NONLINEAR SELF ADJOINTNESS AND EXACT SOLUTION OF FOKAS–OLVER–ROSENAU–QIAO (FORQ) EQUATION

Yıl 2018, Cilt: 67 Sayı: 2, 317 - 326, 01.08.2018

Filiz Taşcan Ömer Ünsal Arzu Akbulut Sait San

Öz

Abstract. Based on Lieís symmetry approach, conservation laws are constructed
for Fokas-Olver-Rosenau-Qiao(FORQ) equation and exact solution
is obtained. Nonlocal conservation theorem is used to carry out the analysis of
conservation process. Nonlinear self adjointness concept is applied to FORQ
equation, it is proved to be strict self adjoint. Characteristic equation and
similarity variable help us fnd exact solution of FORQ equation. Compared
with solutions found in previous papers, our solution is new and important,
since it is not possible to fnd exact solution of FORQ equation quite easily

Anahtar Kelimeler

Conservation laws, symmetry generators, FORQ equation, self adjointness, exact solution

Kaynakça

• Noether E., Invariante Variationsprobleme, 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.
• Steudel H., Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen, Z. Naturforsch 17A (1962) ; 129–132.
• Naz, R., Conservation laws for a complexly coupled KdV system, coupled Burgers’ system and Drinfeld–Sokolov–Wilson system via multiplier approach. Communications in Nonlinear Science and Numerical Simulation, 15(5), (2010), 1177-1182.
• Anco, S.C. and Bluman GW., Direct construction method for conservation laws of partial dif- ferential equations. Part I: examples of conservation law classi…cations, Eur. J. Appl. Math., (2002) ; 545-566.
• Adem, K.R and Khalique, C.M., Exact Solutions and Conservation Laws of a (2+1)- Dimensional Nonlinear KP-BBM Equation, Abstract and Applied Analysis Volume 2013, Article ID 791863, 5 pages
• Naz, R., Conservation laws for some compacton equations using the multiplier approach, Applied Mathematics Letters 25 (2012), 257–261.
• Naz, R., Conservation laws for a complexly coupled KdV system, coupled Burgers’ system and Drinfeld–Sokolov–Wilson system via multiplier approach, Commun Nonlinear Sci Numer Simulat 15 (2010) ; 1177–1182.
• Adem, K.R and Khalique, C.M., Symmetry reductions, exact solutions and conservation laws of a new coupled KdV system, Commun Nonlinear Sci Numer Simulat 17 (2012) ; 3465–3475.
• Kara A.H. and Mahomed, F.M., Relationship between Symmetries and conservation laws, International Journal of Theoretical Physics, 39, (1) (2000) ; 23-40.
• Kara A.H. and Mahomed, F.M., Noether-type symmetries and conservation laws via partial Lagragians, Nonlinear Dynam., 45 (2006) ; 367-383.
• Khalique C.M. and Johnpillai A.G., Conservation laws of KdV equation with time dependent coe¢ cients, Commun Nonlinear Sci Numer Simulat 16 (2011) ; 3081–3089.
• Ya¸sar, E., & Özer, T., Conservation laws for one-layer shallow water wave systems. Nonlinear Analysis: Real World Applications, 11(2) (2010), 838-848.
• Ya¸sar, E., On the conservation laws and invariant solutions of the mKdV equation. Journal of Mathematical Analysis and Applications, 363(1) (2010), 174-181.
• Cheviakov, A.F., GeM software package for computation of symmetries and conservation laws of diğerential equations, Comput Phys Comm. 176 (1) (2007) ; 48-61.
• Camassa, R. and Holm D D., Phys. Rev. Lett. 71 (1993) ; 1661–1664.
• Camassa, R., Holm D.D. and Hyman J.M., Adv. Appl. Mech. 31 (1994) ; 1–33.
• Degasperis A, Holm D.D. and Hone A.N.W., Theor. Math. Phys. 133 (2002) 1461–1472
• Vladimir, N., Generalizations of the Camassa–Holm equation J. Phys. A: Math. Theor. 42 (2009) ;342002.
• Alexandrou H.A., Dionyssios, M., The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation, Nonlinear Analysis 95 (2014), 499–529.
• Olver, P. J., & Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Physical Review E, 53(2), (1996)1900.
• Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: general- izations of the Camassa-Holm equation, Physica D: Nonlinear Phenomena, 95(3), (1996) 243.
• Qiao, Z., A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys. 47 (11) (2006) 112701, 9 pp.
• Ibragimov, N.H., A new conservation theorem. J. Math. Anal. Appl. 2007;333:311–28.
• Avdonina, E.D. and Ibragimov, N.H., Conservation laws and exact solutions for nonlinear diğusion in anisotropic media, Commun Nonlinear Sci Numer Simulat 2013 18 2595–2603.
• Ibragimov, N.H., Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor. (2011) 432002.