On solitary waves and nonlinear oscillations in a stratified fluid with surface tension (Q1329288)

The authors consider nonlinear two-dimensional flows in an ideal stratified liquid. It was previously known that in the case when the linearized equations for this system had not more than two positive eigenvalues, there existed a solution to the full system of nonlinear governing equations in the form of a solitary wave on top of an oscillatory part; the main technical problem was to single out the oscillatory part of the solution. A crucial point in the previous work was the fact that the number of available arbitrary constants in the solution was just sufficient to satisfy all the necessary resolvability conditions. When one has more than three positive eigenvalues in the spectrum of the linearized problem, this is not, generally speaking, possible. However, the existence proof for the solution in the form of the solitary wave on top of oscillations can be extended to this case if the stationary state is subject to a certain condition. This is the subject of the present work. The proof is then extended to the case of an arbitrary number of the positive eigenvalues in the linear spectrum.