# Existence of solitary waves for a full dispersion model

As announced much earlier, I would like to report on a work with Dag Nilsson and Erik Wahlén which is part of an ongoing project studying the properties of a system of evolution equations. In [DNW], we prove the existence of small-amplitude solitary waves to this system. This post is also an opportunity to introduce to some potentially uninformed readers rough ideas of a by-now standard strategy for the construction of solitary waves through minimization problems, which (among other things) hints at why the existence of solitary waves is so ubiquitous in physically motivated nonlinear dispersive equations.

# What is a good model?

The question raised in the title is evidently too vast to be seriously answered in this post, and the reader will only find vague comments on a very narrow portion of the subject. I will however do my best to explain why in my opinion, among the enormous amount of models that have been proposed to describe the propagation of waves at the surface of the ocean, a particular one (spoiler alert, the Serre-Green-Naghdi model) has been consistently considered as a good model.

# Effective potential for the multiscale Schrödinger operator

In a previous post, I explained how, together with Iva Vukićević and Michael Weinstein [DVW14], we introduced an effective potential for the two-scale Schrödinger operator with an oscillatory potential. Roughly speaking, the spectral and scattering properties of the operator

$\displaystyle H_{q_\epsilon}=-\frac{{\rm d}^2}{{\rm d}x^2} +q_\epsilon, \qquad q_\epsilon(x)=\sum_{j\in\mathbb Z}q_j(x)e^{2\pi ijx/\epsilon},$

are well-described by the following effective operator:

$\displaystyle H_{\Lambda_{\rm eff}}=-\frac{{\rm d}^2}{{\rm d}x^2} +\Lambda_{\rm eff}, \qquad \Lambda_{\rm eff}(x)=q_0(x)-\frac{\epsilon^2}{(2\pi^2)}\sum_{j\neq 0}\frac{|q_j(x)|^2}{j^2}.$

Of course, the gain is that the effective operator is much simpler to study than the original one, since the small scale vanished. Specially interesting is the case $q_0(x)\equiv 0$, for which we can deduce the bifurcation of a bound state at a distance $\mathcal O(\epsilon^4)$ from the edge of the continuous spectrum.

It turns out that this reduction is surprisingly powerful, as it applies to more general problems than the above.

# The limit of weak density contrast and the rigid-lid assumption

In previous posts, we motivated our study of internal waves from the fact that the density of water is not uniform in the ocean, due to variations of temperature and salinity (as well as pressure for deep water). Of course, these variations are not that large, so that the dimensionless parameter measuring density contrast is in most situations very small. Thus a natural asymptotic limit to consider is that of the vanishing density contrast, and is at stake in [Duc14,Duc]. As we shall see, this limit is closely related to two widely used approximations in oceanography, namely the rigid-lid and Boussinesq approximations.

# Kelvin-Helmholtz instabilities in shallow water

We come back to the study of internal waves between two layers of immiscible fluid (as described in this post and subsequent ones). A striking property of such setting, in contrast with the water-wave case (namely only one layer of homogeneous fluid with a free surface) is that the Cauchy problem for the governing equations is ill-posed outside of the analytic framework if surface tension is not taken into account. This ill-posedness is caused by the formation of quickly growing, high-frequency (i.e. small wavelength) components of the flow triggered by any non-trivial velocity shear between the two layers: the so-called Kelvin-Helmholtz instabilities. These instabilities may arise in the atmosphere as well as in the ocean, and are easily recognizable as they evolve into billows, as one can see in the video below (borowed from Wikimedia Commons).

# Spectral asymptotics of a broken delta-interaction

The spectrum of the Schrödinger operator with $\delta$-interactions (that is, a potential defined as a Dirac distribution supported on a given geometry) $\delta$ has recently attracted a lot of interest as an alternative to standard quantum graphs. It is not the place here to describe the physical motivations, let me refer to Exner [Exn08] for a review. With Nicolas Raymond, in [DR14], we are precisely interested in the case where the $\delta$-interaction is supported on a broken line, and more precisely in the small angle asymptotic limit.