Modulating pulse solutions to the water-wave problem

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The classical water wave problem concerns the irrotational flow of a perfect fluid of unit density subject to the forces of gravity and surface tension. A modulating pulse is a pulse-like envelope steadily advancing in the laboratory frame and modulating an underlying periodic wavetrain. The existence of modulating pulse solutions to the water-wave equations has been predicted on the basis of simplified model equations and numerics, and there has been speculation concerning their importance in the theory of freak waves. Our objective is to develop a rigorous existence theory for them. More specifically, we wish to carry out a programme of research for water waves which has been successfully used to construct modulating pulse solutions to nonlinear wave equations, namely(i) formulate the problem as an evolutionary equation in which the horizontal spatial coordinate in a frame moving with the pulse plays the role of time (`spatial dynamics');(ii) confirm the presence of an invariant finite-dimensional subspace to an approximating system which contains homoclinic solutions (these are the modulating pulses predicted by simplified model equations);(iii) establish the existence of solutions to our evolutionary system which are approximated by the above homoclinic solutions over very long length scales and for all times.