Bäcklund transformations

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1: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

2: 32.7 Bäcklund Transformations

§32.7 Bäcklund Transformations

… ►With the exception of ${\text{P}}_{\text{I}}$, a Bäcklund transformation relates a Painlevé transcendent of one type either to another of the same type but with different values of the parameters, or to another type. … ►

3: 18.38 Mathematical Applications

… ►It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. …

4: 32.8 Rational Solutions

… ► ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … ►Rational solutions of ${\text{P}}_{\text{II}}$ exist for $\alpha =n(\in \mathbb{Z})$ and are generated using the seed solution $w(z;0)=0$ and the Bäcklund transformations (32.7.1) and (32.7.2). …

5: 32.10 Special Function Solutions

… ►Solutions for other values of $\alpha$ are derived from $w(z;\pm \frac{1}{2})$ by application of the Bäcklund transformations (32.7.1) and (32.7.2). …

6: Bibliography B

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Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.

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External Links:

ISSN 0022-2526, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.10(iv), §32.2(iii), §32.7(iv), §32.8(iv).

Encodings:

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7: Bibliography M

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Bäcklund transformations and solution hierarchies for the third Painlevé equation. Stud. Appl. Math. 98 (2), pp. 139–194.

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External Links:

ISSN 0022-2526, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.10(iii), §32.7(iii), §32.8(iii), §32.9(i).

Encodings:

BibTeX

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