Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

  title={Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation},
  author={Mar{\'i}a Luz Gandarias and M. S. Bruz'on},
  journal={Journal of Nonlinear Mathematical Physics},
We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f (u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived. 

Tables from this paper

Nonclassical and Potential Symmetries for a Boussinesq Equation with Nonlinear Dispersion
In this paper we consider a generalized Boussinesq equation which includes nonlinear dispersion. For this equation nonclassical and potential symmetries are derived. We prove that the nonclassical
Exact solutions of a generalized Boussinesq equation
We analyze a generalized Boussinesq equation using the theory of symmetry reductions of partial differential equations. The Lie symmetry group analysis of this equation shows that the equation has
Classical and nonclassical symmetries and exact solutions for a generalized Benjamin equation
We apply the Lie-group formalism to deduce symmetries of a generalized Benjamin equation. We make an analysis of the symmetry reductions of the equation. In order to obtain travelling wave
Symmetries and conservation laws of a KdV6 equation
In the present work we make an analysis of the Korteweg-de Vries of sixth order. We apply the classical Lie method of infinitesimals and the nonclassical method, due to Bluman and Cole, to deduce new
Conservation laws for a Boussinesq equation.
Abstract In this work, we study a generalized Boussinesq equation from the point of view of the Lie theory. We determine all the low-order conservation laws by using the multiplier method. Taking
Symmetry Analysis and Conservation Laws for Some Boussinesq Equations with Damping Terms
  • M. Gandarias, M. Rosa
  • Mathematics
    Current Trends in Mathematical Analysis and Its Interdisciplinary Applications
  • 2019
In this work, we study some Boussinesq equations with damping term from the point of view of the Lie theory. We derive the classical Lie symmetries admitted by the equation as well as the reduced
Group classification and exact solutions of generalized modified Boussinesq equation
We present symmetry classification and exact solutions of generalized modified Boussinesq (GMB) equation. The direct method of group classification is utilized to determine four different functional
Explicit Solutions of Generalized Nonlinear Boussinesq Equations
By considering the Adomian decomposition scheme, we solve a generalized Boussinesq equation. The method does not need linearization or weak nonlinearly assumptions. By using this scheme, the


Symmetries and differential equations
The knowledge of the maximal Lie group or abstract monoid of symmetries of an ordinary non-singular differential equation (or system of equations) allows us to obtain solutions of them. Traditional
New similarity reductions of the Boussinesq equation
Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is
Classical point symmetries of a porous medium equation
The Lie-group formalism is applied to deduce symmetries of the porous medium equation . We study those spatial forms that admit the classical symmetry group. The reduction obtained from the optimal
Group-invariant solutions of differential equations
We introduce the concept of a weak symmetry group of a system of partial differential equations, that generalizes the “nonclassical” method introduced by Bluman and Cole for finding group-invariant...
Instability and blow-up of solutions to a generalized Boussinesq equation
In this paper we investigate conditions for the finite-time blow-up of solutions of the generalized Boussinesq equation (BQ) \[ u_{tt} - u_{xx} + (f(u) + u_{xx} )_{xx} = 0,\quad x \in {\bf R},t > 0.
Applications of lie groups to differential equations
1 Introduction to Lie Groups.- 1.1. Manifolds.- Change of Coordinates.- Maps Between Manifolds.- The Maximal Rank Condition.- Submanifolds.- Regular Submanifolds.- Implicit Submanifolds.- Curves and
J.Phys.A: Math. Gen
  • J.Phys.A: Math. Gen
  • 1996
SIAM J. Appl. Math
  • SIAM J. Appl. Math
  • 1987
Phys. Lett. A
  • Phys. Lett. A
  • 1986