A new hierarchy of Korteweg–de Vries equations

TLDR

Unlike the standard Lax hierarchy, these new Korteweg–de Vries equations do not conform to the inverse scattering formalism or exhibit typical properties such as Bäcklund transformations. We discovered new Korteweg–de Vries and Boussinesq hierarchies that admit multiple soliton solutions, including an equation with N‑soliton solutions and others with only two‑soliton solutions, exhibit an unusual pattern of conservation laws with every third one missing, and possess bounded oscillating solutions absent in the classical KdV equation.

Abstract

We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the stan­dard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various pro­perties associated with it such as Bäcklund transformations. The most interesting of the new K. de V. equations is ( u nx ≡ ∂ n u /∂ x n ) ( u 4 x + 30 uu 2 x + 60 u 3 ) x + u t = 0. We have proved that this equation has N -soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N -soliton solutions, ( u xx + 6 u 2 ) xx + u xx – u tt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.