NEW EXACT SOLUTIONS OF A GENERALISED BOUSSINESQ EQUATION WITH DAMPING TERM AND A SYSTEM OF VARIANT BOUSSINESQ EQUATIONS VIA DOUBLE REDUCTION THEORY

Justina Ebele Okeke; Rivendra Narain; Keshlan Sathasiva Govinder

doi:10.11948/2018.471

2018 Volume 8 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(2): 471-485

Next Article Previous Article

Justina Ebele Okeke, Rivendra Narain, Keshlan Sathasiva Govinder. NEW EXACT SOLUTIONS OF A GENERALISED BOUSSINESQ EQUATION WITH DAMPING TERM AND A SYSTEM OF VARIANT BOUSSINESQ EQUATIONS VIA DOUBLE REDUCTION THEORY[J]. Journal of Applied Analysis & Computation, 2018, 8(2): 471-485. doi: 10.11948/2018.471

Citation:

Justina Ebele Okeke, Rivendra Narain, Keshlan Sathasiva Govinder. NEW EXACT SOLUTIONS OF A GENERALISED BOUSSINESQ EQUATION WITH DAMPING TERM AND A SYSTEM OF VARIANT BOUSSINESQ EQUATIONS VIA DOUBLE REDUCTION THEORY[J]. Journal of Applied Analysis & Computation, 2018, 8(2): 471-485. doi: 10.11948/2018.471

NEW EXACT SOLUTIONS OF A GENERALISED BOUSSINESQ EQUATION WITH DAMPING TERM AND A SYSTEM OF VARIANT BOUSSINESQ EQUATIONS VIA DOUBLE REDUCTION THEORY

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X 54001, 4000 Durban, South Africa

Abstract

Abstract

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.
- Double reduction theory /
- conservation laws /
- partial Noether approach /
- Lie symmetry method /
- associated symmetry
MSC: 35C07;35Q53

FullText(HTML)

References

[1]	B. Abraham-Shrauner and K. S. Govinder, Provenance of type Ⅱ hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys., 2006, 13(4), 612-622. Google Scholar
[2]	B. Abraham-Shrauner and K. S. Govinder, Master partial differential equations for a type Ⅱ hidden symmetry, J. Math. Anal. Appl., 2008, 343(1), 525-530. Google Scholar
[3]	S. C. Anco, G.W. Bluman, Direct construction method for conservation laws of partial differential equations part Ⅱ:general treatment, Eur. J. Appl. Math., 2002, 13(5), 567-585. Google Scholar
[4]	E. Bessel-Hagen, Uber die Erhaltungss ¨ atze der Elektrodynamik ¨, Math. Ann., 1921, 84(3), 258-276. Google Scholar
[5]	A. H. Bokhari, A. Y. Al-Dweik, F. D. Zaman, A. H. Kara and F. M. Mahomed, Generalization of the double reduction theory, Nonlinear Anal. Real World Appl., 2010, 11(5), 3763-3769. Google Scholar
[6]	P. A. Clarkson, New similarity solutions for the modified Boussinesq equation, J. Phys. A Math. Gen., 1989, 22(13), 2355-2367. Google Scholar
[7]	P. A. Clarkson, Dimensional reductions and exact solutions of a generalized nonlinear Schrodinger equation, Nonlinearity 1992, 5(2), 453-472. Google Scholar
[8]	J. Douglas, Solutions of the inverse problem of the calculus of variations, J.Amer. Math Soc., 1941, 50(1), 71-128. Google Scholar
[9]	E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 2000, 277(4), 212-218. Google Scholar
[10]	Z. Fu, S. Liu and S. Liu, New transformations and new approach to find exact solutions to nonlinear equations, Phys. Lett. A, 2002, 299(5), 507-512. Google Scholar
[11]	M. Ghil and N. Paldor, A model equation for nonlinear wavelength selection and amplitude evolution of frontal waves, J. Nonlinear Sci., 1994, 4(1), 471-496. Google Scholar
[12]	R. K. Gupta and A. Bansal, Painlevé analysis, Lie symmetries and invariant solutions of potential Kadomstev-Petviashvili equation with time dependent coefficients, Appl. Math. Comput., 2013, 219(10), 5290-5302. Google Scholar
[13]	J. L. Hu, A new method for finding exact traveling wave solutions to nonlinear partial differential equations, Phys. Lett. A, 2001, 286(2), 175-179. Google Scholar
[14]	N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 2007, 333(1), 311-328. Google Scholar
[15]	A. H. Kara and F. M. Mahomed, Relationship between symmetries and conservation laws, Int. J. Theor. Phys., 2000, 39(1), 23-40. Google Scholar
[16]	A. H. Kara and F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear Dynam., 2006, 45(3), 367-383. Google Scholar
[17]	B. Kolev, Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 2007, 19(3), 555-574. Google Scholar
[18]	P. S. Laplace, Celestial Mechanics (English translation), Chelsea, New York, 1966. Google Scholar
[19]	B. Li, Y. Chen and H. Q. Zhang, Explicit exact solutions for some nonlinear partial differential equations with nonlinear terms of any order, Czech J. Phys., 2003, 53(4), 283-295. Google Scholar
[20]	S. Y. Lou, Nonclassical symmetry reductions for the dispersive wave equations in shallow water, J. Math. Phys., 1992, 33(12), 4300-4305. Google Scholar
[21]	D. C. Lu and B. J. Hong, New exact solutions for the (2+1)-dimensional generalized Broer-Kaup system, Appl. Math. Comput., 2008, 199(2), 572-580. Google Scholar
[22]	R. Narain and A. H. Kara, On the redefinition of variational and ‘partial’ variational conservation laws in a class of non linear PDEs with mixed derivatives, Math. Comput. Appl., 2010, 15(4), 732-741. Google Scholar
[23]	R. Naz, M. D. Khan and I. Naeem, Conservation laws and exact solutions of a class of non linear regularised long wave equations via double reduction theory and Lie symmetries, Commun. Nonlinear Sci. Numer. Simulat., 2013, 18(4), 826-834. Google Scholar
[24]	R. Naz, F. Mahomed, T. Hayat, Conservation laws for third-order variant Boussinesq system, Appl. Math. Lett., 2010, 23(8), 883-886. Google Scholar
[25]	E. Noether, Invariant variation problems, Transp. Theory. Stat. Phys., 1971, 1(3), 186-207. Google Scholar
[26]	P. J. Olver, Application of Lie groups to differential equations, Springer-Verlag, New York, 1993. Google Scholar
[27]	R. L. Sachs, On the integrable variant of the Boussinesq system:Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy, Physica D, 1988, 30(1-2), 1-27. Google Scholar
[28]	A. Sjöberg, On double reductions from symmetries and conservation laws, Nonlinear Anal. Real World Appl., 2009, 10(6), 3472-3477. Google Scholar
[29]	A. Sjöberg and F. Mahomed, The association of non-local symmetries with conservation laws:applications to the heat and Burgers equations, Appl. Math. Comput., 2005, 168(2), 1098-1108. Google Scholar
[30]	H. Steudel, Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen, Z. Naturforsch A, 1962, 17(2), 129-132. Google Scholar
[31]	M. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, 1995, 199(3-4), 169-172. Google Scholar
[32]	M. Wang, Y. Zhou and Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 1996, 216(1-5), 67-75. Google Scholar
[33]	M. Wazwaz, The tanh and sine-cosine method for compact and noncompact solutions of nonlinear Klein-Gordon equation, Appl. Math. Comput., 2005, 167(2), 1179-1195. Google Scholar
[34]	Z. Y. Yan, Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey-Stewartson-type equation via a new method, Chaos Solitons Fractals, 2003, 18(2), 299-309. Google Scholar
[35]	Z. Y. Yan, F. D. Xie and H. Q. Zhang, Symmetry reductions, integrability and solitary wave solutions to high-order modified Boussinesq equations with damping term, Commun. Theor. Phys., 2001, 36(1), 1-6. Google Scholar
[36]	E. Yasar and BI.B. Giresunlu, Lie symmetry reductions, exact solutions and conservation laws of the third order variant Boussinesq system, Acta Phys. Pol., 2015, 128(3), 243-255. Google Scholar
[37]	J. F. Zhang, Multi-solitary wave solutions for variant Boussinesq equations and Kupershmidt equations, Appl. Math. Mech., 2000, 21(2), 171-175. Google Scholar