Intertwining Laplace Transformations of Linear Partial Differential Equations

Ganzha, E. I.

doi:10.1007/978-3-642-54479-8_4

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…As Type I Darboux transformations do, the transformations of Wronskian type of order one fit the framework developed by E.I. Ganzha in [15]. Given L = CM + cB with [M, B] = 0, we initially assume cB = 0, and write L = X 1 X 2 − H by taking X 1 = C, X 2 = M and H = −cB.…”

Section: Classif Ication Of Darboux Transformations Of F Irst Ordermentioning

confidence: 99%

“…Intertwining Laplace transformations depend on writing L in a particular form, where L = X 1 X 2 − H. In our notation, the definition from [15] then gives M = X 2 , ω = −[X 2 , H]H −1 and N = X 2 +ω, where we would like ω to be a differential operator and not merely pseudo-differential. Substituting M = M 1 for X 2 , we have L = A 1 M 1 + M 2 = X 1 X 2 − H. So it would be natural to try setting X 1 = A 1 , giving H = −M 2 .…”

Section: Continued Type I Darboux Transformationsmentioning

confidence: 99%

“…Laplace transformations have a number of good properties including invertibility. Recently, two different generalizations of Laplace transformations were proposed: "intertwining Laplace transformations" in [15] and "Type I transformations". (The latter are always invertible.)…”

Section: Introductionmentioning

confidence: 99%

“…Intertwining Laplace transformations. These were introduced in [15], and generalize Laplace transformations to linear partial differential operators L ∈ K[∂ t , ∂ x 1 , . .…”

mentioning

confidence: 99%

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Classification of Multidimensional Darboux Transformations: First Order and Continued Type

Hobby¹,

Shemyakova²

2017

SIGMA

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Abstract. We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first-order Darboux transformations. We introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, which was previously introduced. There is also a modification of this type of Darboux transformations, continued Wronskian type, which generalize Wronskian type Darboux transformations.

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Section: Classif Ication Of Darboux Transformations Of F Irst Ordermentioning

confidence: 99%

Section: Continued Type I Darboux Transformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Intertwining Laplace transformations. These were introduced in [15], and generalize Laplace transformations to linear partial differential operators L ∈ K[∂ t , ∂ x 1 , . .…”

mentioning

confidence: 99%

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Classification of Multidimensional Darboux Transformations: First Order and Continued Type

Hobby¹,

Shemyakova²

2017

SIGMA

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“…In [5,6], a generalization for the case of second order operators in three variables is obtained. The idea of generalization of the Darboux transformations in terms of pseudo differential opera tors was suggested in [7].…”

Section: Introductionmentioning

confidence: 99%

Invertible Darboux transformations of type I

Shemyakova

2015

Program Comput Soft

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In the paper, a general category based approach to invertible Darboux transformations is sug gested. For operators of arbitrary dimension and arbitrary order, it is suggested to consider Darboux transfor mations of certain type, namely, of type I. Explicit compact formulas are obtained. In particular, any invert ible Darboux transformations of first order and, for example, Laplace transformations are type I transforma tions. For operators of third order in the plane, analogues of the Darboux transformation chains are obtained.

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Orbits of Darboux Groupoid for Hyperbolic Operators of Order Three

Shemyakova

2015

Trends in Mathematics

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Darboux transformations are viewed as morphisms in a Darboux category. Darboux transformations of type I which we defined previously, make an important subgroupoid consists of Darboux transformations of type I. We describe the orbits of this subgroupoid for hyperbolic operators of order three.We consider the algebras of differential invariants for our operators. In particular, we show that the Darboux transformations of this class can be lifted to transformations of differential invariants (which we calculate explicitly). (2010). Primary 70H06; Secondary 34A26. J 1 = I 1 − T x , J 2 = I 2 + T xy , J 3 = I 3 + I 2 + T xy , J 4 = 0 , J 5 = f − I 4y /2 + I 3x − I 2y + T xxy /2 , where f = I 5 − I 1 I 2 − I 4y /2, T = ln(f ). Here f = 0 and L = CM + f for some C ∈ K[D]. First results on orbits of Darboux groupoid With M of the form M = ∂ y + m: J 1 = I 1 − T y , J 2 = I 2 − T xy , J 3 = 0 , J 4 = I 4 − I 2 + T xy , J 5 = I 1 I 2 − I 1 T xy − I 2 T y − I 3x /2 + I 2x + I 4y + T xyy /2 + T xy T y + f , where f = I 5 − I 3x , T = ln(f ). Here f = 0 and L = CM + f for some C ∈ K[D]. With M of the form M = ∂ x + ∂ y + m: J 1 = I 1 + T x + T y , J 2 = I 2 , J 3 = −I 4 + 2I 3 + I 1y + T xy + T yy , J 4 = I 1x + 2I 4 − I 3 + T xx + T xy , J 5 = I 4y + I 3x + I 1xy /2 − I 1 I 4 − T x I 4 − T y I 4 + T xxy /2 + T xyy /2 + f , Mathematics Subject Classification

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Intertwining Laplace Transformations of Linear Partial Differential Equations

Cited by 7 publications

References 14 publications

Classification of Multidimensional Darboux Transformations: First Order and Continued Type

Classification of Multidimensional Darboux Transformations: First Order and Continued Type

Invertible Darboux transformations of type I

Orbits of Darboux Groupoid for Hyperbolic Operators of Order Three

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