Extended mKP Hierarchy
|Keywords||Integrable system with self-consistent sources Lax representation extended mKP hierarchy pseudodifferential operator|
Soliton equations can be written as the integrable conditions of a couple of linear problems, which is the most fundamental properties. For example, in the 1-dimensional stationary Schr¨odinger equation, if we suppose that the eigen function’s divergence can be given by a pure differential operator N, then the consistent relation between the Schr¨odinger operator L and the operator N will be written as the time evolution of L equivalent to the commutation of L and N. So it’s important to study the integrable systems in a similar way. One of the fundamental problems in the theory of integrable systems is to look for the nonlinear PDE and a couple of operators Land N, such that the nonlinear PDE is just the consistent relation between the operators L and N; in which condition we call the nonlinear PDE is a integrable systems in Lax’s meaning, while L and N are called a Lax pair.Integrable systems with self-consistent sources have been widely used in physics. In order to construct equations of this type, we need generalize the corresponding operators. It’s known that in a given system characterized by L, the supposed existence of the so-called time-flows is of the first importance; the time evolution of every element in the differential algebra is given by the commutation of some corresponding operator and the common one L. The generalization from GD hierarchies to KP hierarchies, and then to mKP hierarchies is realized by extending the operator L.What’s more, for 2+1 dimensional(two temporal variables and one discrete spatial variable) case in, by using symmetry generating functions and treating the constrained equation as the stationary equation for 2-dimensional hierarchy with sources, we obtained 2-d hierarchy with sources and its Lax representation. Enlightened by the idea reflectded in that paper, we now take a view of the constrained flows as exactly some new time flows; at the same time, we may suppose operators in constraint conditions respectively.With the guidance of the idea just described above, we then analyze the properties of the operators in constraint in the frame of a ring made up of pseudodifferential operators, prove the commutable relation of the new and the original time flows, finally construct the so-called extended mKP hierarchy(emKP), the Lax representation is also obtained. As the examples show, some important known hierarchies can be achieved via choosing certain parameters in the emKP hierarchy.