Abstract
Approximate symmetries, which are admitted by the perturbed KdV equation, are obtained. The optimal system of one-dimensional subalgebra of symmetry algebra is obtained. The approximate invariants of the presented approximate symmetries and some new approximately invariant solutions to the equation are constructed. Moreover, the conservation laws have been constructed by using partial Lagrangian method.
Highlights
The partial differential equations (PDEs) with small parameter have been arisen in mathematics, physics, mechanics, etc
The approximate symmetry method, which is developed by Baikov, Gazizov, and Ibragimov [3, 4] in the 1980s, shows an effectiveness to obtain approximate solutions to a perturbed PDEs
The new method maintains the essential features of the standard Lie symmetry method and provides us with the most widely applicable technique to find the approximate solutions to a perturbed differential equations [5, 6]
Summary
The partial differential equations (PDEs) with small parameter have been arisen in mathematics, physics, mechanics, etc For these perturbed equations, the finding of analytic solutions, conservation laws, symmetries, etc. One approach, based on generalizations of the Noether’s theorem to perturbed equations, is given to get approximate conservation laws via the approximate Noether symmetries associated with a Lagrangian of the perturbed equations [5, 9] This method depends on the existence of the Lagrangian functional for underlying differential equations. Partial Lagrangian method can construct approximate conservation laws of perturbed equations via approximate operators that are not necessarily approximate symmetry operators of the underlying system of equations. This method is used for more general equations as even the equations do not admit essential Lagrangian.
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