Abstract
Two non-isospectral modified KdV equations with self-consistent sources are derived, which correspond to the time-dependent spectral parameter λ satisfying λt = λ and λt = λ 3 , respectively. Gauge transformation between the first non-isospectral equation (corresponding to λt = λ) and its isospectral counterpart is given, from which exact solutions and conservation laws for the non-isospectral one are easily listed. Besides, solutions to the two nonisospectral modified KdV equations with self-consistent sources are derived by means of the Hirota method and the Wronskian technique, respectively. Nonisospectral dynamics and source effects, including one-soliton characteristics in non-uniform media, two-solitons scattering and special behaviours related to sources (for example, the ‘ghost’ solitons in the degenerate two-soliton case), are investigated analytically.
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