Compatibility of stagnation-point assumptions with Gardner-equation validity

Determine whether the parameter assumptions that yield stagnation points beneath solitary waves for large linear-shear parameters \(\gamma\) and \(\gamma_1\) are compatible with the asymptotic assumptions required for the validity of the Gardner equation for interfacial internal waves.

Background

The paper notes that the velocity field can be analytically extended into the fluid domains and that underlying shear currents may produce stagnation points beneath solitary waves. Prior work is cited as demonstrating such stagnation points for various shear currents, including linear shear currents with large values of $\gamma$ and $\gamma_1$.

The unresolved issue is whether parameter values sufficiently large to generate these stagnation points remain consistent with the small-amplitude, long-wave, and related asymptotic assumptions under which the Gardner equation was derived and is valid. Resolving this compatibility would establish whether the stagnation-point scenarios have relevance within the paper's Gardner-model regime.