Sato–Bäcklund transformations and string equations of the mKP hierarchy

2019 ◽  
Vol 34 (25) ◽  
pp. 1950142 ◽  
Author(s):  
Huizhan Chen ◽  
Lumin Geng ◽  
Jipeng Cheng

Additional symmetry is an important kind of symmetries depending explicitly on the time and space variables, which can be expressed through Sato–Bäcklund transformations. In this paper, we construct Sato–Bäcklund transformations of the modified KP hierarchy and its constrained cases. Then the string equations of the [Formula: see text]-reduced modified KP hierarchy are established by requiring the system independent on some additional symmetry flows, which are expressed by the Lax operator [Formula: see text] and the Orlov–Shulman’s operator [Formula: see text]. At last, we obtain the negative Virasoro constraint on the two tau functions of the 2-reduced modified KP hierarchy satisfying the string equations.

2018 ◽  
Vol 32 (17) ◽  
pp. 1850181 ◽  
Author(s):  
Haifeng Wang ◽  
Chuanzhong Li

In this paper, in order to generalize the Painlevé equations, we give a [Formula: see text]-Painlevé IV equation which can apply Bäcklund transformations to explore. And these Bäcklund transformations can generate new solutions from seed solutions. Similarly, we also introduce a Frobenius Painlevé I equation and Frobenius Painlevé III equation. Then, we find the connection between the Frobenius KP hierarchy and Frobenius Painlevé I equation by the Virasoro constraint. Further, in order to seek different aspects of Painlevé equations, we introduce the Lax pair, Hirota bilinear equation and [Formula: see text] functions. Moreover, some Frobenius Okamoto-like equations and Frobenius Toda-like equations can also help us to explore these equations.


1993 ◽  
Vol 08 (03) ◽  
pp. 507-543 ◽  
Author(s):  
OLIVIER BABELON ◽  
DENIS BERNARD

We reconsider the construction of solitons by dressing transformations in the sine-Gordon model. We show that the N-soliton solutions are in the orbit of the vacuum, and we identify the elements in the dressing group which allow us to build the N-soliton solutions from the vacuum solution. The dressed τ functions can be computed in two different ways: either using adjoint actions in the affine Lie algebra [Formula: see text], and this gives the relation with the Bäcklund transformations, or using the level-one representations of the affine Lie algebra [Formula: see text], and this directly gives the formulae for the τ functions in terms of vertex operators.


1993 ◽  
Vol 08 (20) ◽  
pp. 3457-3478 ◽  
Author(s):  
SUDHAKAR PANDA ◽  
SHIBAJI ROY

We show directly in the Lax operator approach how the Virasoro and W-constraints on the τ-function arise in the p-reduced KP hierarchy or generalized KdV hierarchy. In particular, we consider the KdV and the Boussinesq hierarchy to show that the Virasoro and the W-constraints follow from the string equation by expanding the "additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies.


1993 ◽  
Vol 182 (1) ◽  
pp. 71-76 ◽  
Author(s):  
Yi Cheng ◽  
Walter Strampp ◽  
You-Jin Zhang

2017 ◽  
Vol 72 (4) ◽  
pp. 331-337 ◽  
Author(s):  
Zhao-Wen Yan

AbstractThe Heisenberg supermagnet model is an important supersymmetric integrable system in (1+1)-dimensions. We construct two types of the (2+1)-dimensional integrable Heisenberg supermagnet models with the quadratic constraints and investigate the integrability of the systems. In terms of the gage transformation, we derive their gage equivalent counterparts. Furthermore, we also construct new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.


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