On the variable depth Kawahara approximation

The Kawahara equation is a higher-request Korteweg-de Vries equation with an extra fifth order derivative term. It was inferred by Hasimoto [Water waves, Kagaku, 40:401-408, 1970 (Japanese)] as a model of the gravity waves in a vastly long channel over a flat bottom in a long wave with surface tension. In 2008, Iguchi [Bull. Inst. Math. Acad. Sin. (N.S.), 2:179-220, 2008] gave a mathematically rigorous justification of this modeling and demonstrated that the solution of the Kawahara equation approximates the full Euler problem of cappilary-gravity waves and he considered the situation where the bottom is not flat and its fluctuation is small and determined coupled Kawahara type equations whose solutions approximate the Euler problem. In this paper, we derived the Kawahara-type equation over an uneven bottom that generalizes the Kawahara equation in the previously mentioned work by Iguchi and we prove its consistency with the Euler system. After that, we propose a regularized-approximate version up to the good order (i.e. order of derivation of the Kawahara equation) and then we prove its unconditional well-posedness. Finally, we perform a numerical simulation on the Kawahara equation and its approximation equation and we verify numerically the coherence of the order of approximation.