Similarity Reductions of a Generalized Double Dispersion Equation

Abstract

We apply the Lie-group formalism to deduce symmetries of a generalized double dispersion equation. We derive the ordinary differential equation to which the equation is reduced. We obtain exact solutions which can be expressed by various single and combine nondegenerative Jacobi elliptic function solutions and their degenerative solutions.

Conservation laws for a Boussinesq equation.

“…The symmetries were determined by using a direct method and cannot be obtained by using the Lie group method for finding group-invariant solutions. Bruzón and Gandarias applied the theory of groups transformations and the nonclassical method to derive exact solutions of some Boussinesq equations [9][10][11][12][13]. Symmetry reductions and exact solutions have several different important applications in the context of differential equations.…”

Lie Symmetry Reductions and Exact Solutions of a Multidimensional Double Dispersion Equation

“…In [37], the authors applied the Lie-group formalism to deduce symmetries of a generalized double dispersion equation…”

“…So that Equation (23) is solvable in terms of Jacobi elliptic function, following the method described in [37]. …”

Exponential Attractor for the Boussinesq Equation with Strong Damping and Clamped Boundary Condition

“…When < 0, (4) is called "good" Boussinesq equation, when > 0, (4) is called "bad" Boussinesq equation. There have been lots of research on the well-posedness, blowup, and other properties of solutions for both the "good" and the "bad" Boussinesq equation of type (1) (see [2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein). While for the investigation on the global attractor to (1), one can see [15][16][17][18][19] and references therein.…”

“…with Ω ⊂ R 2 and the clamped boundary condition (2). Here > 0 is the damping parameter and the mapping 0 : R 2 → R 2 and the smooth functions 1 and 2 represent (nonlinear) feedback forces acting upon the plate, in particular, 2 Discrete Dynamics in Nature and Society relaxation time) sufficiently small, (5) becomes the modified Cohn-Hilliard equation + − Δ (−Δ + ( )) = , (7) which is proposed by Galenko et al [22][23][24] to model rapid spinodal decomposition in nonequilibrium phase separation processes.…”