不同取值与相应的F(ξ)值之间的关系,从解的浓缩公式中,得到了丰富的显式精确解,特别是以两个不同的Jacobi椭圆函数表示的精确解。在极限的情况下,即当模m→1或m→0时,这些解退化为相应的类孤立波解和三角函数表示的精确解。该方法具有直接、简洁的特点,可以用来求解更多的在数学物理、自然科学和应用科学等领域出现的非线性偏微分方程的精确解。

The mKdV equation with variable coefficients is reduced to an ordinary differential equation through a traveling wave transformation. With the aid of the symbolic computation software Mathematica as well as the extended F-expansion method recently proposed on the basis of the analogical method, the homogeneous balance principle and Jacobian elliptic function method, the concentrated formulas of exact solutions are derived if the coefficients of the mKdV equation satisfy some specific constraint conditions. By using the relations between values of P, Q, R and corresponding solutions F（s~） for the first kind of elliptic equation, from the concentrated formulae of solutions, a large number of explicit exact solutions, especially, the solutions expressed in two different Jacobian elliptic functions are obtained. In the limit cases, that is, when the module approaches 1 or 0, these explicit exact solutions degenerate into the soliton-like solutions and the exact solutions in the form of trigonometric functions, respectively. It is worthwhile to mention that the method used here is straightforward, concise and powerful and can be used for solving many other similar nonlinear partial differential equations which would appear in the fields of mathematical physics, natural sciences and applied sciences.