The method of inverse scattering is extended to solve the initial value problem of a variable coefficient and nonisospectral Korteweg–de Vries equation with time varying boundary condition. One- and two-soliton solutions are examined in detail. By an appropriate decomposition, soliton interactions and asymptotic behaviors are investigated. Oscillating and asymptotically standing two-soliton solutions are discussed.
A hierarchy of first-degree time-dependent symmetries is proposed for Dirac soliton hierarchy and their commutator relations with time-dependent symmetries are exhibited. Meantime, a hereditary structure of Dirac soliton hierarchy is elucidated and a Lax operator algebra associated with Virasoro symmetry algebra is given.
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