A unified explicit form of Bäcklund transformations for generalized hierarchies of KdV equations
In this paper, the authors show that the explicit form of Bäcklund transformations of the KdV equation is universal for the whole hierarchies of KdV equations. These equations are of \((2L+1)th\) order for any integer \(L>0\) and their coefficients may depend on the time variable t. The theorem of permutability and the superposition formulas is obtained in a universal manner. The proof is simpler than that obtained in the conventional way. Hence, the authors obtain a universal expression of solitons whose speed is not generally a constant. Some new facts are presented for \(L>1:\) For instance, the soliton with a large amplitude may have a lower speed. There are solitons which vibrate periodically. The interactions of two solitons may have many modes. Beside the usual case, instead of crossing, there are reflections of solitons or a gluing of solitons. Two solitons may have infinite collisions or keep apart forever.
scientific article; zbMATH DE number 3977560