# Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

@article{Gandarias1998ClassicalAN, 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}, year={1998}, volume={5}, pages={8-12} }

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.

## 50 Citations

Nonclassical and Potential Symmetries for a Boussinesq Equation with Nonlinear Dispersion

- Mathematics
- 2011

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

- Mathematics
- 2009

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

- Mathematics
- 2015

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

- Mathematics
- 2017

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…

On double reductions from symmetries and conservation laws for a damped Boussinesq equation

- Mathematics
- 2016

Conservation laws for a Boussinesq equation.

- Mathematics
- 2017

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

- MathematicsCurrent 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

- Mathematics
- 2015

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

- Mathematics
- 2001

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…

## References

SHOWING 1-10 OF 13 REFERENCES

Symmetries and differential equations

- Mathematics
- 1981

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

- Mathematics
- 1989

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

- Mathematics
- 1996

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

- Mathematics
- 1987

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

- Mathematics
- 1995

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

- Mathematics
- 1986

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