Self-adjoint sub-classes of third and fourth-order evolution equations

doi:10.1016/j.amc.2011.04.041

Applied Mathematics and Computation

Volume 217, Issue 22, 15 July 2011, Pages 9467-9473

https://doi.org/10.1016/j.amc.2011.04.041 Get rights and content

Abstract

In this work a class of self-adjoint quasilinear third-order evolution equations is determined. Some conservation laws of them are established and a generalization on a self-adjoint class of fourth-order evolution equations is presented.

Introduction

In this paper we consider the problem on self-adjointness condition of equation $u_{t} = r (u) u_{xxx} + p (u) u_{xx} + q (u) u_{x}^{2} + a (u) u_{x} + b (u),$ where $r, p, q, a, b : R \to R$ are arbitrary smooth functions.

Eq. (1) includes important evolution equations employed in mathematical physics and in mathematical biology, for instance, inviscid Burgers equation, Burgers equation, potential Burgers equation, Fisher equation, Korteweg-de Vries (KdV) equation, Gardner equation and so on, see [4], [5], [9]. It can be used to describe shallow watter waves, collisionless-plasma magnetohydrodynamics waves and ion acoustic waves, among other physical or biological phenomena, see also [12], [15].

Similar work has been performed by Bruzón et al. [3] regarding equation $u_{t} + f (u) u_{xxxx} + g (u) u_{x} u_{xxx} + h (u) u_{xx}^{2} + d (u) u_{x}^{2} u_{xx} - p (u) u_{xx} - q (u) u_{x}^{2} = 0 .$ However, in (2) source terms and nonlinearities type a(u)u_x and r(u)u_xxx were not taken into account. So we shall complement the results previously obtained by them including these terms.

Ibragimov [7] has recently established a new conservation theorem for equations without Lagrangians. If (1) is self-adjoint it is possible to construct conservation laws D_tC⁰ + D_xC¹ = 0 for it, where the components C⁰ and C¹ depend on t, x, u and its derivatives.

The purpose of this paper is to determine the self-adjoint equations type (1) and, by using the recent result [7], establish some nontrivial conservation laws for some of these equations. The results on self-adjointness condition of Eq. (2) obtained in [3] is also generalized by including dispersive, convective and source terms.

The paper is organized as the follows: in the Section 2 we revisit some results regarding Lie point symmetries and conservation laws for differential equations. Section 3 is devoted to find the self-adjoint equations type (1). We comment some results presented in [3] in Section 4.

Section snippets

Preliminaries

This section contains a brief discussion on the space of differential functions $A$ , Lie–Bäcklund operators, self-adjoint equations and conservation laws for differential equations. For more details, see [6], [3], [7], [8]. In the following the summation over repeated indices is understood.

Let x = (x¹, … , xⁿ) be n independent variables and u = (u¹, … , u^m) be m dependent variables with partial derivatives $u_{i}^{α} = \frac{\partial u^{α}}{\partial x^{i}}, u_{ij}^{α} = \frac{\partial^{2} u^{α}}{\partial x^{i} \partial x^{j}}$ , etc. The total differentiation operators are given by $D_{i} = \frac{\partial}{\partial x^{i}} + u_{i}^{α} \frac{\partial}{\partial u^{α}} + u_{ij}$

The class of self-adjoint equations type (1)

By applying the Euler–Lagrange operator (8) to (12), where F is given by (3) and equating to 0, we obtain the adjoint equation to (1).

Supposing that F is self-adjoint, Eq. (9) holds, for some $ϕ \in A$ . Comparing the coefficient of u_t, we obtain ϕ = −1 and $3 r^{″} + {ur}^{‴} = 0, 3 {ur}^{″} + 6 r^{'} = 0, - {up}^{'} - p = - uq, {uq}^{'} - 2 p^{'} - {up}^{″} + q = 0, {ub}^{'} = - b .$ Solving the system (14), we obtain $r = a_{1} + \frac{a_{2}}{u},$ $q = \frac{(up)^{'}}{u}$ and $b = \frac{a_{3}}{u},$ where a₁, a₂ and a₃ are arbitrary constants.

The following theorem is proved.

Theorem 1

Eq. (1) is self-adjoint if and only if it has the form $u_{t} = (a_{1} + a)$

Self-adjoint adjoint equations of fourth-order

Concerning Eq. (2), in [3] is proved that Eq. (2) is self-adjoint if and only if $g = h + \frac{1}{u} (uf)^{'}, d = \frac{c_{1}}{u} + \frac{1}{u} (uh)^{'}$ and $q = \frac{1}{u} [c_{2} + (up)^{'}],$ where f, h and p are arbitrary functions of u (see [3], Theorem 3.2, p.p 310).

From Theorem 1 we conclude that Eqs. (27), (16) cannot be compatible whenever c₂ ≠ 0. In fact, the correct statement is.

Theorem 2

Eq. (2) is self-adjoint if and only if g and d are given by (26) and q is given by (16), where f, h and p are arbitrary functions of u and c₁ is an arbitrary constant.

Proof

From the

Conclusions

In this paper the self-adjoint subclasses of Eq. (1) was obtained. Thanks to the recently proposed conservation theorem due to Ibragimov, some conservation laws of particular self-adjoint equations type (18) were established. Further examples can be found in [3], [7], [9], [14].

A comment in a recently published result (see [3], Theorem 3.2) was given in Section 4. In particular the self-adjointness condition obtained by Bruzón, Gandarias and Ibragimov to Eq. (2) was generalized to Eq. (34). Eq.

Acknowledgement

The author would like to thank the referees for reading carefully this paper and for useful comments.

References (15)

Y. Bozhkov et al.
Special conformal groups of a riemannian manifold and the Lie point symmetries of the nonlinear Poisson equations
J. Diff. Eqn.
(2010)
M.S. Bruzón et al.
Self-adjoint sub-classes of generalized thin film equations
J. Math. Anal. Appl.
(2009)
N.H. Ibragimov
A new conservation theorem
J. Math. Anal. Appl.
(2007)
I.L. Freire
On the paper “Symmetry analysis of wave equation on sphere” by H. Azad and M.T. Mustafa
J. Math. Anal. Appl.
(2010)
R. Naz et al.
Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics
Appl. Math. Comput.
(2008)
C. Qu
Symmetries and solutions to the thin film equations
J. Math. Anal. Appl.
(2006)
E. Yasar
On the conservation laws and invariant solutions of the mKdV equation
J. Math. Anal. Appl.
(2010)

There are more references available in the full text version of this article.

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Self-adjoint sub-classes of third and fourth-order evolution equations

Abstract

Introduction

Section snippets

Preliminaries

The class of self-adjoint equations type (1)

Self-adjoint adjoint equations of fourth-order

Conclusions

Acknowledgement

J. Diff. Eqn.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Appl. Math. Comput.

J. Math. Anal. Appl.

J. Math. Anal. Appl.