Exact PsTd invariant and PsTd symmetric breaking solutions, symmetry reductions and Bäcklund transformations for an AB-KdV system

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=fˆA for suitable fˆ operators. A nonlocal AB-KdV system with shifted-parity (Ps, parity with a shift), delayed time reversal (Td, time reversal with a delay) symmetry where B=PsˆTdˆA is constructed directly from the normal KdV equation to describe two-area physical event. The exact solutions of the AB-KdV system, including PsTd invariant and PsTd symmetric breaking solutions are shown by different methods. The PsTd 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.