Classical and nonclassical symmetry classifications of nonlinear wave equation with dissipation

Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

“…As we did in the previous section, we can simplify these determining equations. Solving (22) and (23) for u t , u x , v t , and v x gives…”

“…where c 1 and c 2 are arbitrary constants showing that the nonclassical symmetries of the single Equation (11b) contain the nonclassical symmetries of the equivalent system (22), and are in fact more general.…”

“…The nonclassical method, first introduced by Bluman and Cole [8] (see, for example, [2] or [3]), seeks invariance of a given partial differential equation (PDE) augmented with the invariant surface condition. As the determining equations for these nonclassical symmetries are nonlinear, there seemed to be little hope for this new method; however, with the development of computer algebra systems, the nineties saw a huge explosion of interest as several authors took interest in the nonclassical method and continues today to be an active area of interest (e.g., [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and references within).…”

Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations

“…To facilitate the algorithm, symbolic manipulation programs by computer algebra systems such as Mathematica and Maple have been developed by many researchers [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In solving DTEs, Cartan's exterior forms approach [18], Janet-Riquier theory [8,9], the Gröbner base algorithm [11,12], the Rosenfeld-Gröbner algorithm [14], the Ritt-Kolchin differential algebra method [5], formal power series analysis [6], the characteristic set method [19,20], and so on are used to reduce the DTEs into a simpler form, such as the normal, orthonomic, involutive, and passive forms, so that the reduced system is easily manageable. However, from the point of view of the algorithm, producing DTEs (Equation (5)) from Equation 4for a symmetry is restricted by an imposed requirement on the considered PDEs that the PDEs have to admit a "solved-and-triangular"structure (see a description about the structure of the implementation of the Lie algorithm on pages 372-375 in [16]).…”

Some geometric structures of wave equations on manifolds of `neutral signatures'

“…The symmetry classification problem for a number of wave equations has been studied in flat space [17,18,2,9] and non flat space (non-zero constant curvature) [1,8].…”