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Existence and non existence of solitons for a 1D Benney-Luke model of higher order. (English) Zbl 1328.35178

Summary: We shall establish the existence and non existence of solitons (travelling waves of finite energy) for a Benney-Luke equation of higher order, which includes models for long water waves with small amplitude. Following a variational approach, solitons are characterized as critical points of the action functional. Existence of solitons follows by the Concentration-Compactness principle by P.-L. Lions [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109–145 (1984; Zbl 0541.49009); ibid. 1, 223–283 (1984; Zbl 0704.49004)], applied to an appropriated minimization problem. It is also shown that solitons are smooth.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
76B25 Solitary waves for incompressible inviscid fluids
35C08 Soliton solutions
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Full Text: Euclid