This work has been published as:
Transverse instability of the line solitary water-waves. Invent. Math. 184, (2011) 257-388.
The line solitary waves for the water-waves system (in the presence of surface tension) were constructed in the article:
C. Amick and K. Kirchgassner, A theory of solitary water-waves in the presence of surface tension. Arch. Ration. Mech. Anal. 105, (1989) 1-49.
These solitary waves are constructed in the KdV regime and are thus closely related to the famous KdV solitons.
The KdV equation itself is a one dimensional asymptotic model obtained from the water-waves system. So are the KP models which are two dimensional asymptotic models. The KP-II model corresponds to weak surface tension while the KP-I model corresponds to stronger surface tension.
The transverse instability of the KdV soliton as a solution of the KP-I equation was obtained in the following remarkable paper by Zakharov:
V. Zakharov, Instability and nonlinear oscillations of solitons. JEPT Lett. 22, (1975) 172-173.
Zakharov's proof heavily exploits the integrability properties of the KP equations. Our result with F. Rousset can be seen as an analogue of the Zakharov result for the full water-waves system for which integrability systems methods do not seem efficient (at least are not developed).
Our work on the transverse instability for the line solitary water-waves was preceded by the papers:
F. Rousset, N. Tzvetkov, Transverse nonlinear instability for some Hamiltonian PDE's. J. Math. Pures Appl. 90, (2008) 550-590.
F. Rousset, N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models. Ann. Inst. Henri Poincare, Anal. Non Lineaire 26, (2009) 477-496,
where on simpler models we developed an intuition how to handle the transverse stability of the solitary water-waves.
In our proof of the transverse instability of the line solitary water-waves, we relied on the following previous contributions:
D. Lannes, Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, (2005) 605-654.
E. Grenier, On the nonlinear instability of Euler and Prandtl equations. Commun. Pures Appl. Math. 53, (2000) 1067-1091.
A. Mielke, On the energetic stability of solitary water waves. Philos. Trans. R. Soc. Lond. A 360, (2002) 2337-2358.
The work by Lannes made the water-waves problem accessible to people with basic PDE background and we were (among many others) to benefit of this.
The work by Grenier gave us the basic idea how to construct an accurate approximate solution for times long enough, so that we can observe the transverse instability of the solitary wave. The work by Mielke was of importance in the detailed study of the linearised about the solitary wave operator.
During our work on the transverse instability of the line solitary water-waves, we realised that one may get a very simple abstract criterium for the linear transverse instability of a solitary wave for a Hamiltonian partial differential equation. This work appeared as :
F. Rousset, N. Tzvetkov, A simple criterion of transverse linear instability for solitary waves. Math. Res. Lett. 17 (2010), 157-169.
The analysis of our work on the transverse instability for the line solitary water-waves was the starting point of a work in which we construct a multi-soliton for the water-waves system. Previous works on related problems considered semi-linear problems and it looks that our work is the first one to perform such a construction in the context of a quasilinear problem (i.e. with severe derivative losses). This work has been published as:
M. Ming, F. Rousset, N. Tzvetkov, Multi-solitons and related solutions for the water-waves system. SIAM J. Math. Anal. 47 (2015), 897-954.