Non-isospectral modelling in probing nonlinear water waves

Reducing nonlinear evolution equations into integrable systems has emerged as a practical approach to probing nonlinear water waves. As a widely-accepted result, rogue waves can generate from the state changes of other types of wave solutions, for instance, a breather whose period approaches infinity. Crucially, the physical parameters varying across time and space can induce state transfers between different nonlinear wave solutions. More precisely, the coefficients of integrable systems are closely related to the real-world physical context. Especially in oceanography, the physical parameters concerned have large scale distribution in space and time, which implies that the simple iso-spectral models are out of reach.

 

The non-isospectral generalization of the (2+1)-dimensional Gardner equation adequately describes the nonlinear oceanic waves' abundant phenomena and characteristics. For this sake, we propose an extension of the Bell polynomial approach to derive its integrability features. Moreover, we obtain the explicit solution of corresponding non-isospectral solitary interactions, which indicates the drifting and alternating mechanisms of the wave envelopes in non-isospectral kink collisions.