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Self-adjointness, symmetries, and conservation laws for a class of wave equations incorporating dissipation. (English) Zbl 1275.35016

Summary: We study the nonlinear self-adjointness and conservation laws for a class of wave equations with a dissipative source. We show that the equations are nonlinear self-adjoint. As a result, from the general theorem on conservation laws proved by Ibragimov and the symmetry generators, we find some conservation laws for this kind of equations.

MSC:

35B06 Symmetries, invariants, etc. in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
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[1] Barone, A.; Esposito, F.; Magee, C. G.; Scott, A. C., Theory and applications of the sine-gordon equation, La Rivista del Nuovo Cimento, 1, 2, 227-267 (1971) · doi:10.1007/BF02820622
[2] Ames, W. F.; Lohner, R. J.; Adams, E., Group properties of \(u_{t t} = \left``[f \left(u\right) u_x\right``]_x\), International Journal of Non-Linear Mechanics, 16, 5-6, 439-447 (1981) · Zbl 0503.35058
[3] Torrisi, M.; Valenti, A., Group properties and invariant solutions for infinitesimal transformations of a nonlinear wave equation, International Journal of Non-Linear Mechanics, 20, 3, 135-144 (1985) · Zbl 0572.35070 · doi:10.1016/0020-7462(85)90007-1
[4] Torrisi, M.; Valenti, A., Group analysis and some solutions of a nonlinear wave equation, Atti del Seminario Matematico e Fisico dell’Università di Modena, 38, 2, 445-458 (1990) · Zbl 0717.73005
[5] Ibragimov, N. H., Lie Group Analysis of Differential Equations-Symmetries, Exact Solutions and Conservation Laws (1994), Boca Raton, Fla, USA: CRC, Boca Raton, Fla, USA · Zbl 0864.35001
[6] Oron, A.; Rosenau, P., Some symmetries of the nonlinear heat and wave equations, Physics Letters A, 118, 4, 172-176 (1986) · Zbl 1020.35501 · doi:10.1016/0375-9601(86)90250-1
[7] Kingston, J. G.; Sophocleous, C., Symmetries and form-preserving transformations of one-dimensional wave equations with dissipation, International Journal of Non-Linear Mechanics, 36, 6, 987-997 (2001) · Zbl 1345.35063 · doi:10.1016/S0020-7462(00)00064-0
[8] Ibragimov, N. H., A new conservation theorem, Journal of Mathematical Analysis and Applications, 333, 1, 311-328 (2007) · Zbl 1160.35008 · doi:10.1016/j.jmaa.2006.10.078
[9] Bruzón, M. S.; Gandarias, M. L.; Ibragimov, N. H., Self-adjoint sub-classes of generalized thin film equations, Journal of Mathematical Analysis and Applications, 357, 1, 307-313 (2009) · Zbl 1170.35439 · doi:10.1016/j.jmaa.2009.04.028
[10] Freire, I. L., Conservation laws for self-adjoint first-order evolution equation, Journal of Nonlinear Mathematical Physics, 18, 2, 279-290 (2011) · Zbl 1219.35228 · doi:10.1142/S1402925111001453
[11] Freire, I. L., Self-adjoint sub-classes of third and fourth-order evolution equations, Applied Mathematics and Computation, 217, 22, 9467-9473 (2011) · Zbl 1219.35048 · doi:10.1016/j.amc.2011.04.041
[12] Ibragimov, N. H., Quasi-self-adjoint differential equations, Archives of ALGA, 4, 55-60 (2007)
[13] Ibragimov, N. H.; Torrisi, M.; Tracinà, R., Quasi self-adjoint nonlinear wave equations, Journal of Physics A, 43, 44 (2010) · Zbl 1206.35174 · doi:10.1088/1751-8113/43/44/442001
[14] Ibragimov, N. H.; Torrisi, M.; Tracinà, R., Self-adjointness and conservation laws of a generalized Burgers equation, Journal of Physics A, 44, 14 (2011) · Zbl 1216.35115 · doi:10.1088/1751-8113/44/14/145201
[15] Torrisi, M.; Tracinà, R., Quasi self-adjointness of a class of third order nonlinear dispersive equations, Nonlinear Analysis, 14, 3, 1496-1502 (2013) · Zbl 1261.35131 · doi:10.1016/j.nonrwa.2012.10.013
[16] Gandarias, M. L., Weak self-adjoint differential equations, Journal of Physics A, 44, 26 (2011) · Zbl 1223.35203 · doi:10.1088/1751-8113/44/26/262001
[17] Gandarias, M. L.; Redondo, M.; Bruzón, M. S., Some weak self-adjoint Hamilton-Jacobi-Bellman equations arising in financial mathematics, Nonlinear Analysis, 13, 1, 340-347 (2012) · Zbl 1238.35048 · doi:10.1016/j.nonrwa.2011.07.041
[18] Ibragimov, N. H., Nonlinear self-adjointness and conservation laws, Journal of Physics A, 44 (2011) · Zbl 1270.35031
[19] Ibragimov, N. H., Nonlinear self-adjointness in constructing conservation laws, Archives of ALGA, 7/8, 1-90 (2011)
[20] Freire, I. L.; Sampaio, J. C. S., Nonlinear self-adjointness of a generalized fifth-order KdV equation, Journal of Physics A, 45, 3 (2012) · Zbl 1234.35221 · doi:10.1088/1751-8113/45/3/032001
[21] Freire, I. L., New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order, Communications in Nonlinear Science and Numerical Simulation, 18, 3, 493-499 (2013) · Zbl 1286.35075 · doi:10.1016/j.cnsns.2012.08.022
[22] Olver, P. J., Applications of Lie Groups to Differential Equations (1986), New York, NY, USA: Springer, New York, NY, USA · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2
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