Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity

Year 2019, Volume: 2 Issue: 2, 180 - 185, 20.12.2019

Hatice Taşkesen Mohanad Alaloush

https://doi.org/10.33401/fujma.587740

Abstract

In this paper, the conservation laws for a model with both quadratic and cubic nonlinearity 

\begin{eqnarray*}

m_{t}=bu_{x}+\frac{1}{2}a\left[ \left( u^{2}-u_{x}^{2}\right) m\right] _{x}+%

\frac{1}{2}c\left( 2m\cdot u_{x}+m_{x}\cdot u\right) ;\text{ \ \ }m=u-u_{xx}

\end{eqnarray*}%

are considered for the six cases of coefficients. By using a variational derivative approach, conservation laws were constructed. The computations to derive  multipliers and conservation law fluxes are conducted by using a Maple-based package which is called GeM.

Keywords

Conservation laws, Integrable equations, Multiplier

References

• [1] T. Wolf, A comparison of four approaches to the calculation of conservation laws Eur. J. Appl. Math., 13(2) (2002), 129-152.
• [2] P. D. Lax, Shock wave and entropy, in Contributions to Functional Analysis, ed. EA Zarantonello, Academic Press, New York, 1971.
• [3] R. J. DiPerna, Decay of solutions of hyperbolic systems of conservation laws with a convex extension, Arch. Ration. Mech. An., 64(1) (1977), 1-46.
• [4] B. A. Bilby, K. J. Miller, J. R. Willis, Fundamentals of Deformation and Fracture, Cambridge University Press, Cambridge, 1985.
• [5] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993.
• [6] P. S. Laplace, Traite de Mecanique Celeste, Tome I, Paris, 1798.
• [7] E. Noether, Invariante variations probleme, Nachr. Konig. Gesell. Wiss. Gottingen Math. Phys. Kl. Heft 2 (1918), 235-257, English translation in Transport Theory Statist. Phys. 1(3) (1971), 186-207.
• [8] H. Steudel, Uber die Zuordnung zwischen lnvarianzeigenschaften und Erhaltungssatzen, Z. Naturforsch., 17A(2) (1962), 129-132.
• [9] Ü. Göktaş, W. Hereman, Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symb. Comput., 24(5) (1997), 591-622.
• [10] P. J. Adams, W. Hereman, TransPDEDensityFlux.m: Symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities, Scientific Software, 2002.
• [11] L. D. Poole , W. Hereman, ConservationLawsMD.m: A Mathematica package for the symbolic computation of conservation laws of polynomial systems of nonlinear PDEs in multiple space dimensions, Scientific Software, 2009, available at http://inside.mines.edu/~whereman/.
• [12] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Commun., 176(1) (2007), 48-61.
• [13] A. F. Cheviakov, Computation of fluxes of conservation laws, J. Eng. Math., 66(1-3) (2010), 153-173.
• [14] I. M. Anderson, E. S. Cheb-Terrab, Differential geometry package, Maple Online Help, 2009.
• [15] T. M. Rocha Filho, A. Figueiredo, [SADE] A Maple package for the symmetry analysis of differential equations, Comput. Phys. Commun., 182(2) 2011), 467-476.
• [16] B. Xia, Z. Qiao, J. Li, An integrable system with peakon, complex peakon, weak kink, and kink peakon interactional solutions, Commun. Nonlinear Sci. Numer. Simul., 63 (2018), 292-306.
• [17] G. W. Bluman, A. F. Cheviakov, S. C. Anco, Applications of symmetry methods to partial differential equations (First Edition), Springer, New York, 2010. [18] R. Naz, Symmetry solutions and conservation laws for some partial differential equations in fluid mechanics, Ph.D. Thesis, University of the Witwatersrand, Johannesburg, South Africa, 2008.