Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity

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Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity. / Groves, M. D.; Wahlén, Erik.

I: Proceedings of the Royal Society of Edinburgh. Section A, Vol. 145, Nr. 4, 2015, s. 791-883.

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TY - JOUR

T1 - Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity

AU - Groves, M. D.

AU - Wahlén, Erik

PY - 2015

Y1 - 2015

N2 - We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy H subject to the constraint I = 2 mu, where I is the wave momentum and 0 < mu << 1. Since H and I are both conserved quantities, a standard argument asserts the stability of the set D-mu of minimizers: 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 described by solutions of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrodinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as mu down arrow 0.

AB - We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy H subject to the constraint I = 2 mu, where I is the wave momentum and 0 < mu << 1. Since H and I are both conserved quantities, a standard argument asserts the stability of the set D-mu of minimizers: 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 described by solutions of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrodinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as mu down arrow 0.

KW - water waves

KW - solitary waves

KW - vorticity

KW - calculus of variations

U2 - 10.1017/S0308210515000116

DO - 10.1017/S0308210515000116

M3 - Article

VL - 145

SP - 791

EP - 883

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 4

ER -