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Bäcklund transformations for fourth Painlevé hierarchies. (English) Zbl 1080.34073

Summary: Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here, we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé \((P_{\text{IV}})\) hierarchies recently found by the authors [Publ. Res. Inst. Math. Sci. 37, 327–347 (2001; Zbl 0997.35094)]. We show how the known pattern of BTs for \(P_{\text{IV}}\) can be extended to our \(P_{\text{IV}}\) hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water wave hierarchy. We also obtain the important result that the fourth Painlevé equation has only one nontrivial fundamental BT, and not two such as is frequently stated.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
33E17 Painlevé-type functions
35Q51 Soliton equations
35Q58 Other completely integrable PDE (MSC2000)

Citations:

Zbl 0997.35094
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References:

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