Recursion Operator for a Constrained Bkp System
Recursion Operator for a Constrained Bkp System
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DOI：
10.1142/9789812817587_0045
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发表时间：
2000
期刊：
影响因子：
1.7
通讯作者：
I. Loris;M. Boiti;L. Martina;F. Pempinelli;B. Prinari;G. Soliani
中科院分区：
文献类型：
--
作者：
I. Loris;M. Boiti;L. Martina;F. Pempinelli;B. Prinari;G. Soliani
The 2+ 1 dimensional Kadomtsev-Petviashvili (KP) hierarchy and its B-, C-, multi-component, etc. variants contain, by way of dimensional reductions, many (if not all) examples of 1+ 1 dimensional integrable systems. The KdV equation can eg be obtained from a standard Ot₂u= 0 reduction of KP; the NLS Schrödinger equation can be obtained via a symmetry constraint on KP. In this note, I shall discuss a symmetry constraint of the BKP hierarchy. The BKP hierarchy is obtained from the standard KP hierarchy by imposing an extra condition between the Lax operator and its adjoint ¹. A wellknown standard reduction of this hierarchy is the Sawada-Kotera equation2. The symmetry constraint on which this note focuses, is introduced with the help of a special'eigenfunction symmetry'defined in terms of a pair of BKP eigenfunctions³. This symmetry can be shown to be expressible as a ratio of two BKP ta u-functions. The result of the symmetry constraint (constraining the eigenfunction symmetry to the translational symmetry of the BKP equation) is a third order system in two fields. It contains as a special case the KdV equation (showing that the reduction of 1+ 1 dime nsional systems from KP type hierarchies is not unique). A Hirota (bilinear) form for the reduced. system is given.The last section is used to derive a recursion operator for the 1-constrained BKP system. With this operator, the higher order flows of the corresponding hierarchy can be easily generated. The adjoint of the recursion operator is used to generate a sequen ce of conserved covariants, from which conservation laws are derived (not all conservation laws can be found in this manner). The story starts with the introduction of the KP hierarchy. The KP hierarchy1, 4 can be introduced with the help of the pseudodifferential Lax operator L=+ µ2Ə− ¹+ uçƏ− ²+.... On this operator λ
