A global theory of internal solitary waves in two-fluid systems

Abstract

The problem analyzed is that of two-dimensional wave motion in a heterogeneous, inviscid fluid confined between two rigid horizontal planes and subject to gravity g g . It is assumed that a fluid of constant density ρ + {\rho _ + } lies above a fluid of constant density ρ − > ρ + > 0 {\rho _ - } > {\rho _ + } > 0 and that the system is nondiffusive. Progressing solitary waves, viewed in a moving coordinate system, can be described by a pair ( λ , w ) (\lambda ,w) , where the constant λ = g / c 2 \lambda = g/{c^2} , c c being the wave speed, and where w ( x , η ) + η w(x,\eta ) + \eta is the height at a horizontal position x x of the streamline which has height η \eta at x = ± ∞ x = \pm \infty . It is shown that among the nontrivial solutions of a quasilinear elliptic eigenvalue problem for ( λ , w ) (\lambda ,w) is an unbounded connected set in R × ( H 0 1 ∩ C 0 , 1 ) {\mathbf {R}} \times (H_0^1 \cap {C^{0,1}}) . Various properties of the solution are shown, and the behavior of large amplitude solutions is analyzed, leading to the alternative that internal surges must occur or streamlines with vertical tangents must occur.