利用对称性研究 Gardner 方程的完全可积性、孤立波解与孤立子
Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation
AI总结：
本文利用尺度对称性方法计算多项式非线性偏微分方程的守恒律与 Lax 对等以证明其完全可积性，并以 Gardner 方程为例推导其精确孤立波解，揭示了立方非线性符号对解性质的影响。
摘要：
利用尺度对称性，展示了如何计算多项式非线性偏微分方程的多项式守恒律、广义对称性、递归算子、Lax 对和双线性形式，从而确立其完全可积性。选择 Gardner 方程作为关键示例，因为它同时包含 Korteweg-de Vries 和 modified Korteweg-de Vries (mKdV) 方程。连接这些方程的 Gardner 和 Miura 变换也通过尺度齐性概念进行计算。利用 Hirota 方法和其它直接方法推导了 Gardner 方程的精确孤立波解和孤立子。这些解的性质取决于 Gardner 方程中立方项的符号以及底层的 mKdV 方程。结果表明，大振幅的平坦（平顶）波仅在立方非线性符号为负（散焦情形）时出现，而聚焦 Gardner 方程具有标准的弹性碰撞孤立子。本文旨在综述 Gardner 方程的可积性性质和解，并说明尺度对称性方法的适用性。本文使用的方法和算法已在 Mathematica 中实现，但可适配于主要的计算机代数系统。
英文摘要：
Using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations thereby establishing their complete integrability. The Gardner equation is chosen as the key example for it comprises both the Korteweg-de Vries and modified Korteweg-de Vries (mKdV) equations. The Gardner and Miura transformations which connect these equations are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota's method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case) whereas the focusing Gardner equation has the standard elastically colliding solitons. The paper's aim is to provide a review of integrability properties and solutions of the Gardner equation and illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica but can be adapted for major computer algebra systems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
