Abstract
In this paper, nonlinear self-adjointness and conservation laws for the variable coefficient combined KdV equation with a forced term are studied. We discuss its self-adjointness and find that the equation is nonlinearly self-adjoint. At the same time, the formal Lagrangian for the equation is obtained. Having performed Lie symmetry analysis for the equation, we derive several nontrivial conservation laws for the equation by using a general theorem on conservation laws, given by Ibragimov.
Highlights
The notion of conservation laws plays an important role in the study of nonlinear science [ – ]
For a self-adjoint nonlinear equation, its adjoint equation is equivalent with the original equation after replacing the nonlocal variable with the dependent variable in the original equation
Ut = –a(t)uux – m(t)u ux – b(t)uxxx + R(t), we obtain a system of over-determined partial differential equations (PDEs) with respect to ξ, τ, and η:
Summary
The notion of conservation laws plays an important role in the study of nonlinear science [ – ]. Equation ( ) is the special case of the equation ut + f (t, u)uxxxxx + r(t, u)uxxx + g(t, u)uxuxx + h(t, u)u x + a(t, u)ux + b(t, u) = , nonlinear self-adjointness for the equation has been considered in [ ], conservation laws of the time dependent KdV equation, In Section , after performing Lie symmetry analysis, nontrivial conservation laws of Eq ( ) are derived making use of the obtained formal Lagrangian and Lie symmetries.
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