Variational principles for some nonlinear partial differential equations with variable coefficients
Introduction
Recently variational principles for nonlinear partial differential equations with variable coefficients have come to play an important role in a branch of mathematics, which have served as a basis for development of variety of approximate methods of analysis and various numerical techniques [1], [2]. Khater et al. [2] obtained some variational principles and conservation laws by the Noether’s theorem and one-parameter group of transformation for the generalized Korteweg–de Vries equation and nonlinear Schrödinger’s equation. The derivation is clear but it is a very tedious work. In this paper, we will apply the semi-inverse method [3] to the search for various variational principles and conservation laws for the discussed problems, emphasis is put on the generalized KdV equation and nonlinear Schrödinger’s equation in order to compare with Khater et al.’s results by Noether’s theorem [2].
Section snippets
Semi-inverse method
The semi-inverse method [3] is a powerful tool to the search for various variational principles for physical problems [4], [5], [6], [7], [8], [9] directly from field equations and boundary conditions. For better illustration of the basic procedure of the semi-inverse method, making the underlying idea clear and not darkened by the unnecessarily complicated form of mathematical expressions, we consider a simple example.The equation describes a nonlinear oscillator with damp
The generalized Korteweg–de Vries equation (KdV)
The general form of KdV equation can be expressed as [2]where a, b, c, α, β and d are arbitrary constants. In case α=0, Eq. (9) reduces to that in Ref. [2]. For d=0,1 and 2 correspond respectively to plane, cylindrical and spherical cases in 1+1-dimensions. For different values of a, b, c, α, β and d, Eq. (9) governs different physical phenomena and reduces to one of the well-known forms in the hierarchy of KdV equation [2].
Introducing a potential function v
The variable coefficients nonlinear Schrödinger’s equation
The variable coefficients nonlinear Schrödinger’s equation can be expressed as [2]:where ψ=ψ(x,t) is a complex valued function of x and t, m and n are real numbers and a and b are constant parameters.
On substituting ψ(x,t)=u(x,t)+iv(x,t), where u(x,t) and v(x,t) are real functions of x and t, in Eq. (23), we get:which leads to a system of two second-order equations expressed as
Conclusion
We suggest an attracting method to the search for various variational principles for physical problems. The most interesting features of the proposed method, compared with the Noether’s theorem, are its extreme simplicity and concise results for a wide range of nonlinear problems.
Applying the Lie group transformation methods [2], same results can be obtained with those in Ref. [2] with reduced derivation for the forms of variational functionals obtained by the present theorem are much more
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