Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type equations and nonlinear Schrödinger equations

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Abstract
Group theoretic properties of nonlinear time evolution equations have been studied from the standpoint of a generalized Lie transformation. It has been found that with each constant of motion of the KdV type equation fxxx+a (f) fx+ft=0 and of the coupled nonlinear Schrödinger equation fxx +a (f,g)+ift=0, gxx+a (g,f) −igt=0 one invariance group of the equations is always associated. The well‐known series of constants of motion of the KdV equation and the cubic Schrödinger equation will be recovered from the invariance groups of the equations. The doublet solution of the KdV equation will be characterized as the invariant solution of one of the groups. In a more general context, it will be shown that the well‐known equation of quantum mechanics (d/dt) 〈U〉=〈[iH,U] +∂U/∂t〉 can be generalized to a class of nonlinear time evolution equations and that if U is a generator of an invariance group of the equation then (d/dt) 〈U〉=0. The class includes equations such as the KdV, the cubic Schrödinger, and the Hirota equations.

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