The special class of second integrals of the KdV equation

Abstract The special class of second integrals, distinguished for its infinite dimension, emerges naturally when generalizing the second integrals of ordinary differential equations to partial differential equations. The conserved quantities of the KdV equation are a special class of second integrals. We proved its uniqueness under the assumption on the cofactor operator. The special class is so peculiar that to find it is almost an algorithm. Thus we managed to generalize the special class of the conserved quantities of the KdV equation to a new 2-parameter special class. Among the special classes, the special class of nonlocal second integrals plays an extra role. As an example, the special class that generates the multi-soliton solutions of the KdV equation is presented.

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