Nonclassical Symmetries, Nonlinear Self-adjointness, Conservation Laws and Some New Exact Solutions of Cylindrical KdV Equation

This paper is devoted to obtain some new exact solutions to the cylindrical KdV equation using nonclassical symmetry analysis. Classical Lie point symmetries and nonclassical symmetries are useful when searching for exact solutions to differential equations. The advantage of nonclassical symmetry analysis over classical symmetry analysis is that it is possible to obtain some more new solutions, those cannot be obtained by using usual symmetry analysis, to the given partial differential equation(s). Here, we perform the nonclassical analysis and find nonclassical symmetries of cylindrical KdV equation and obtain some new solutions to the cylindrical KdV equation. These solutions (nonclassical solutions) do not arise as invariant solutions of the cylindrical KdV equation with respect to any of its classical symmetries. It is remarked that, all the previously derived group invariant solutions available in the literature arising from some classical symmetries of this equation have been recovered as some special case of nonclassical solutions. Moreover, we study physical significance of some of the obtained solutions those represent some singularities. Several conservation laws of the cylindrical KdV equation are constructed by using nonlinear self-adjointness property of the equation.