Conservation Laws by using the Multiplier Method for a Fifth-Order Kdv Equation with Time-Dependent Coefficients and Linear Damping
DOI:
https://doi.org/10.5890/JAND.2017.12.003Abstract
In this paper we consider a family of fifth-order Korteweg-de Vries equations with time-dependent coefficients and linear damping term. By using the multiplier method of Anco and Bluman we determine all the low order conservation laws.References
[1] Boyd, J.P. (1991), Weakly Non-Local Solitons for Capillary-Gravity Waves: Fimh-Degree Korteweg-de VriesEquation, Physica D, 48, 129-146.
[2] Hunter, J.K. and Scheurle, J. (1988), Existence of perturbed solitary wave solutions to a model equation forwater waves, Physica D, 32, 253-268.
[3] Grimshaw, R. and Joshi, N. (1995), Weakly nonlocal solitary waves in a singularly perturbed KortewegdeVries Equation, SIAM J. Appl. Math., 55, 124-135.
[4] Xia, X. and Shen, H.T. (2002), Nonlinear Interaction of Ice Cover with Shallow Water Wave in Channels, J.Fluid Mech., 467, 259-268.
[5] Yun, X., Gao, Y.T., Sun, Z.Y., and Liu, Y. (2010), N-soliton solutions, B¨acklund transformation and Lax pairfor a generalized variable-coefficient fifth-order Korteweg-de Vries equation, Phys. Scr., 81, 045402-045408.
[6] Champneys, A.R. and Groves, M.D. (1997), A global investigation of solitary-wave solutions to a twoparametermodel for water waves, J. Fluid Mech., 342, 199-229.
[7] Kirchgässner, K. (1988), Nonlinearly Resonant SurfaceWaves and Homoclinic Bifurcation, Adv. Appl. Math.,26, 135-181.
[8] Chen, B. and Xie, Y.C. (2005), Exact solutions for generalized stochastic Wick-type KdV-mKdV equations,Chaos Solitons Fractals, 23, 281-287.
[9] de la Rosa, R., Gandarias, M.L., and Bruz′on M.S. (2015), Symmetries and conservation laws of a fifth-orderKdV equation with time-dependent coefficients and linear damping, Nonlinear Dyn..
[10] Noether, E. (1918), Invariante variations probleme, In: Nachrichten von der K¨oniglichen Gesellschaft derWissenschaften zu G¨ottingen, 234–57.
[11] Anco, S.C. and Bluman, G. (1997), Direct construction of conservation laws from field equations, Phys. Rev.Lett., 78, 2869-2873.
[12] Anco, S.C. and Bluman, G. (2002), Direct constrution method for conservation laws of partial differentialequations Part I: Examples of conservation law classifications, Euro. Jnl of Applied mathematics, 13, 545–566.
[13] Anco, S.C. and Bluman, G. (2002), Direct constrution method for conservation laws of partial differentialequations Part II: General treatment, Euro. Jnl of Applied mathematics, 13, 567–585.
[14] Anco, S.C. (2016), Generalization of noether theorem in modern form to non-variational partial differentialequations, in Recent progress and Modern Challenges in Applied Mathematics, Modeling and ComputationalScience, Fields Institute Communications.
[15] Bluman, G.W. and Kumei S. (1989), Symmetries and differential equations, Berlin: Springer.
[16] Olver, P. (1993), Applications of Lie groups to differential equations Springer-Verlag: New York.
[17] Anco, S.C. and Khalique, C.M. (2016), Conservation laws of coupled semilinear wave equations, Int. J.Modern Physics B, 1640004. doi: 10.1142/S021797921640004X.
[18] Bruzón, M.S., Gandarias, M.L., and de la Rosa, R. (2015), Conservation Laws of a Gardner Equation withTime-dependent Coefficients, Journal of Applied Nonlinear Dynamics, 4(2), 169–180.
[19] Gandarias, M.L. and Bruz′on, M.S. (2012), Conservation laws for a class of quasi self-adjoint third orderequations, Appl. Math. and Comp., 219, 668-678.
[20] Tracinà, R., Bruzón, M.S., and Gandarias, M.L. (2016), On the nonlinear self-adjointness of a class of fourthorderevolution equations. Appl. Math. Comput., 275, 299–304 .
[21] Anco, S.C., et al. (2016), Symmetries and conservation laws of the generalized Krichever-Novikov equation,J. Phys. A: Math. Theor., 49 105201-105230. doi:10.1088/1751-8113/49/10/105201
[22] Anco, S.C. (2016), Symmetry properties of conservation laws, Int. J. Mod. Phys. B, 30 1640003.
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