Exact $P_s T_d$ Invariant and $P_s T_d$ Symmetric Breaking solutions, Symmetry Reductions and Bäcklund Transformations For An AB-KdV System

M Jia; S Y Lou

doi:10.1016/j.physleta.2018.02.036

paper

Exact $P_{s} T_{d}$ Invariant and $P_{s} T_{d}$ Symmetric Breaking solutions, Symmetry Reductions and Bäcklund Transformations For An AB-KdV System

arXiv:1612.00546 · doi:10.1016/j.physleta.2018.02.036

Abstract

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, $B = f^A$ for suitable $f^$ 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 $B = P_{s}^T_{d}^A$ 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.

20 pages

Exact $P_{s} T_{d}$ Invariant and $P_{s} T_{d}$ Symmetric Breaking solutions, Symmetry Reductions and Bäcklund Transformations For An AB-KdV System

Abstract

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Exact $P_{s} T_{d}$ Invariant and $P_{s} T_{d}$ Symmetric Breaking solutions, Symmetry Reductions and Bäcklund Transformations For An AB-KdV System