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ISRN High Energy Physics
Volumeย 2012ย (2012), Article IDย 503621, 12 pages
http://dx.doi.org/10.5402/2012/503621
Research Article

On the Algebra of ๐‘ž-Deformed Pseudodifferential Operators

Dรฉpartement de Physique, Laboratoire des Hautes Energies, Sciences de l'Ingรฉnierie et Rรฉacteurs (LHESIR), Facultรฉ des Sciences, Universitรฉ Ibn Tofail, Kรฉnitra, Morocco

Received 14 December 2011; Accepted 4 January 2012

Academic Editor: A.ย Belhaj

Copyright ยฉ 2012 Abderrahman EL Boukili and Moulay Brahim Sedra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

While basing on the study that we we achieved on pseudodifferential operators in the works [arXiv:0708.4046 and hep-th/0610056 ], we interest in this paper to the construction of the algebra of ๐‘ž-deformed pseudodifferential operators. We use this algebraic structure to study in particular ๐‘ž-Burgers and ๐‘ž-KdV differential operators by the Lax generating technique. We give ๐‘ž-deformed Lax equations as well as the report between these equations through the ๐‘ž-deformed Burgers-KdV mapping.

1. Basic Notions

1.1. ๐‘ž-Pseudodifferential Operators

We start this part with defining the ๐‘ž-derivation. For it, we are going to introduce the general case to know the ๐›ผ-derivation that is defined by๐œ•๐›ผ(๐‘“๐‘”)=๐›ผ(๐‘“)๐‘‘๐›ผ(๐‘”)+๐‘‘๐›ผ(๐‘“)๐‘”,(1.1) where the two functions ๐‘“ and ๐‘” are polynomials in an indeterminant ๐‘ฅ and its inverse ๐‘ฅโˆ’1.

In (1.1), ๐›ผ is a linear mapping. An example of the ๐›ผ-derivation is given by Jackson's ๐‘ž-differential operator ๐œ•๐‘ž, such as [1]๐œ•๐‘ž(๐‘“)=๐‘“(๐‘ž๐‘ฅ)โˆ’๐‘“(๐‘ฅ),(๐‘žโˆ’1)๐‘ฅ(1.2) which gives the following form for (1.1):๐œ•๐‘ž(๐‘“๐‘”)=๐œ‚๐‘ž(๐‘“)โ‹…๐œ•๐‘ž(๐‘”)+๐œ•๐‘ž(๐‘“)โ‹…๐‘”.(1.3)

The ๐‘ž-shift operator ๐œ‚๐‘ž is given by๐œ‚๐‘ž(๐‘“(๐‘ฅ))=๐‘“(๐‘ž๐‘ฅ).(1.4)

One can define the commutation relation as follows:[]๐‘“,๐‘”=๐‘“โˆ˜๐‘”โˆ’๐‘”โˆ˜๐‘“,(1.5) where the multiplication law โ€œโˆ˜โ€ is๐œ•๐‘žโˆ˜๐‘“=๐œ‚๐‘ž(๐‘“)๐œ•๐‘ž+๐œ•๐‘ž๐œ•๐‘“,๐‘žโˆ’1โˆ‘โˆ˜๐‘“=๐‘˜โชฐ0(โˆ’1)๐‘˜๐‘žโˆ’๐‘˜(๐‘˜+1)/2๐œ‚๐‘žโˆ’๐‘˜โˆ’1๎€ท๐œ•๐‘˜๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’๐‘˜โˆ’1.(1.6) The last equation are obtained by using the following relation:๐œ•๐‘žโˆ’1โˆ˜๐œ•๐‘žโˆ˜๐‘“=๐œ•๐‘žโˆ˜๐œ•๐‘žโˆ’1โˆ˜๐‘“=๐‘“,(1.7) where ๐œ•๐‘žโˆ’1 is the formal inverse of ๐œ•๐‘ž.

We should note that ๐œ‚๐‘ž does not commute with ๐œ•๐‘ž,๐œ•๐‘ž๎€ท๐œ‚๐‘˜๐‘ž๎€ธ(๐‘“)=๐‘ž๐‘˜๐œ‚๐‘˜๐‘ž๎€ท๐œ•๐‘ž๐‘“๎€ธ,๐‘˜โˆˆโ„ค(1.8) or in the following general case:๐œ•๐‘š๐‘ž๎€ท๐œ‚๐‘˜๐‘ž๎€ธ(๐‘“)=๐‘ž๐‘˜+๐‘š๐œ‚๐‘˜๐‘ž๎€ท๐œ•๐‘š๐‘ž๐‘“๎€ธ,๐‘˜,๐‘šโˆˆโ„ค.(1.9)

Note that (1.6) can be unified as follows:๐œ•๐‘›๐‘ž๎“โˆ˜๐‘“=๐‘˜โ‰ฅ0โŽ›โŽœโŽœโŽ๐‘›๐‘˜โŽžโŽŸโŽŸโŽ ๐‘ž๐œ‚๐‘ž๐‘›โˆ’๐‘˜๎€ท๐œ•๐‘˜๐‘ž๐‘“๎€ธ๐œ•๐‘ž๐‘›โˆ’๐‘˜,(1.10) for all ๐‘›. In the last equation, the ๐‘ž-binomials take the formโŽ›โŽœโŽœโŽ๐‘›๐‘˜โŽžโŽŸโŽŸโŽ ๐‘ž=(๐‘›)๐‘ž(๐‘›โˆ’1)๐‘žโ‹ฏ(๐‘›โˆ’๐‘˜+1)๐‘ž(1)๐‘ž(2)๐‘žโ‹ฏ(๐‘˜)๐‘ž,(1.11) and the ๐‘ž-numbers are given by(๐‘›)๐‘ž=๐‘ž๐‘›โˆ’1,๐‘žโˆ’1(1.12) where the conventionโŽ›โŽœโŽœโŽ๐‘›0โŽžโŽŸโŽŸโŽ ๐‘ž=1,(1.13) is taken.

We can write out several explicit forms of (1.10) for ๐‘ž-derivative ๐œ•๐‘›๐‘ž and ๐œ•๐‘žโˆ’๐‘›(๐‘›โ‰ฅ0) as๐œ•๐‘ž๎€ท๐œ•โˆ˜๐‘“=๐‘ž๐‘“๎€ธ+๐œ‚๐‘ž(๐‘“)๐œ•๐‘ž,๐œ•2๐‘ž๎€ท๐œ•โˆ˜๐‘“=2๐‘ž๐‘“๎€ธ+(๐‘ž+1)๐œ‚๐‘ž๎€ท๐œ•๐‘ž๐‘“๎€ธ๐œ•๐‘ž+๐œ‚2๐‘ž(๐‘“)๐œ•2๐‘ž,๐œ•3๐‘ž๎€ท๐œ•โˆ˜๐‘“=3๐‘ž๐‘“๎€ธ+๎€ท๐‘ž2๎€ธ๐œ‚+๐‘ž+1๐‘ž๎€ท๐œ•2๐‘ž๐‘“๎€ธ๐œ•๐‘ž+๎€ท๐‘ž2๎€ธ๐œ‚+๐‘ž+12๐‘ž๎€ท๐œ•๐‘ž๐‘“๎€ธ๐œ•2๐‘ž+๐œ‚3๐‘ž(๐‘“)๐œ•3๐‘ž,๐œ•๐‘žโˆ’1โˆ˜๐‘“=๐œ‚๐‘žโˆ’1(๐‘“)๐œ•๐‘žโˆ’1โˆ’๐‘žโˆ’1๐œ‚๐‘žโˆ’2๎€ท๐œ•๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’2+๐‘žโˆ’3๐œ‚๐‘žโˆ’3๎€ท๐œ•2๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’3โˆ’๐‘žโˆ’6๐œ‚๐‘žโˆ’4๎€ท๐œ•3๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’4+1๐‘ž10๐œ‚๐‘žโˆ’5๎€ท๐œ•4๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’5+โ‹ฏ+(โˆ’1)๐‘˜๐‘žโˆ’(1+2+3+โ‹ฏ+๐‘˜)๐œ‚๐‘žโˆ’๐‘˜โˆ’1๎€ท๐œ•๐‘˜๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’๐‘˜โˆ’1๐œ•+โ‹ฏ,๐‘žโˆ’2โˆ˜๐‘“=๐œ‚๐‘žโˆ’2(๐‘“)๐œ•๐‘žโˆ’2โˆ’1๐‘ž2(2)๐‘ž๐œ‚๐‘žโˆ’3๎€ท๐œ•๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’3+1๐‘ž(2+3)(3)๐‘ž๐œ‚๐‘žโˆ’4๎€ท๐œ•2๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’4โˆ’1๐‘ž(2+3+4)(4)๐‘ž๐œ‚๐‘žโˆ’5๎€ท๐œ•3๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’5+(+โ‹ฏโˆ’1)๐‘˜๐‘ž(2+3+โ‹ฏ+๐‘˜+1)(๐‘˜+1)๐‘ž๐œ‚๐‘žโˆ’2โˆ’๐‘˜๎€ท๐œ•๐‘˜๐‘ž๐‘“๎€ธ๐œ•๐‘žโˆ’2โˆ’๐‘˜+โ‹ฏ.(1.14) We also add that the residue of the symbol โ„’(๐‘ฅ,๐œ•๐‘ž) can be written as๎ƒฉ๐‘๐ž๐ฌ๐‘๎“๐‘–=โˆ’โˆž๐‘ข๐‘–(๐‘ฅ)๐œ•๐‘–๐‘ž๎ƒช=๐‘ขโˆ’1(๐‘ฅ),(1.15) and its Tr-functional is๎ƒฉ๐“๐ซ๐‘๎“๐‘–=โˆ’โˆž๐‘ข๐‘–(๐‘ฅ)๐œ•๐‘–๐‘ž๎ƒช=๎€œ๐‘†1๐‘ขโˆ’1(๐‘ฅ)๐‘‘๐‘ฅ.(1.16)

1.2. Algebraic Structure of ๐‘ž-PDO

Now let us introduce the ๐‘ž-pseudodifferential operators algebra ๐‘ž-PDO. The latter is characterized by the relation [1]:๐‘ž-PDO=๎ƒฏโ„’๎€ท๐‘ฅ,๐œ•๐‘ž๎€ธ=๐‘๎“๐‘–=โˆ’โˆž๐‘ข๐‘–(๐‘ฅ)๐œ•๐‘–๐‘ž๎ƒฐ.(1.17)

We can noted this space in the following way ๐”ฎ๐’œโ‰ก๐‘žโˆ’ฮจDO is seen as being the algebra of all local and nonlocal ๐‘ž-differential operators of arbitrary conformal spins and arbitrary degrees, this spaces can be seen as being the ๐‘ž-deformation of pseudodifferential algebra ๐’œ that we saw in [2โ€“11]. One may expand ๐”ฎ๐’œ as๐”ฎ๐’œ=โŠ•๐‘šโ‰ค๐‘›๐”ฎ๐’œ(๐‘š,๐‘›)=โŠ•๐‘šโ‰ค๐‘›โŠ•๐‘ โˆˆโ„ค๐”ฎ๐’œ๐‘ (๐‘š,๐‘›),๐‘š,๐‘›,๐‘ ,โˆˆโ„ค,(1.18) where we have denoted by (๐‘š,๐‘›) the lowest and the highest degrees, respectively, and by ๐‘  the conformal spin. To be explicit, consider the space ๐”ฎ๐’œ๐‘ (๐‘š,๐‘›) of ๐‘ž-differential operators:โ„’๐‘ (๐‘š,๐‘›)=๐‘›๎“๐‘–=๐‘š๐‘ข๐‘ โˆ’๐‘–(๐‘ง)๐œ•๐‘ž๐‘–.(1.19)

The vector space ๐”ฎ๐’œ(๐‘š,๐‘›) of ๐‘ž-differential operators with given degrees (๐‘š,๐‘›) but undefined spin๐”ฎ๐’œ(๐‘š,๐‘›)=โŠ•๐‘ โˆˆโ„ค๐”ฎ๐’œ๐‘ (๐‘š,๐‘›)(1.20) exhibits a Lie algebra's structure with respect to the Lie bracket for ๐‘šโ‰ค๐‘›โ‰ค1.

In fact, It's straightforward to check that the commutator of two operators of ๐”ฎ๐’œ๐‘ (๐‘,๐‘ž) is an operator of conformal spin 2๐‘  and degrees (๐‘,2๐‘žโˆ’1). Since the Lie bracket [โ‹…,โ‹…] acts as[]โ‹…,โ‹…โˆถ๐”ฎ๐’œ๐‘ (๐‘š,๐‘›)ร—๐”ฎ๐’œ๐‘ (๐‘š,๐‘›)โŸถ๐”ฎ๐’œ(๐‘š,2๐‘›โˆ’1)2๐‘ ,(1.21) imposing the closure, one gets strong constraints on the spin ๐‘  and the degrees parameters (๐‘š,๐‘›), namely,๐‘ =0,๐‘šโ‰ค๐‘›โ‰ค1.(1.22) From these equations, we learn in particular that the spaces ๐”ฎ๐’œ0(๐‘š,๐‘›),๐‘šโ‰ค๐‘›โ‰ค1 admit a Lie algebra's structure with respect to the bracket (1.5) provided that the Jacobi identity is fulfilled. This can be ensured by showing that the Leibnitz product is associative.

The spaces ๐”ฎ๐’œ0(๐‘š,๐‘›),๐‘šโ‰ค๐‘›โ‰ค1 as well as the vector space ๐”ฎ๐’œ0(0,1) are in fact subalgebra of the Lie algebra ๐”ฎ๐’œ0(โˆ’โˆž,1) which can be decomposed as๐”ฎ๐’œ0(โˆ’โˆž,1)=๐”ฎ๐’œ0(โˆ’โˆž,โˆ’1)โŠ•๐”ฎ๐’œ0(0,1).(1.23)๐”ฎ๐ด0(โˆ’โˆž,โˆ’1) is nothing but the Lie algebra of Lorentz scalar pure ๐‘ž-pseudodifferential operators of higher degree ๐‘›=โˆ’1 and ๐”ฎ๐’œ0(0,1) is the central extension of the Lie algebra ๐”ฎ๐’œ0(1,1) of vector fields Di๏ฌ€(๐‘†1):๐”ฎ๐’œ0(0,1)=๐”ฎ๐’œ0(0,0)โŠ•๐”ฎ๐’œ0(1,1),(1.24) and where ๐”ฎ๐’œ0(0,0)โ‰Š๐’œ0(0,0) is the one dimensional trivial ideal.

The infinite dimensional huge space ๐”ฎ๐’œ is the algebra of ๐‘ž-differential operators of arbitrary spins and arbitrary degrees. Itโ€™s obtained from the space ๐”ฎ๐’œ(๐‘š,๐‘›) by summing over all allowed degrees ๐”ฎ๐’œ:๐”ฎ๐’œ=โŠ•๐‘šโ‰ค๐‘›๐”ฎ๐’œ(๐‘š,๐‘›)=โŠ•๐‘šโˆˆโ„ค๎‚ธโŠ•๐‘˜โˆˆโ„•๐”ฎ๐’œ(๐‘š,๐‘š+๐‘˜)๎‚น=โŠ•๐‘šโˆˆโ„ค๎‚ธโŠ•๐‘˜โˆˆโ„•๎‚ธโŠ•๐‘ โˆˆโ„ค๐”ฎ๐’œ๐‘ (๐‘š,๐‘š+๐‘˜).๎‚น๎‚น(1.25) This infinite dimensional space which is the combined conformal spin and degrees tensor algebra is closed under the Lie bracket without any constraint.

A remarkable property of ๐”ฎ๐’œ is that it can splits into six infinite subalgebras ๐”ฎ๐’œ๐‘—+ and ๐”ฎ๐’œ๐‘—โˆ’,๐‘—=0,ยฑ1 related to each others by conjugation of the spin and degrees. Indeed given two integers ๐‘š and ๐‘›โ‰ฅ๐‘š, it is not difficult to see that the vector spaces ๐”ฎ๐’œ(๐‘š,๐‘›) and ๐”ฎ๐’œ(โˆ’๐‘›โˆ’1,โˆ’๐‘šโˆ’1) are dual with respect to the pairing product (โ‹…,โ‹…) defined as๎€ทโ„’(๐‘š,๐‘›),โ„’(๐›ผ,๐›ฝ)๎€ธ=๐›ฟ0,1+๐‘š+๐›ฝ๐›ฟ0,1+๐‘›+๐›ผ๎€บโ„’๐‘๐ž๐ฌ(๐‘š,๐‘›)โˆ˜โ„’(๐›ผ,๐›ฝ)๎€ป,(1.26) where ๐‘‘(๐›ผ,๐›ฝ) are ๐‘ž-differential operators with fixed degrees (๐›ผ,๐›ฝ;๐›ฝโ‰ฅ๐›ผ) but arbitrary spin and where the residue operation res is defined as:๎€ท๐œ•๐‘๐ž๐ฌ๐‘–๐‘ž๎€ธ=๐›ฟ0,๐‘–+1.(1.27) This equation shows that the operation res exhibits a conformal spin ฮ”=1. Using the properties of this operation and the pairing product (1.26), one can decompose ๐”ฎ๐’œ as follows:๐”ฎ๐’œ=๐”ฎ๐’œ+โŠ•๐”ฎ๐’œโˆ’(1.28) with๐”ฎ๐’œ+=โŠ•๐‘šโ‰ฅ0๎‚ธโŠ•๐‘˜โˆˆโ„•๐”ฎ๐’œ(๐‘š,๐‘š+๐‘˜)๎‚น,(1.29)๐”ฎ๐’œโˆ’=โŠ•๐‘šโ‰ฅ0๎‚ธโŠ•๐‘˜โˆˆโ„•๐”ฎ๐’œ(โˆ’๐‘šโˆ’๐‘˜โˆ’1,โˆ’๐‘šโˆ’1)๎‚น.(1.30) The indices + and โˆ’ carried by ๐”ฎ๐’œ+ and ๐”ฎ๐’œโˆ’ refer to the positive (local) and negative (nonlocal) degrees respectively. On the other hand one can decomposes the space ๐”ฎ๐’œ(๐‘š,๐‘š+๐‘˜),๐‘˜โ‰ฅ0 as๐”ฎ๐’œ(๐‘š,๐‘š+๐‘˜)=๐‘žฮฃโˆ’(๐‘š,๐‘š+๐‘˜)โŠ•๐‘žฮฃ0(๐‘š,๐‘š+๐‘˜)โŠ•๐‘žฮฃ+(๐‘š,๐‘š+๐‘˜).(1.31)๐‘žฮฃโˆ’(๐‘š,๐‘š+๐‘˜) and ๐‘žฮฃ+(๐‘š,๐‘š+๐‘˜) denote the spaces of ๐‘ž-differential operators of negative and positive definite spin. They are read as๐‘žฮฃโˆ’(๐‘š,๐‘š+๐‘˜)=โŠ•๐‘ >0๐”ฎ๐’œ(๐‘š,๐‘š+๐‘˜)โˆ’๐‘ ,(1.32)๐‘žฮฃ0(๐‘š,๐‘š+๐‘˜)=๐”ฎ๐’œ0(๐‘š,๐‘š+๐‘˜),(1.33)๐‘žฮฃ+(๐‘š,๐‘š+๐‘˜)=โŠ•๐‘ >0๐”ฎ๐’œ๐‘ (๐‘š,๐‘š+๐‘˜).(1.34)๐‘žฮฃ0(๐‘š,๐‘š+๐‘˜) is just the vector space of Lorenz scalar ๐‘ž-differential operators. Combining (1.28)โ€“(1.34), one sees that ๐”ฎ๐’œ decomposes into 6=3ร—2 subalgebras๐”ฎ๐’œ=โŠ•๐‘—=0,+,โˆ’๎€บ๐”ฎ๐’œ๐‘—+โŠ•๐”ฎ๐’œ๐‘—โˆ’๎€ป(1.35) with๐”ฎ๐’œ๐‘—+=โŠ•๐‘šโ‰ฅ0๎‚ธโŠ•๐‘˜โˆˆโ„•๐‘žฮฃ๐‘—(๐‘š,๐‘š+๐‘˜)๎‚น,๐”ฎ๐’œ๐‘—โˆ’=โŠ•๐‘šโ‰ฅ0๎‚ธโŠ•๐‘˜โˆˆโ„•๐‘žฮฃ๐‘—(โˆ’๐‘šโˆ’๐‘˜โˆ’1,โˆ’๐‘šโˆ’1)๎‚น.(1.36) The duality of these 6=3ร—2 subalgebras is described by the combined scalar product โŸจโŸจโ‹…,โ‹…โŸฉโŸฉ built out of the product equation(1.26) and conformal spin pairing:โŸจ๐‘ข๐‘˜,๐‘ข๐‘™๎€œโŸฉโˆถ=๐‘‘๐‘ง๐‘ข๐‘˜(๐‘ง)๐‘ข1โˆ’๐‘˜(๐‘ง)๐›ฟ๐‘˜+๐‘™,1,(1.37) as follows [2, 3]:โ„’๎‚ฌ๎‚ฌ๐‘ (๐›ผ,๐›ฝ),โ„’๐‘Ÿ(๐‘š,๐‘›)๎‚ญ๎‚ญโˆถ=๐›ฟ0,๐‘Ÿ+๐‘ ๐›ฟ0,1+๐‘›+๐›ผ๐›ฟ0,1+๐‘š+๐›ฝ๎€œ๐‘‘๐‘งres๎‚ƒโ„’๐‘ (๐›ผ,๐›ฝ)โˆ˜โ„’(โˆ’๐›ฝโˆ’1,โˆ’๐›ผโˆ’1)โˆ’๐‘ ๎‚„(1.38) with respect to this new product, ๐”ฎ๐’œ++, ๐”ฎ๐’œ0+, and ๐”ฎ๐’œโˆ’+ behave as the dual algebras of ๐”ฎ๐’œโˆ’โˆ’, ๐”ฎ๐’œ0โˆ’, and ๐”ฎ๐’œ+โˆ’, respectively, while ๐”ฎ๐’œ0โˆ’ is just the algebra of Lorenz scalar pure ๐‘ž-pseudo operators. This algebra and its dual ๐”ฎ๐’œ0+, the space of Lorenz scalar local ๐‘ž-differential operators, are very special subalgebras as they are systematically used to construct new realizations of the ๐‘ค๐‘–-symmetry, ๐‘–โ‰ฅ2 by using scalar ๐‘ž-differential operators typeโ„’(๐‘˜)(๐‘Ž)=๐‘Žโˆ’๐‘˜(๐‘Ž)๐œ•๐‘˜๐‘ž.(1.39)

We note that the space ๐”ฎ๐’œ++ is the algebra of local ๐‘ž-differential operators of positive definite spins and positive degrees. ๐”ฎ๐’œโˆ’โˆ’, however, is the Lie algebra of pure ๐‘ž-pseudodifferential operators of negative degrees and spins.

2. ๐‘ž-Deformed Lax Generating Technique

The aim of this section is to present some results related to the Lax representation in its ๐‘ž-deformed version. Using the convention notations and the analysis presented previously, we perform consistent algebraic computations, based on the Pseudodifferential analysis, to derive explicit Lax pair operators of some integrable systems in the ๐‘ž-deformation framework.

We underline that the present formulation is based on the (๐‘ž-pseudo) operators ๐œ•๐‘›๐‘ž and ๐œ•๐‘žโˆ’๐‘› instead of the (pseudo) operators ๐œ•๐‘› and ๐œ•โˆ’๐‘› used in several works. We note also that the obtained results are shown to be compatible with the ones already established in literature [12โ€“16] in the case of ๐‘ž=1.

The basic idea of the Lax formulation consists first in considering a noncommutative integrable system which possesses the Lax representation:[โ„’,๐œ•๐‘กโˆ’๐ต]๐‘ž=0,(2.1) with ๐œ•๐‘กโ‰ก๐œ•/๐œ•๐‘ก et [๐‘“,๐‘”]๐‘ž=๐‘“โˆ˜๐‘”โˆ’๐‘“โˆ˜๐‘”.

Equation (2.1) and the associated pair of operators (โ„’,๐ต) are called the Lax ๐‘ž-differential equation and the Lax pair, respectively. The ๐‘ž-differential operator โ„’ defines the integrable system which we should fix from the beginning.

Note that the ๐‘ ๐‘™๐‘›-KdV hierarchy in the ๐‘ž-deformed version is defined as:๐œ•โ„’๐œ•๐‘ก๐‘˜=[(โ„’๐‘˜/2)+,โ„’]๐‘ž,(2.2) and the way with which ones to writes the Lax ๐‘ž-differential equation as in (2.1) is equivalent to the following equation:[โ„’,๐œ•๐‘กโˆ’๐ต]๐‘žโ‰ก[โ„’,๐œ•๐‘ก๐‘˜โˆ’(โ„’๐‘˜/2)+]๐‘ž=0,(2.3) where the operator ๐ต is the analogue of (โ„’๐‘˜/2)+ describing then an ๐‘ž-differential operator of conformal spin ๐‘˜.

Now, let us apply the ๐‘ž-deformation Lax-pair generating technique. We need to find an appropriate operator ๐ต which satisfies (2.1), for this we have to make some constraints on the operator ๐ต, namely,

Ansatz for the operator ๐ต:๐ต=๐œ•๐‘›๐‘žโˆ˜โ„’๐‘š+๎‚๐ต,(2.4) with ๐œ•๐‘›๐‘ž is the ๐‘ž-differential operator which acts on โ„’๐‘š according to (1.10) and ๎‚๐ต is another operator of same conformal weight than ๐ต. Then, with this ansatz, the problem reduces to find the operator ๎‚๐ต.

To understand the situation, we will study two interesting examples to know ๐‘ž-KdV and ๐‘ž-Burgers equations.

2.1. ๐‘ž-Deformed Burgers Equations

The โ„’-operator for the ๐‘ž-deformed Burgers equation is given byโ„’๐‘žโˆ’burgers=๐œ•๐‘ž+๐‘ข1(2.5) withโ„’๐‘žโˆ’burgersโˆˆ๐”ฎ๐’œ1(0,1).(2.6) Let's consider the constraint ๐‘›=1=๐‘š, for the ๐‘ž-deformed Burgers operator ๐ต can be written, from the ansatz (2.4), as follows:๐ต=๐œ•๐‘ž๎‚๐ตโˆ˜โ„’+=๐œ•2๐‘ž+๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ๐œ•๐‘ž+๐œ•๐‘ž๎€ท๐‘ข1๎€ธ+๎‚๐ต.(2.7) Simply algebraic computations give๎‚ƒ๎‚๐ต๎‚„=๎€ท๐œ‚โ„’,๐‘ž๎€ท๐‘ข1๎€ธโˆ’๐‘ข1๎€ธ๐œ•2๐‘ž+๎‚ƒ๐‘ž๐œ‚๐‘ž๎€ท๐œ•๐‘ž๎€ท๐‘ข1+๎€ท๐œ‚๎€ธ๎€ธ๐‘ž๎€ท๐‘ข1๎€ธ๎€ธ2+๐œ•๐‘ž๎€ท๐‘ข1๎€ธโˆ’๐œ•๐‘ž๎€ท๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ๎€ธโˆ’๐‘ข1๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ๎‚„๐œ•๐‘ž+๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ๐œ•๐‘ž๎€ท๐‘ข1๎€ธโˆ’ฬ‡๐‘ข1,(2.8) where ฬ‡๐‘ข1=๐œ•๐‘ข1/๐œ•๐‘ก.

Now, our goal is to extract, from (2.1) and (2.8), the Lax equation called ๐‘ž-deformed Burgers or just ๐‘ž-Burgers equation. For this we will follow the following procedure:

Ansatz for the operator ๎‚๐ต:๎‚๐ต=๐›ผ๐œ•๐‘ž+๐›ฝ,(2.9) where ๐›ผ and ๐›ฝ are arbitrary functions on ๐‘ข and its derivatives. one finds๎‚ƒ๎‚๐ต๎‚„=๎€ท๐œ‚โ„’,๐‘ž๎€ธ๐œ•(๐›ผ)โˆ’๐›ผ2๐‘ž+๎€บ๐œ•๐‘ž(๐›ผ)+๐œ‚๐‘ž(๐›ฝ)+๐‘ข1๐›ผโˆ’๐›ผ๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ๎€ป๐œ•โˆ’๐›ฝ๐‘ž+๐œ•๐‘ž(๐›ฝ)โˆ’๐›ผ๐œ•๐‘ž๎€ท๐‘ข1๎€ธ.(2.10) While identifying the two equations (2.8) and (2.10) we finds๐‘Ž๐œ•2๐‘ž๎€ท๐‘ข1๎€ธ+๎€บ๐œ‚(๐‘โˆ’1)๐‘ž๎€ท๐‘ข1๎€ธ๐œ•๐‘ž๎€ท๐‘ข1๎€ธ+๐‘ข1๐œ•๐‘ž๎€ท๐‘ข1๎€ธ๎€ป+ฬ‡๐‘ข1=0,(2.11) with ๐‘Ž and ๐‘ are arbitrary real constant.

Equation (2.11) is called ๐‘ž-deformed Burgers equation or ๐‘ž-Burgers equation. the characteristic of this equation is that it is linear for ๐‘=1 and that for ๐‘ž=1. (i.e.,๐œ‚๐‘ž(๐‘ข1)=๐‘ข1) we recover the same equation gotten in works [4, 5, 9]๐‘Ž๐‘ข1๎…ž๎…ž+2(๐‘โˆ’1)๐‘ข1๐‘ข๎…ž1+ฬ‡๐‘ข1=0.(2.12)

2.2. ๐‘ž-Deformed KdV Equations

In this second example, we go worked on an ๐‘ž-differentials operator of conformal weight 2, this operator is given by the KdV Lax operatorโ„’๐‘ž-KdV=๐œ•2๐‘ž+๐‘ข2.(2.13)

We are going to follow the same method of the previous example, therefore the Ansatz for the operator ๐ต is๐ต=๐œ•๐‘ž๎‚๐ตโˆ˜โ„’+=๐œ•3๐‘ž+๐œ‚๐‘ž๎€ท๐‘ข2๎€ธ๐œ•๐‘ž+๐œ•๐‘ž๎€ท๐‘ข2๎€ธ+๎‚๐ต(2.14) and the associated Lax equation:๎€บ๐œ•๐‘ก๎€ปโˆ’๐ต,โ„’๐‘ž=0(2.15) after a calculation, one finds[]โ„’,๐ต๐‘ž=โˆ’ฬ‡๐‘ข1,(2.16) by the same way of the case of Burgers, we finds the following ๐‘ž-KdV equation:ฬ‡๐‘ข2=๎€บ๐‘ข2+๐œ‚๐‘ž๎€ท๐‘ข2๐œ•๎€ธ๎€ป๐‘ž๎€ท๐‘ข2๎€ธ+๐œ•2๐‘ž๎€บ๐œ•๐‘ž๎€ท๐‘ข2๎€ธ+๐œ‚๐‘ž๎€ท๐œ•๐‘ž๐‘ข2๎€ธ๎€ป(2.17) as for ๐‘ž=1, we finds the standard KdVฬ‡๐‘ข2=๐‘ข2๐‘ข๎…ž2+๐‘ข2๎…ž๎…ž๎…ž.(2.18)

2.3. ๐‘ž-Deformed Burgers-KdV Mapping

In this section, we present an approach to define the correspondence between integrables systems ๐‘ž-deformed-type Burgers and integrables systems ๐‘ž-deformed-type KdV. such correspondence named ๐‘ž-deformed Burgers-KdV mapping that is considered like a generalization of the Burgers-KdV mapping studied in works [7, 8, 11, 17].

We illustrate this idea with the example of KdV and Burgers equation and then we are going to make a generalization for cameraman ๐‘ž-differentials-operators-type ๐‘ ๐‘™๐‘›-KdV.

Let's consider the Burgers ๐‘ž-differential operator (2.5):โ„’๐‘ž-burgers=๐œ•๐‘ž+๐‘ข1โˆˆ๐”ฎ๐’œ1(0,1)(2.19) and the KdV ๐‘ž-differential operator (2.13):โ„’๐‘ž-KdV=๐œ•2๐‘ž+๐‘ข2โˆˆ๐”ฎ๐’œ2(0,2)๐”ฎ๐’œ2(1,1).(2.20)

Proposition 2.1 (๐‘ž-deformed Miura transformation). If one considers the two previous ๐‘ž-di๏ฌ€erential operators, one can make the following decomposition: โ„’๐‘ž-KdV๎€ท๐‘ข2๎€ธ=โ„’๐‘ž-burgers๎€ท๐‘ข1๎€ธโˆ˜โ„’๐‘ž-burgers๎€ท๐‘ฃ1๎€ธ,(2.21) with ๐ฏ1=โˆ’๐œผ๐‘ž(๐‘ข1) and ๐ฎ2=๐œ•๐‘ž(โˆ’๐œ‚๐‘ž(๐‘ข1))โˆ’๐ฎ1๐œผ๐‘ž(๐‘ข1). This decomposition is called ๐‘ž-deformed Miura transformation. one can see this mapping under the following form: โ„’๐‘ž-burgers๎€ท๐‘ข1๎€ธโ†ชโ„’๐‘ž-KdV๎€ท๐‘ข2๎€ธ=โ„’q-burgers๎€ท๐‘ข1๎€ธโˆ˜โ„’๐‘ž-burgers๎€ทโˆ’๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ๎€ธwith๐‘ข2=๐œ•๐‘ž๎€ทโˆ’๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ๎€ธโˆ’๐‘ข1๐œ‚๐‘ž๎€ท๐‘ข1๎€ธ.(2.22)

Proposition 2.2. As basing on the conforms weights of the operators derivatives: [๐œ•๐‘ก๐‘ž-KdV]=3 and [๐œ•๐‘ก๐‘ž-Burgers]=2, one can make the following correspondence: ๐œ•๐‘ก๐‘ž-Burgersโ†ช๐œ•๐‘ก๐‘ž-KdV=๐›ผ๐œ•๐‘žโˆ˜๐œ•๐‘ก๐‘ž-Burgers+๐›ฝ๐œ•3๐‘ž,(2.23) where ๐›ผ and ๐›ฝ are arbitrary real constants.

Proposition 2.3 (Gรฉnรฉralisation). Being given an ๐‘ž-deformed Burgers operator โ„’๐‘ž-Burgers and an ๐‘ž-deformed sl๐‘›-KdV operator of type: โ„’๐‘ž-๐‘ ๐‘™๐‘›-KdV=๐œ•๐‘›๐‘ž+๐‘ข2๐œ•๐‘ž๐‘›โˆ’2+๐‘ข3๐œ•๐‘ž๐‘›โˆ’3+โ‹ฏ+๐‘ข๐‘›,(2.24) then we can make the following decomposition: โ„’๐‘ž-๐‘ ๐‘™๐‘›โˆ’KdV=โ„’๐‘ž-Burgers๎€ท๐‘ฃ1๎€ธโˆ˜โ„’๐‘ž-Burgers๎€ท๐‘ฃ2๎€ธโˆ˜โ‹ฏโˆ˜โ„’๐‘ž-Burgers๎€ท๐‘ฃ๐‘›๎€ธ,(2.25) where ๐‘ฃ๐‘–,๐‘–=1,โ€ฆ,๐‘› are the fields of conformal weight 1 and which can be written in functions of the fields ๐‘ข๐‘—,๐‘—=2,โ€ฆ,๐‘› and their ๐‘ž-derivatives.

3. Conclusion

The importance of the theory of pseudodifferential operators in the study of nonlinear integrable systems is point out. Principally, the algebra of nonlinear (local and nonlocal) differential operators acte on the ring of analytic functions ๐‘ข๐‘ (๐‘ฅ,๐‘ก).

In This paper, we have devoted to a brief account of the basic properties of the space of ๐‘ž-pseudo differential Lax operators in the bosonic case. Presently, we know that any ๐‘ž-pseudodifferential operator is completely specified by a conformal spin ๐‘ ,๐‘ โˆˆโ„ค, two integers ๐‘, and ๐‘ž=๐‘+๐‘›,๐‘›โ‰ฅ0 defining the lowest and the highest degrees, respectively, and finally (1+๐‘žโˆ’๐‘)=๐‘›+1 analytic fields ๐‘ข๐‘—(๐‘ง). We recall that the space ๐”ฎ๐’œ of all local and nonlocal ๐‘ž-pseudodifferential operators admits a Lie algebra's structure with respect to the commutator buildout of the Leibnitz product. Moreover, we find that A splits into 3ร—2=6 subalgebr as ๐”ฎ๐’œ๐‘—+ and ๐”ฎ๐’œ๐‘—โˆ’,๐‘—=0,ยฑ1 related to each others by two types of conjugations, namely, the spin.

Finally, we have focused in this work to present the basics steps towards constructing the ๐‘ž-deformed integrable systems and the associated Lax generating technique. Particular interest is devoted to the ๐‘ž-Burgers and the ๐‘ž-KdV systems and their underlying mapping.

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