| Abstract: |
| We study two-dimensional internal travelling waves featuring both a free surface and an internal interface, propagating under the combined effects of gravity and surface and interfacial tension. By applying the method of spatial dynamics, the governing equations are reformulated as an infinite-dimensional dynamical system in which the unbounded spatial variable assumes the role of time:
\[u_x=Lu+N(u)\]
We focus on the special case in which the imaginary part of the spectrum of $L$ consists of the eigenvalues $0$, $\pm\mathrm{i}\kappa$, each with algebraic multiplicity two. A centre manifold reduction is then carried out, leading to a finite-dimensional system of reduced equations. At leading order, this reduced system takes the form of a coupled KdV-NLS system and we discuss the existence of homoclinic solutions of this system. |
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