**AUTHORS:**Meknani Bassem, Messaoudi Rima, Talaat Abdelhamid, Nasserdine Kechkar, Ehab S. Selima

**Download as PDF**

**ABSTRACT:**
In this work, we consider the quadratic generalized Kortewegde Vries (QGKdV) equation that is a
mathematical model of waves on shallow water surfaces. Numerical solution of a Cauchy boundary-value problem
with known exact solution is developed in details. Discretization is first accomplished by means of a quadratic finite
element method. Then, the obtained system of first-order ordinary differential equations is discretized through
a backward finite difference formula. Finally, the derived non linear algebraic system is solved by Newton’s
method with the Gauss elimination method as the inner iteration solver. Numerical results are presented in order to
illustrate the efficiency of the present numerical treatment. In addition, a general form of multiple-soliton solution
of QGKdV equation is obtained using the simplest equation method with Burgers equation as simplest equation

**KEYWORDS:**
KdV equation, Finite element method, Finite difference method.

**REFERENCES:**

[1] M.E Alexander and J.Ll Morris. Galerkin methods applied to some model equations for nonlinear dispersive waves. Journal of Computational Physics, 30(3):428 – 451, 1979.

[2] G.B. Whitham B. Fornberg. A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. Roy. Soc., 289(240):373–404., 1978.

[3] T. B. Benjamin. The stability of solitary waves. Proceedings Mathematical Physical Engineering Sciences, 328, 05 1972.

[4] J. Bona. On the stability theory of solitary waves. Proceedings Mathematical Physical Engineering Sciences, 344, 07 1975.

[5] Luis Vega Carlos E. Kenig, Gustavo Ponce. Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle. Communications on Pure and Applied Mathematics, 46, 1993.

[6] E.N. Aksan; A. O¨zdes. Numerical solution of kortewegde vries equation by galerkin b-spline finite element method. Applied Mathematics and Computation, 175, 2006.

[7] S. Kutluay; A.R. Bahadir; A. O¨zdes. A small time solutions for the kortewegde vries equation. Applied Mathematics and Computation, 107, 2000.

[8] S. O¨zer; S. Kutluay. An analytical–numerical method for solving the kortewegde vries equation. Applied Mathematics and Computation, 164, 2005.

[9] Katuhiko Goda. On stability of some finite difference schemes for the korteweg-de vries equation. Journal of the Physical Society of Japan, 39(1):229–236, 1975.

[10] A.J. Khattak and Siraj ul Islam. A comparative study of numerical solutions of a class of kdv equation. Applied Mathematics and Computation, 199(2):425 – 434, 2008.

[11] D. J. Korteweg and G.de Vries. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philo.Mag., 39(1895):422–443.

[12] S. Kutluay, A.R. Bahadir, and A. AzdeA˙ A ˇ small time solutions for the kortewegede vries equation. Applied Mathematics and Computation, 107(2):203 – 210, 2000.

[13] N.J.Zabuskay. synergetic approach to problem of nonlinear dispersive wave propagation and interaction. in: W. Ames (Ed.), Proc. Symp. Nonlinear Partial Diff. Equations, Academic Press,, Academic Press(240):223–258., 1967.

[14] Ehab S Selima, Aly R Seadawy, Xiaohua Yao, and FA Essa. Integrability of the coupled cubic–quintic complex ginzburg–landau equations and multiple-soliton solutions via mathematical methods. Modern Physics Letters B, page 1850045, 2018.

[15] Ehab S. Selima, Xiaohua Yao, and Abdul-Majid Wazwaz. Multiple and exact soliton solutions of the perturbed Korteweg de Vries equation of long surface waves in a convective fluid via painleve analysis, factorization, and simplest e- ´ quation methods. Phys. Rev. E, 95:062211, Jun 2017.

[16] A. C. Vliegenthart. On finite-difference methods for the korteweg-de vries equation. Journal of Engineering Mathematics, 5(2):137–155, Apr 1971.

[17] Abdul-Majid Wazwaz. Multiple-front solutions for the Burgers equation and the coupled Burgers equations. Applied mathematics and computation, 190(2):1198–1206, 2007.

[18] Michael I. Weinstein. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Communications on Pure and Applied Mathematics, 39, 1986.

[19] Frank Merle Yvan Martel. Asymptotic stability of solitonsfor subcritical generalized kdv equations. Archive for Rational Mechanics and Analysis, 157, 04 2001.

[20] Tai-Peng Tsai Yvan Martel, Frank Merle. Stability and asymptotic stability for subcritical gkdv equations. Communications in Mathematical Physics, 231, 12 2002.

[21] N. J. Zabusky and M. D. Kruskal. Interaction of ”solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15:240–243, Aug 1965.