Alice & Bob

S. Lou, Fei-Long Huang

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 correlated if one event is determined by another, say, B=fˆA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{B}}=\hat{{\boldsymbol{f}}}A$$\end{document} for suitable fˆ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{{\boldsymbol{f}}}$$\end{document} operators. Taking KdV and coupled KdV systems as examples, we can find some types of models (AB-KdV systems) to exhibit the existence on the correlated solutions linked with two events. The idea of this report is valid not only for physical problems related to KdV systems but also for problems described by arbitrary continuous or discrete models. The parity and time reversal symmetries are extended to shifted parity and delayed time reversal. The new symmetries are found to be useful not only to establish AB-systems but also to find group invariant solutions of numerous AB-systems. A new elegant form of the N-soliton solutions of the KdV equation and then the AB-KdV systems is obtained. A concrete AB-KdV system derived from the nonlinear inviscid dissipative and barotropic vorticity equation in a β-plane channel is applied to the two correlated monople blocking events which is responsible for the snow disaster in the winter of 2007/2008 happened in Southern China.

Alice Bob, Alice David, Bob Greg, E. Fred

S. Lou

To describe two-place physical problems, many possible models named Alice-Bob (AB) systems are proposed. To find and to solve these systems, the Parity (P), time reversal (T), charge conjugation (C), shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) and their possible combinations such as PT, PC, $P_sC$, $P_sT_d$ and $P_sT_dC$ etc. can be successively used. Especially, some special types of $P_s$-$T_d$-$C$ group invariant multi-soliton solutions for the KdV-KP-Toda type, mKdV-sG type, NLS type and discrete $H_1$ type AB systems are explicitly constructed.

Congcong Li, S. Lou, M. Jia

To describe two-place events, Alice–Bob systems have been established by means of the shifted parity and delayed time reversal in the preprint arXiv:1603.03975v2 [nlin.SI], (2016). In this paper, we mainly study exact solutions of the integrable Alice–Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation is constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton, and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.

S. Lou 楼, Z. Qiao 乔

We study the Alice–Bob peakon system generated from an integrable peakon system using the strategy of the so-called Alice–Bob non-local KdV approach [Scientific Reports 7 (2017) 869]. Nonlocal integrable peakon equations are obtained and shown to have peakon solutions.