Squared Eigenfunctions of the (modiied) Kp Hierarchy and Scattering Problems of Loewner Type
نویسنده
چکیده
It is shown that products of eigenfunctions and (integrated) adjoint eigenfunctions associated with the (modiied) Kadomtsev-Petviashvili (KP) hierarchy form generators of a symmetry transformation. Linear integro-diierential representations for these symmetries are found. For special cases the corresponding nonlinear equations are the compatibility conditions of linear scattering problems of Loewner type. The examples include the 2+1-dimensional sine-Gordon equation with space variables occuring on an equal footing introduced recently by Konopelchenko and Rogers. This equation represents a special squared eigenfunction symmetry of the Ishimori hierarchy. 1 Introduction 2 The generalized Loewner system 3 The multicomponent (modiied) KP hierarchy 3.1 The KP hierarchy: examples 3.2 The modiied KP hierarchy: examples 4 The symmetry generated by squared eigenfunctions 4.1 Results for the KP hierarchy 4.2 Results for the modiied KP hierarchy 5 Linear representation of the squared eigenfunction symmetry 5.1 Results for the KP hierarchy 5.2 Results for the modiied KP hierarchy 6 Scattering problems of Loewner type 6.1 Results for the KP hierarchy 6.2 Results for the modiied KP hierarchy 7 Reductions 7.1 Results for the KP hierarchy 7.2 Results for the modiied KP hierarchy 8 Examples 8.1 The KP hierarchy: examples 8.1 The modiied KP hierarchy: examples 9 Conclusions and perspectives References Appendix
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