Abstract
The main result of this work is the construction of multi-solitons solutions, that is, solutions that are time asymptotics to a sum of decoupling solitary waves for the full water-waves system with surface tension. Our approach uses the construction of a precise approximate solution that is controlled by using spectral information for each solitary wave and a bootstrap argument for the control of the remainder. For this stage, we need only few properties about the nonlinear Cauchy problem, namely, local well-posedness for very smooth data. We also use a similar construction to refine our previous result [Invent. Math., 184 (2011), pp. 257--388] about the nonlinear instability of one-dimensional solitary waves in the two-dimensional model: we prove the existence of semiglobal solutions that depend nontrivially of the transverse variable and that tend to the line solitary wave as time goes to infinity.
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