A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation
In the first part of the paper, a generalized AKNS hierarchy is constructed which includes the kdV, mKdV equations as well as certain systems of these. The construction starts from a \(4\times 4\) spectral problem \(\phi_x = U\phi\) on the real line, where \(U=U(u,\lambda)\), \(\lambda\) is the spectral parameter and \(u=(u_1, \dots, u_4)\) is the potential. A sequence of operators \(V^{(n)}\) is found with the corresponding time-dependent problems \(\phi_{t_n}=V^{(n)}\phi\) so that the compatibility conditions of each time-dependent problem and the spectral problem represent completely integrable equations for \(u\). Special choices of the parameters and the symmetry reductions lead to the kdV, mKdV equations as well as systems of these.NEWLINENEWLINEIn the second part, a Darboux transformation is found for the generalized AKNS hierarchy using a gauge transformation with a Darboux matrix. With the help of this transformation, solutions of the hierarchy can be generated. The paper provides several explicit solutions. These include a periodic solution (in both time and space) of the mKdV, which the author calls a `positon', as well as a `negaton' and a `complexiton' solution.
scientific article; zbMATH DE number 6056882
scientific article