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.


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