Symmetry Analysis and Solutions for a Generalization of a Family of BBM Equations

Abstract

In this paper, the family of BBM equation with strong nonlinear dispersive B(m, n) is considered. We apply the classical Lie method of infinitesimals. The symmetry reductions are derived from the optimal system of subalgebras and lead to systems of ordinary differential equations. We obtain for special values of the parameters of this equation, many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions (soliton, kink and compactons).

Nonlinearly Self-Adjoint, Conservation Laws and Solutions for a Forced BBM Equation

“…In [5], it was proved that these conservation laws are the only conservation laws admitted by the BBM equation. In [6], a family of BBM equations with strong nonlinear dispersive term was considered from the point of view of symmetry analysis. The symmetry reductions were derived from the optimal system of subalgebras and lead to systems of ordinary differential equations.…”

Traveling wave solutions of the BBM-like equations

“…These kinds of nonlinear equations have recently attracted much attention due to the existence of solitary wave solutions with compact support called compactons. In the literature, there are many studies dealing with the traveling wave solutions of the K(m, n) and B(m, n) equations, but in general they are restricted to the case m = n [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Our aim here is just to search for the traveling wave solutions of the B(m, n) equations, with m = n, by means of the factorization technique [17][18][19][20][21].…”

“…We remark that this method allows us to get a wider set of solutions, compared with other methods, such as the sine-cosine and the tanh methods, the extended Riccati equations method, Jacobian elliptic function expansion method [11,12,[22][23][24], used to solve BBM equations. In [14], the symmetry reductions of B(m, n) equations have been derived and also some exact solutions have been obtained for special values of the parameters. It is clear that one of the symmetry reductions of B(m, n) supplied in [14] coincides with the traveling wave reduction of the B(m, n) equations.…”

“…In [14], the symmetry reductions of B(m, n) equations have been derived and also some exact solutions have been obtained for special values of the parameters. It is clear that one of the symmetry reductions of B(m, n) supplied in [14] coincides with the traveling wave reduction of the B(m, n) equations. However, here we will get a wider class of solutions, for this reduction, than those presented in [14].…”

Similarity Reductions of a Generalized Double Dispersion Equation

“…Now we will determine the exponents and coefficients of equations (7), following the method described in [2]. So that equation (7) is solvable in terms of Jacobi elliptic function, we compare the exponents and the coefficients of equations (7) and (6).…”