A. SAFA, H. LE MEUR, J-P. CHEHAB, R. TALHOUK. Asymptotic expansion of the solutions to a regularized Boussinesq system (theory and numeric), Acta Applicandae Mathematicae 191:12 (2024). https://doi.org/10.1007/s10440-024-00660-3.
Abstract. We consider the propagation of surface water waves described by the Boussinesq system. Following (Molinet et al. in Nonlinearity 34:744-775, 2021), we introduce a regularized Boussinesq system obtained by adding a non-local pseudo-differential operator. In this paper, we display a twofold approach: first, we study theoretically the existence of an asymptotic expansion for the solution to the Cauchy problem associated to this regularized Boussinesq system with respect to the regularizing parameter. Then, we compute numerically the function coefficients of the expansion and verify numerically the validity of this expansion up to order 2. We also check the numerical L2 stability of the numerical algorithm.
S. ISRAWI, A. SAFA, R. TALHOUK, I. ZAITER. On the variable depth Kawahara approximation (Submitted in Octobre 2024).
Abstract. 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 -1970) 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 (in Bull. Inst. Math. Acad. Sin. (N.S.) 2 -2008) gave a mathematically rigorous justification of this modeling and demonstrate that the solution of the Kawahara equation approximates the full Euler problem of cappilary-gravity waves and he consider the situation where the bottom is not flat and its fluctuation is small and determine coupled Kawahara type equations whose solutions approximates the Euler problem. In this paper, we derived the Kawahara-type equation over an uneven bottom that generalize the Kawahara equation of 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.