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On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water. (English) Zbl 1270.35365
Summary: This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy $${{\mathcal E}}$$ subject to the constraint $${{\mathcal I}=\sqrt{2}\mu}$$, where $${{\mathcal I}}$$ is the wave momentum and $$0 <\mu \ll 1$$ . Since $${{\mathcal E}}$$ and $${{\mathcal I}}$$ are both conserved quantities a standard argument asserts the stability of the set $$D_\mu$$ of minimisers: solutions starting near $$D_\mu$$ remain close to $$D_\mu$$ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schrödinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as $${\mu \downarrow 0}$$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76B25 Solitary waves for incompressible inviscid fluids 76B45 Capillarity (surface tension) for incompressible inviscid fluids 35Q51 Soliton equations
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