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Periodic-wave and semirational solutions for the $$(2 + 1)$$-dimensional Davey-Stewartson equations on the surface water waves of finite depth. (English) Zbl 1439.35411
Summary: The $$(2 + 1)$$-dimensional Davey-Stewartson equations concerning the evolution of surface water waves with finite depth are studied. We derive the periodic-wave solutions through the Kadomtsev-Petviashvili hierarchy reduction. We obtain the growing-decaying periodic wave and three kinds of breathers via those solutions. We obtain the periodic wave takes on the growing and decaying property. Taking the long-wave limit on the periodic-wave solutions, we derive the semirational solutions describing the interaction of the rogue wave, lump, breather and periodic wave. We illustrate the lump and rogue wave and find that the rogue wave (lump) is the long-wave limit of the periodic wave (breather).

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35N05 Overdetermined systems of PDEs with constant coefficients 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 76B25 Solitary waves for incompressible inviscid fluids
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