The Kawahara equation is a higher-order Korteweg-de Vries equation with an additional fifth order derivative term. It was derived by Hasimoto as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime when the Bond number is nearly one third. In this paper, we give a mathematically rigorous justification of this modeling and show that the solution...

In this paper we establish the nonlinear stability of solitary travelling-wave solutions for the Kawahara-KdV equation ut + uux + uxxx − γ1uxxxxx = 0, and the modified Kawahara-KdV equation ut + 3u 2ux + uxxx − γ2uxxxxx = 0, where γi ∈ R is a positive number when i = 1, 2. The main approach used to determine the stability of solitary travelling-waves will be the theory developed by Albert in [1].

Hajime Kawahara, Kohji Yoshikawa, Shin Sasaki, Yasushi Suto, Nobuyuki Kawai, Kazuhisa Mitsuda, Takaya Ohashi, and Noriko Y. Yamasaki 1Department of Physics, School of Science, The University of Tokyo, Tokyo 113-0033 [email protected] 2Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397 3Department of Physics, Tokyo Institute of Techn...

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the sm...

Hajime Kawahara, Kohji Yoshikawa, Shin Sasaki Yasushi Suto, Nobuyuki Kawai, Kazuhisa Mitsuda, Takaya Ohashi, and Noriko Y. Yamasaki 1Department of Physics, School of Science, The University of Tokyo, Tokyo 113-0033 [email protected] 2Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397 3Department of Physics, Tokyo Institute of Techno...

In this article, we analysed the approximate solutions of time-fractional Kawahara equation and modified equation, which describe propagation signals in transmission lines formation nonlinear water waves long wavelength region. An efficient technique, namely natural transform decomposition method, is used present study. Fractional derivatives are considered Caputo, Caputo–Fabrizio, Atangana–Bal...

We introduce a new solution for Kawahara-KdV equations. The Lie group analysis is used to carry out the integration of this equations. The similarity reductions and exact solutions are obtained based on the optimal system method.

In this paper, we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiationmatrix and discrete Fourier transform. The relationship is crucial for implementing the scheme efficiently. By using the relationship, we can apply the Fast Fourier transform to solve the Kawahara equation. The effecti...

In this paper, we study the computability of the solution operator of the initial problem for the nonlinear Kawahara equation, which is based on the Type-2 Turing machines. We will prove that in Sobolev space 2 ( ) s H R , for 0 s ≥ , the solution operator: 2 : ( ) s R K H R → ( ; C R 2 ( )) s H R is ( ,[ s H δ ρ → ]) s H δ -computable. The conclusion enriches the theory of computability. Key w...