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, $${\boldsymbol{B}}=\hat{{\boldsymbol{f}}}A$$B = f ˆ A for suitable $$\hat{{\boldsymbol{f}}}$$f ˆ 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.

Soliton molecules can be formed in some possible mechanisms both theoretically and experimentally. In this paper, we introduce a new mechanism, namely the velocity resonant, to find soliton molecules. Under the velocity resonance mechanism, two solitons can form a kink-antikink molecule, an asymmetric soliton, a two-peak soliton and/or a far away bounded molecule depended on the selections of the wave numbers and the distance between two solitons of a molecule. The results are exhibited via three well known fifth order integrable systems which serve as a general fluid model, as well as models many other physical fields.

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 ($\widehat{P}$ ), time reversal ($\widehat{T}$ ), charge conjugation (Ĉ), and their possible combinations such as $\widehat{P}\widehat{T}$ , $\widehat{P}Ĉ$ , and $\widehat{P}\widehat{T}Ĉ$ , etc., can be successively applied. Especially, some special types of $\widehat{P}-\widehat{T}-Ĉ$ group invariant multi-soliton solutions for the KdV-KP-Toda type, mKdV-sG type, and nonlinear Schrödinger equations (NLS) type AB systems are explicitly constructed. The possible $\widehat{P}\widehat{T}$ symmetry breaking solutions of two special ABKdV systems are explicitly given. Applying the $\widehat{P}-\widehat{T}-Ĉ$ symmetries to coupled Ablowitz-Kaup-Newell-Segur systems, some four-place nonlocal NLS systems are also derived.

The nonlinear Schrödinger (NLS) equation is widely used in natural science. Various nonlinear excitations of the NLS equation have been found by many methods. However, except for the soliton-soliton interactions, it is very difficult to find interaction solutions between different types of nonlinear excitations. In this paper, the symmetry reduction method is further developed to find interaction solutions between solitons and other types of NLS waves. Especially, the soliton-cnoidal wave interaction solutions are explicitly studied in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integrals. Some special concrete interaction solutions and their asymptotic behaviors are discussed both in analytical and graphical ways.