AB-KdV系统的精确$P_s T_d$不变量和$P_s T_d$对称破缺解、对称约化和贝克隆变换
Exact $P_s T_d$ Invariant and $P_s T_d$ Symmetric Breaking solutions, Symmetry Reductions and Bäcklund Transformations For An AB-KdV System
摘要：
在自然科学和社会科学中，发生在不同时空的许多事件可能是密切相关的。如果一个事件是由另一个事件确定的，例如，由合适的$\hat{f}$运算符的$B=\hat{f} A$确定的，则两个事件A（Alice）和B（Bob）被定义为相关的。一个具有移位宇称（$P_s$，parity with a shift）、延迟时间反转（$T_d$，time reversal with a delay）对称性的非定域AB-KdV系统，其中$B=\hat{P_s}\hat{T_d} A$直接由正规KdV方程构造，用以描述两个区域的物理事件.用不同的方法给出了AB-KdV系统的精确解，包括$P_s T_d$不变解和$P_s T_d$对称破缺解. $P_s T_d$不变解表明发生在$A$的事件也将发生在$B$。这些解，如单孤子解、无穷多个奇异孤子解、孤子-椭圆余弦波相互作用解、对称约化解等，表明AB-KdV体系具有丰富的结构。通过对剩余对称性的局部化，提出了一种与剩余对称性有关的特殊Bäcklund变换，从而得到了孤子与其他类型AB-KdV系统的相互作用解.
英文摘要：
In natural and social science, many events happened at different space-times may be closely correlated. Two events, A (Alice) and B (Bob) are defined as correlated if one event is determined by another, say, $B=\hat{f} A$ for suitable $\hat{f}$ operators. A nonlocal AB-KdV system with shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) symmetry where $B=\hat{P_s}\hat{T_d} A$ is constructed directly from the normal KdV equation to describe two-area physical event. The exact solutions of the AB-KdV system, including $P_s T_d$ invariant and $P_s T_d$ symmetric breaking solutions are shown by different methods. The $P_s T_d$ invariant solution show that the event happened at $A$ will happen also at $B$. These solutions, such as single soliton solutions, infinitely many singular soliton solutions, soliton-cnoidal wave interaction solutions, and symmetry reduction solutions etc., show the AB-KdV system possesses rich structures. Also, a special Bäcklund transformation related to residual symmetry is presented via the localization of the residual symmetry to find interaction solutions between the solitons and other types of the AB-KdV system.
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