Dynamics of three nonisospectral nonlinear Schrödinger equations (NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrödinger equation (NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover, we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes, velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.

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In this paper we explain how space-time localized waves can be generated by introducing nonisospectral effects which are usually related to non-uniformity of media. The nonisospectral Korteweg–de Vries, modified Korteweg–de Vries and the Hirota equations are employed to demonstrate the idea. Their solutions are presented in terms of Wronskians and double Wronskians and space-time localized dynamics are illustrated.

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