Quasideterminant Darboux solutions of Noncommutative Equations of Langmuir Oscillations
aa r X i v : . [ n li n . S I] F e b Quasideterminant Darboux solutions of NoncommutativeEquations of Langmuir Oscillations
Irfan MahmoodCentre for High Energy Physics, University of the Punjab, Lahore 54590e-mail: [email protected] MahmoodCollege of Engineering, Ch. Engg Department, King Saud University, Kingdom of Saudi ArabiaFebruary 24, 2021
Abstract
This article encloses some results on nonncommutative analogue of nonabelian equations ofLangmuir oscillations. One of the main contributions of this work is to construct the Darbbouxtransformation for the solution of that equation in noncommutative framework incorporating asso-ciated discrete Lax system. Further the standard Darboux transformation on arbitrary eigenfunc-tions of the Lax system are presented in quasideterminants for few index values. Moreover, thesecomputations include the derivation of noncommutative version of nonabelian discrete nonlinearSchr¨ o dinger which coincides with its classical model under commutative limit. The end portionof this article reveals the identity of noncommutative formalism incorporating a derivation of anequation of motion which coincides with its existing commutative form in background zero valueof spectral parameter. Keywords:
Darboux transformation, Quasideterminats, Riccati equation, Noncommutative discretenonlinear Schr¨ o dinger equation The differential difference equation u nt = u n ( u n − − u n +1 ) (1)appears in analysis of spectrum structure associated to the Langmuir oscillations in plasma whichcompletely integrable in classical framework as possesses Lax representation [1] and also connected tothe well know integrable systems. The non-abelian analogue of equation (1) u nt = u n − u n − u n u n +1 (2)presented in [2] arrise from the compatibility of following linear system u n ψ n +1 = λψ n − ψ n − (3) ψ ( n ) t = − u n u n +1 ψ n +2 . (4)1nd an ingenious method was adopted to connect that system with discrete nonlinear Schr¨ o dingerequation [3, 4].. Moreover an efficient integrable approach the Darboux method has incorporated toconstruct the Darboux solutions of nonabelian equations of Langmuir oscillation. The Darboux trans-formation (DT) was proposed at the end of the last century [5] as a method to find transformationson potential and the eigenfunctions of the Schr¨ o dinger equation simultaneously. After that some moreremarkable results on DT enclosed in [6] to ensure its importance in theory of integrable system, wherethe Backlund transformation demonstrated for the integrable Korteweg-de Vries (KdV) equation de-duced from the Darboux transformations through its corresponding linear problem. Later on modernideas on DT with its implementations on various nonlinear physical systems were developed f by V.Matveev [7]. The Darboux transformation method is a very simplest way to construct a large classof solutions such as solution with permanent profile the soliton, hierarchy of rational solutions, kinksand breather like solution of various nonlinear differential equations with background of arbitrary seedsolution without applying comparatively complex inverse method. Recently the Darboux transfor-mation has got considerable attention in the modern theory of integrable systems to explore theirvarious algebraic and geometrical aspects. The successful implementations of these transformationshave been shown in the analysis of various mathematical features of graphene [8] and also applied incavity quantum electrodynamics [9, 10] for the dynamical analysis of the propagation of associateddisturbance. One of the remarkable achievement due to the DT in modern theory integrable systemsis to construct the solitonic solutions of discrete systems incorporating their corresponding Lax repre-sentations [11]. Moreover these transformations significantly extended to construct the determinantalsolutions of noncommutative integrable systems such as in case of noncommutative Painlev´e secondequation [12] where its N -fold Darboux solutions have been expressed in terms of quasideterminats.and also for its associated nocommutative Toda equation [13] in case of special index value. Morerecently one of the remarkable application of these transformation has been elaborated in finding theDarboux Wronskian solutions of Painlev´e second equation [14] in classical framework and generaliseto N -fold representation.In this article, the NC extensions of results on nonabelian equations of Langmuir oscillations [2] arepresented with some new findings such as its NC Darboux transformtion holds for all solutions. Whereas in case of nonabelian version (2) its Darbouxb transformations [2] holds only for a constant nonzero seed solution which does not satisfy directly equation (2) as its trivial solution. This work alsoencloses the reduction of equation (2) to NC analogue of discrete nonlinear Schr¨ o dinger equation inNC framework. In case of noncommutative extension of equation (2) the field and variables are considered purelynoncommuting objects such as [ u i , t ] = 0 and fields with their derivatives are also noncommuting.From the compatibility condition of linear systems (3) and (4) , we obtain NC version u nt = u n − u n − u n u n +1 (5)Now the standard Darboux transformation [15] on arbitrary function ψ n in NC framework can beexpressed as ψ n [1] = ψ n − ϕ n ϕ − n +1 ψ n +1 (6)and under this transformation, the linear system (3) can be written in following form u n [1] ψ n +1 [1] = λψ n [1] − ψ n − [1] (7)2ow substituting the values for transformed eigenfuctions from (6) into above transformed expressionand then with the help of system (4) the resulting expression yields Darboux transformation on u n asbelow u n [1] = u n − λϕ n ϕ − n +1 . (8)we can also express above result as follow, taking λϕ n = ϕ ′ n u n [1] = u n − ϕ ′ n ϕ − n +1 . (9)The above transformation connecting new solution u n [1] to old solution u n of equation (2) throughthe particular solutions of linear systems (3) and (4). Here the comparison of Darboux solution (9) andwith result on Darboux solution obtained in [2] shows a difference, here transformations are additiveholds for all seed solution even for u n = 0 also satisfies equation (5) where as the transformation on u n in [2] in multiplicative form holds only for non-zero seed solutions.The one fold Darboux transformation (6) with its second iteration can be expressed in form ofquasideterminat as below with setting ψ = ψ and ϕ = ψ so on ψ n [1] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ n +1 ψ n +1) ψ n ) ψ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (10)and the two fold NC Darboux transformation can be evaluated as ψ n [2] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ n +2) ψ n +2) ψ n +2) ψ n +1) ψ n +1) ψ n +1) ψ n ) ψ n ) ψ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (11)respectively and can be generalised to N -fold Darboux transformation in terms of quasideterminantsin the same way as done for the case of NC discrete Toda equation [13]. The above two fold DT canbe generalised into N -th form as below ψ n [ N ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ N ( n + N ) ψ N − n + N ) · · · ψ n + N ) ... ... ... ... ψ N ( n + N − ψ N − n + N − · · · ψ n + N − ψ N ( n ) ψ N − n ) · · · ψ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12)here N is non-negative integer as N = 0 , , , , ... and For the reduction to NC discrete NLS equation, let us apply the setting (21) for u n from [2] into NCversion (5) and then using values B n = (cid:20) − i i (cid:21) n (cid:20) q ∗ n q n (cid:21) in resulting expression keeping theconstraint of noncommutativity during the computations, we obtain end result as follow iq nt = q n − + q n +1 + q n − q ∗ n q n + q ∗ n q n q n +1 . (13)3ere the above equation (13) is noncommutative analogue of the nonabelian NLS equation, this canbe shown that under commutative limit equation (13) reduces to iq nt = q n − + q n +1 + | q n | ( q n − + q n +1 ) . (14)which is nonabelian discrete NLS equation presented in [2] . Let us start with setting S n = iψ n − ψ − n where ψ n are the solutions of system (3) and (4) provided that u n = S n S ∗ n +1 . (15)Now taking the time derivation of above expression S ′ n = ψ ′ n − ψ − n − ψ n − ψ − n ψ ′ n ψ − n in noccommutative case the derivative of inverse function is defined as ∂ t ψ − = ψ − n ψ ′ n ψ − n and here ′ stands for time derivative as S ′ n = ∂ t S n . Above result with the help of linear systems (3) and (4) willreduce to inhomogeneous NC Riccati equation as S ′ n = − λu n − + u n − S n + λ S n − λS n + S n u n The last result In background of λ = 0 can be considered as the NC analogue of the expression S nt = | S n | ( S n − − S n +1 ) derived in [2] with same background with constraint S ∗ n +1 = − S n +1 and isgiven by S nt = S n − S ∗ n S n − S n S ∗ n S n +1 (16)under the commutative limit expression (16) concedes with non-abbelian version S nt = | S n | ( S n − − S n +1 ).Now in commutative non-abelian framework with trivial solution u n = 0 (a seed solution) for equation(5), our DT (9) yields second solution as u n [1] = − λϕ n ϕ − n +1 . (17)That solution calculated from our newly introduced Darboux solution for the equation (5) holds forall solutions of that equation incorporating trivial solution u n = 0 this v-can be shown that our result(17) coincides to the solution (15) of same system presented in [2] for the specific value of λ = 0 as u n [1] = S n S ∗ n +1 . It is straight forward to calculate explicit solution, one soliton, with help of linearsystems (3) and (4) as u n [1] = u coshκ n + cosh ζ n coshκ n + coshζ n here u is constant and angles κ n and ζ n posses the same values given in [2] .4 Conclusion:
In this paper, the noncommutative analogue of nonabelian equations of Langmuir oscillation has beenensured with its connection to purly noncommutative discrete Schr¨ o dinger equation. This work alsoenclosed the derivation of Darboux transformation in additive structure as most of the integrable possesin noncommutative as well as in classical framework. This Darboux expression is applicable to generateall solitonic solutions even starting with more simplest zero trivial solution as seed solution. Furtherthese darboux solutions are presented in term of quasideterminants and through the noncommutativericcati equation in background of spectral parameter λ = 0 purly noncommutative version tfor theequation of motion (27) given in [2] has also been constructed. Acknowledgments: