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Multiple double-valley and single-valley dark solitons in the complex modified Korteweg-de Vries equation: shape-preserving collisions and shape-altering collisions

The shape-preserving and shape-altering collisions of dark solitons are investigated in the complex modified Korteweg-de Vries equation. The obtained dark soliton solutions are classified into two distinct types, referred to as type-I and type-II dark solitons, which exhibit fundamentally different structural and dynamical characteristics. A single type-I dark soliton is symmetric about its center and admits three distinct valley profiles, namely single-valley, double-valley, and flat-bottom structures, whereas a single type-II dark soliton only supports a single-valley profile. These two types of dark solitons differ in their phase behaviors as the spatial variable varies from $-\infty$ to $+\infty$, as well as in their velocity--amplitude relations. For multiple pure type-I dark solitons, collisions are shape-preserving; however, a nontrivial collective effect is revealed in which modifying the parameters of one soliton can induce changes in the profiles and amplitudes of the other solitons, even though all collisions remain elastic in nature. In contrast, multiple pure type-II dark solitons behave independently, undergoing only phase shifts without any modification of their shapes or amplitudes. When type-I and type-II dark solitons coexist, their interactions lead to genuine shape-altering collisions, where the valley structures and amplitudes of type-I dark solitons are modified, while type-II solitons remain unaffected except for phase shifts. Asymptotic analysis further shows that the influence of type-II dark solitons on type-I dark solitons is confined to the pre-collision stage and disappears after the interaction.