计算非线性偏微分方程递归算子的符号算法
A Symbolic Algorithm for Computing Recursion Operators of Nonlinear PDEs
摘要：
递归算子是将非线性偏微分方程的广义对称映射到新的对称的积分微分算子。因此，递归算子的存在保证了偏微分方程有无穷多个高阶对称，这是完全可积性的一个关键特征。完全可积的非线性偏微分方程具有双哈密顿结构和拉克斯对，它们也可以用逆散射变换求解，并允许任何阶的孤子解。给出了（1+1）维非线性偏微分方程多项式递归算子的一种直接符号计算方法。基于守恒密度和广义对称性，由待定系数的标度不变项的线性组合构造了一个候选递归算子。将候选递归算子代入其定义方程，求解待定系数的线性系统。该方法是算法，并在Mathematica中实现。生成的符号包PDERecursionOperator.m可用于测试多项式PDE的完全可积性，这些PDE可写成非线性演化方程。利用PDERecursionOperator.m，得到了数学物理和孤子理论中几个著名的非线性偏微分方程的递推算子.
英文摘要：
A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear PDE to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order. A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved. The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.
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