An integrable evolution equation for surface waves in deep water

Abstract

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative methods, an asymptotic model for small-aspectratio waves is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical irrotational results is performed.

The Reduced Ostrovsky Equation: Integrability and Breaking

“…Thus, clearly not all solutions will break, although Boyd [2] showed numerically that a large class of solutions will break, even when the amplitude is very small provided that the length scale is correspondingly also very small. However, it has been shown by Vakhnenko and Parkes [24] that (4) is integrable, with soliton solutions, see also Vakhnenko [23], Parkes [21], and Kraenkel et al [16]. This would seem to predict that breaking is prevented, contrary to the numerical results of Boyd [2], and also to the proven nonexistence of solitary waves.…”

“…Indeed, the key result (15) can be established directly from (11) and (17). Next, we note that using (16) and the definition (14), the expression (17) takes either of the equivalent forms…”

“…In an apparently alternative approach, Kraenkel et al [16] have shown that the original Equation (4) in the (x, t) coordinate system, also has a Lax pair, expressed in terms of a function F where…”

“…In the transformed variables this becomes (Fφ) T = 0, so that Fφ = F 0 (X ), where F(X, T 0 ) = F 0 (X ), (16) where we have used the boundary condition that φ = 1 at T = T 0 , and we note that at T = T 0 , X = x, so that F 0 (x) is determined by the initial conditions in the original variables. Further, we find that…”

“…This is to be contrasted with the regularization of the Hopf equation by the KdV Equation (1) when it is known that breaking is replaced by the emergence of internal solitary waves, often in the form of internal undular bores. The reduced Ostrovsky Equation (4) (also known variously as the Ostrovsky-Hunter equation, or the Vakhnenko equation) has been previously studied numerically and theoretically, notably by Hunter [15] Vakhnenko [23], Parkes [21], Vakhnenko and Parkes [24], Boyd [1], [2], Stepanyants [22], Liu et al [19], and Kraenkel et al [16]. We also note that it is readily shown that Equation (4) does not have any smooth solitary wave solutions (see Liu et al [19] and the appendix of Grimshaw and Helfrich [10], where the argument produced there still applies when λ = 0), but does support a family of smooth periodic traveling wave solutions (Ostrovsky,[20]).…”