Invariant solutions of a nonlinear wave equation with a small dissipation obtained via approximate symmetries
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Invariant solutions of a nonlinear wave equation with a small dissipation obtained via approximate symmetries A. M. Grundland1,2
· A. J. Hariton1
Received: 24 August 2019 / Revised: 17 December 2019 © Università degli Studi di Napoli "Federico II" 2020
Abstract In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra of this equation and identify the conjugacy classes of the one-dimensional subalgebras of this Lie algebra. We show that the subalgebra classification of the integro-differential form of the nonlinear wave equation is much larger than the one obtained from the original wave equation. A systematic use of the symmetry reduction method allows us to find new invariant solutions of this wave equation. Keywords Symmetry reduction method · Approximate symmetries · Wave equation · Small dissipation Mathematics Subject Classification 35L60 · 20F40
1 Introduction A systematic computational method for constructing an approximate symmetry group for a given system of partial differential equations (PDEs) has been extensively developed by many authors, see e.g. [1–3]. A broad review of recent developments in this subject can be found in such books as Bluman and Kumei [4], Olver [5], Sattinger and Weaver [6], Rozdestvenskii and Janenko [7] and Baikov et al. [8,9]. Recently, Ruggieri and Speciale [10] determined the Lie algebras of approximate symmetries of
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A. M. Grundland [email protected] A. J. Hariton [email protected]
1
Centre de Recherches Mathématiques, Université de Montréal, Succ. Centre-ville, C.P. 6128, Montréal, QC H3C 3J7, Canada
2
Université du Québec, C.P. 500, Trois-Rivières, QC G9A 5H7, Canada
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A. M. Grundland, A. J. Hariton
nonlinear wave equations admitting a small perturbative dissipation. They discussed the generators of four different versions of the system of equations associated with the nonlinear wave equation (1) u tt = [ f (u)u x ]x , where u(t, x) is a function of t and x. They considered the following second-order PDE with a small dissipative term: u tt = [ f (u)u x ]x + ε [λ(u)u t ]x x ,
(2)
where ε 0. In the specific case where λ0 = 0, t0 = 0 and either s = −2 or s = 21 , we obtain G = C1 t 3 + C2 t −2 , so the solution is −2 p − q, u 0 = x 2 p ε f 0 (t − t0 )
u 1 = x 2s C1 t 3 + C2 t −2 .
(50)
In the specific case where λ0 = 0, t0 = 0 and either s = 1 or s = − 25 , we obtain G = C1 t 4 + C2 t −3 , so the solution is −2 p u 0 = x 2 p ε f 0 (t − t0 ) − q,
u 1 = x 2s C1 t 4 + C2 t −3 .
(51)
These solutions involve combinations of powers of x and t, and each solution admits a pole and is unbounded for large values of x. 11. For the subalgebra {X 3 + a X 4 }, we get u 0 = t 2ap F(ξ ) − q,
u 1 = t 2as−1 G(ξ ),
(52)
(a + 1)2 p 2 p ξ p f0 and G = Rξ 2s , where R is a constant. Here, the following conditions have to be satisfied: where the self-similar invaria
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