A. SAFA, H. LE MEUR, J-P. CHEHAB and 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 and I. ZAITER. On the variable depth Kawahara approximation, COMMUN. MATH. SCI. Vol. 24, No. 5, pp. 1373–1402 (2026).
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.
A. SAFA and S. ISRAWI. Analytical solution of the variable-depth Kawahara equation with surface tension: theoretical and numerical results (Submitted in December 2025).
Abstract. We here consider the propagation of capillary–gravity waves in shallow water over a non-flat bottom, modeled by a generalized Kawahara equation (including fifth-order dispersion). Accounting for surface tension, we derive an analytical travelling-wave solution by combining the solution of the equation for the flat-bottom case with that of a transport equation, resulting in a final solution formed from the composition of these two solutions. To validate the proposed solution, we perform numerical simulations. The comparisons reveal excellent agreement between the analytical and numerical profiles, confirming that the model accurately captures the interplay between nonlinearity, higher-order dispersion, and bottom topography. These results provide an effective framework for the study of nonlinear waves in coastal environments with complex topography.