^{1}and Gérard Iooss

^{1}

### Abstract

We review the mathematical results on traveling waves in one or several superposed layers of potential flow, subject to gravity, with or without surface and interfacial tension, where the *bottom layer is infinitely deep*. The problem is formulated as a “spatial dynamical system,” and it is shown that the linearized operator of the resulting reversible system has an *essential spectrum filling the real line*. We consider three cases where bifurcation occurs. (i) The first case is when, in moving a parameter, two pairs of imaginary eigenvalues merge into one pair of double eigenvalues, and then split into four symmetric complex conjugate eigenvalues. (ii) The second case is when one pair of imaginary eigenvalues meet in 0, and disappear; (iii) the third case is when the phenomenon described in (ii) is superposed to the presence of another pair of imaginary eigenvalues sitting at finite distance from 0. We give a physical example for each case and more specially study the solitary waves and generalized solitary waves, emphasizing the differences, in the methods and in the results, between these cases and the finite depth case.

**A case of great physical interest is the infinite depth limit. In such a case, the classical reduction technique to a small-dimensional center manifold fails because the linearized operator possesses an**

*horizontal space variable plays the role of a “time.”***, and adapted tools are necessary. We give a method and the results for different types of systems. A first case is with a single infinitely deep layer, with surface tension at the free surface, where the bifurcation occurs when two pairs of imaginary eigenvalues meet and split into two pairs of complex eigenvalues. This case leads to solitary waves with polynomially (instead of exponentially) damped oscillations at infinity.**

*essential spectrum filling the whole real axis*^{1}Another case is with two superposed layers, the bottom one being infinitely deep, with no surface tension at the interface. For a strong enough surface tension at the free surface, the bifurcation occurs when a pair of imaginary eigenvalues merge at 0, which is part of the essential spectrum, and disappears when a parameter is varying. In the case of no surface tension at the free surface, there is in addition an oscillating mode. In both cases the bifurcating solutions are ruled at main order by the Benjamin-Ono nonlocal differential equation, coupled, in the latter case with an oscillatory mode. The case of strong surface tension leads to a one-parameter family of solitary waves,

^{2}and a two-parameter family of periodic waves,

^{3}forming a phase portrait analogous to the one for the corresponding three-dimensional reversible bifurcation case, except the asymptotics at infinity, which is now polynomial for the solitary waves. In the absence of surface tension, this spatial dynamics is coupled with a nonlinear oscillator, and leads to the bifurcation of a family of generalized solitary waves, tending at infinity towards periodic waves.

^{4}The amplitude of these limiting periodic waves cannot vanish in general, and their minimal size is exponentially small.

^{5}

I. INTRODUCTION II. SYSTEM 1. A SINGLE FLUID LAYER WITH SURFACE TENSION AT THE FREE SURFACE III. SYSTEM 2. TWO SUPERPOSED LAYERS WITH NO INTERFACIAL TENSION AND STRONG SURFACE TENSION IV. SYSTEM 3. TWO SUPERPOSED LAYERS WITH NO SURFACE TENSION

### Key Topics

- Eigenvalues
- 50.0
- Surface tension
- 23.0
- Free surface
- 18.0
- Periodic solutions
- 18.0
- Bifurcations
- 15.0

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