Ovsyannikov equations describe long-wavelength dynamics of nonlinear waves in two-layer shallow water systems. We suggest their simple generalisations which model qualitatively dispersive properties of such systems. These generalised equations are completely integrable by the known methods of nonlinear physics and we present their periodic and soliton solutions. The theory is illustrated by concrete examples.

We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schrödinger (NLS) equation, namely, $iu_t - \Delta^2 u - 2a \Delta u + |u|^{\alpha} u = 0, ~ x,a \in \mathbb R$, $\alpha \gt 0$, and investigate the dynamics of its solutions for various powers of $\alpha$, including the ground state solutions and their perturbations, leading to scattering or blow-up dichotomy when $a \leq 0$, or to a trichotomy when $a \gt 0$. Ground state solutions are numerically constructed, and their stability is studied, finding that the ground state solutions may form two branches, stable and unstable, which dictates the long-term behaviour of solutions. Perturbations of the ground states on the unstable branch either lead to dispersion or the jump to a stable ground state. In the critical and supercritical cases, blow-up in finite time is also investigated, and it is conjectured that the blow-up happens with a scale-invariant profile (when $a=0$) regardless of the value of $a$ of the lower dispersion. The blow-up rate is also explored.