Published online by Cambridge University Press: 05 February 2016
In this contribution we present High-Order Perturbation of Surfaces (HOPS) methods as applied to the spectral stability problem for traveling water waves. The Transformed Field Expansion method (TFE) is used for both the traveling wave and its spectral data. The Lyapunov-Schmidt reductions for simple and repeated eigenvalues are compared. The asymptotics of modulational instabilities are discussed.
The water wave stability problem has a rich history, with great strides made in the late sixties in the work of Benjamin and Feir  and in the ensuing development of Resonant Interaction Theory (RIT) [2–5]. The predictions of RIT have since been leveraged heavily by numerical methods; the influential works of MacKay and Saffman  and McLean  led to a taxonomy of water wave instabilities based on RIT (Class I and Class II instabilities). The most recent review article is that of Dias & Kharif ; since the publication of this review a number of modern numerical stability studies have been conducted [9–13].
In these lecture notes, we explain how the spectral data of traveling water waves may be computed using a High-Order Perturbation of Surfaces (HOPS) approach, which numerically computes the coefficients in amplitude-based series expansions . For the water wave problem, a crucial aspect of any numerical approach is the method used to handle the unknown fluid domain. Just as in the traveling waves lecture of this short course, numerical results will be presented from the Transformed Field Expansion (TFE) method, whose development for the spectral stability problem appears in [13, 15–17].
The TFE method computes the spectral data as a series in wave slope/ amplitude, and thus relies on analyticity of the spectral data in amplitude. A large number of studies of the spectrum have been made that do not make such an assumption [9, 10, 18, 19]. On the other hand, it is known that the spectrum is analytic for all Bloch parameters at which eigenvalues are simple in the zero amplitude limit . Numerically it has been observed that the spectrum is analytic in amplitude at Bloch parameters for which there are eigenvalue collisions, but that the disc of analyticity is discontinuous in Bloch parameter. This discontinuity in radius is due to modulational instabilities, as explained in .