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Bäcklund and Darboux Transformations
2002This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on transformations of privileged ...
C. Rogers, W. K. Schief
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Higher Rank Darboux Transformations
1994The Darboux transformation has been discovered several times in history (cf. references in [EK] and in [G3]): the reason for this is its deep geometric significance. In this note we explore some geometric aspects of the transformation and their relevance to the difficult problems of describing explicitly higher-rank commutative algebras of ordinary ...
Geoff Latham, Emma Previato
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On Darboux and singular Darboux transformations ofG-unitons
Science China Mathematics, 2002We present a unified method for constructing generalized flag transformations of unitons into a Lie groupG via singular Darboux transformations. These flag transformations provide the possibility to establish some factorizations forG-unitons.
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Bäcklund Transformations and Darboux Transformations
1995In 1883, the German geometer A. V. Backlund found an interesting property of the sine-Gordon equation in studying surfaces of constant negative curvature.[2]
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Darboux Transformation of the Laguerre Operator
Complex Analysis and Operator Theory, 2018For \(\alpha>-1\), let \(L_n(x,\alpha)\), \(n\) a non-negative integer, denote the classical Laguerre polynomials orthogonal in \(L^2(\mathbb R_+,w_\alpha)\), where \(w_\alpha(x)=x^\alpha \cdot e^{-x}\). For \(\alpha>-1\), the classical Laguerre operator \(L_\alpha\) is defined on \(L^2(\mathbb R_+,w_\alpha)\) by \[ L_\alpha(y):= xy^{\prime\prime ...
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Double Darboux Transformation for AKNS hierarchy
Acta Mathematica Sinica, 1990Summary: In this paper we construct a double Darboux Transformation for AKNS hierarchy and give its decomposition theorem. A remarkable characteristic of the double Darboux Transformation - incommutativity - is proved.
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