Deformation of Hamiltonian operator and its classifications. I (Q2140127)

By means of linear transformations of dependent and independent variables, the authors study a classification problem of integrable generalizations of the KdV, the modified KdV and the Boussinesq equations, based on characteristics of Hamiltonian pairs. The so-called border equation is introduced to isolate different classes of integrable equations from generalized ones. The main result is the formulation of three classes of Hamiltonian operators containing arbitrary constants, which cover the KdV, the modified KdV and the Boussinesq equations. A general case of the AKNS Hamiltonian pair, which involves dependent variables nonlinearly, is also discussed. More generally, multiple component KdV cases are considered in [\textit{W. Ma}, Acta Math. Appl. Sin. 13, No. 4, 484--496 (1990; Zbl 0725.58020)] and a general two-component case, similar to the Boussinesq system case is given in [\textit{X. Gu} and \textit{W.-x. Ma}, Math. Methods Appl. Sci. 41, No. 10, 3779--3789 (2018; Zbl 1397.35004)]. A detailed study on \(\bar u\)-nonlinear Hamiltonian operators is contained in [\textit{W. Ma}, J. Shanghai Jiaotong Univ. 21, No. 5, 99--108 (1987; Zbl 0661.58037)].