Transverse instability of solitary-wave states of the water-wave problem

Abstract

Transverse stability and instability of solitary waves correspond to a class of perturbations that are travelling in a direction transverse to the direction of the basic solitary wave. In this paper we consider the problem of transverse instability of solitary waves for the water-wave problem, from both the model equation point of view and the full water-wave equations. A new universal geometric condition for transverse instability forms the backbone of the analysis. The theory is first illustrated by application to model PDEs for water waves such as the KP equation, and then it is applied to the full water-wave problem. This is the first theory proposed for transverse instability of solitary waves of the full water-wave problem. The theory suggests the introduction of a new functional for water waves, whose importance is suggested by the mathematical structure. Without explicit calculation, the theory is used to argue that the basic class of solitary waves of the water-wave problem, which bifurcate at Froude number unity, are likely to be stable to transverse perturbations, even at large amplitude.

Transverse instability of gravity–capillary solitary waves

“…It is straightforward to show (see 2) that the form (1.3), in terms of the total energy E, of the transverse-instability condition (1.2) obtained by Bridges [3] remains valid in the case of gravity-capillary solitary waves as well. This suggests that the solitary waves of interest here, which exist below the minimum gravity-capillary phase speed, are unstable to transverse perturbations, as E is expected to increase when the wave speed V is decreased.…”

“…is the quantity that enters the instability condition (1.2) derived by Bridges [3]. Note that, in water of finite depth, C = 0 so I is distinct from the true impulse in general.…”

“…In addition, it is possible to deduce the associated instability growth rate in the long-wave limit from the theory of Bridges [3]. For the latter purpose, rather than specializing the general formalism to the problem at hand, we find it more instructive to directly tackle the stability eigenvalue problem via a long-wave expansion procedure analogous to the one followed by Kataoka & Tsutahara [6] for pure gravity solitary waves.…”

“…When this condition is satisfied, moreover, the KP equation admits fully localized solitary-wave solutions, commonly referred to as 'lumps', which thus become the asymptotic states of the initial-value problem in two spatial dimensions [2]. In more recent work, Bridges [3] obtained a condition for transverse instability to long-wave perturbations of solitary-wave solutions of Hamiltonian partial differential equations formulated as multi-symplectic systems. When applied to the water-wave problem with gravity and possibly surface tension present, this condition implies instability if ∂I ∂V < 0, (1.2) where I is a certain quantity related to the total horizontal linear momentum, or impulse, of the solitary wave and V denotes the wave speed.…”

“…When applied to the water-wave problem with gravity and possibly surface tension present, this condition implies instability if ∂I ∂V < 0, (1.2) where I is a certain quantity related to the total horizontal linear momentum, or impulse, of the solitary wave and V denotes the wave speed. (Bridges [3] refers to I as the impulse but this ignores the circulation of the solitary wave which is non-zero in general; see 2 below.) In the case of pure gravity solitary waves, it was shown by Longuet-Higgins [4] that V ∂I/∂V = ∂E/∂V , E being the total energy of the solitary wave, so an alternative form of the instability condition (1.2) is…”