Cauchy problems related to integrable deformations of pseudo differential operators

JOURNAL OF GEOMETRY AND PHYSICS

卷:85

Article

Helminck, G. F.¹  Helminck, A. G.²  Panasenko, E. A.³ 

[1] Univ Amsterdam, Korteweg de Vries Inst, NL-1098 XH Amsterdam, Netherlands

[2] N Carolina State Univ, Raleigh, NC 27695 USA

[3] Derzhavin State Univ, Tambov 39200, Russia

关键词: Cauchy problem;    Integrable deformation;    Pseudo differential operators;    (Strict) KP hierarchy;    Zero curvature relations;   

DOI  :  10.1016/j.geomphys.2014.05.004

来源: Elsevier

【 摘 要 】

In this paper we discuss the solvability of two Cauchy problems in the pseudo differential operators. The first is associated with a set of pseudo differential operators of negative order, the prominent example being the set of strict integral operator parts of the different powers of a solution of the KP hierarchy. We show that it can be solved, provided the setting possesses a compatibility completeness. In such a setting all solutions of the KP hierarchy are obtained by dressing with the solution of the related Cauchy problem. The second Cauchy problem is slightly more general and links up with a set of pseudo differential operators of order zero or less. The key example here is the collection of integral operator parts of the different powers of a solution of the strict KP hierarchy. This system is solvable as soon as exponential and compatibility completeness holds. Also under these circumstances, all solutions of the strict KP hierarchy are obtained by dressing with the solution of the corresponding Cauchy problem. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

Free