A Model for Large-Amplitude Internal Waves with Finite-Thickness Pycnocline

We derive a new model of internal solitary wave propagation governed by the Euler equations for a continuously stratified, incompressible fluid in the Boussinesq approximation. The model is derived by a systematic joint asymptotic expansion for long wavelengths and thin pycnoclines. The resulting solutions extend those of the strongly nonlinear two-layer system (Choi and Camassa in J. Fluid Mech. 396:1-36, 1999) to continuous stratifications, and capture non-trivial features of the internal pycnocline structure. These solutions can be used, among other applications, to perform shear instability calculations in analytic form.