On the Integrability of the Discrete Nonlinear Schroedinger Equation

a r X i v :0808.0837v 1 [m a t h -p h ] 6 A u g 2008On the Integrabilityof the Discrete Nonlinear Schr¨o dinger EquationDecio Levi ∗Dipartimento di Ingegneria Elettronica,Universit`a degli Studi Roma Tre and Sezione INFN Roma Tre,Via della Vasca Navale 84,00146Roma,ItalyMatteo Petrera †and Christian Scimiterna ‡Dipartimento di Fisica,Universit`a degli Studi Roma Tre andSezione INFN Roma Tre,Via della Vasca Navale 84,00146Roma,Italy(Dated:August 6,2008)In this letter we present an analytic evidence of the non-integrability of the discrete nonlinear Schr¨o dinger equation,a well-known discrete evolution equation which has been obtained in various contexts of physics and biology.We use a reductive perturbation technique to show an obstruction to its integrability.PACS numbers:02.30.Ik,47.11.StINTRODUCTIONThe nonlinear Schr¨o dinger (NLS)equation i ∂t f +∂xx f =σ|f |2f,f =f (x,t ),σ=±1,(1)is a universal nonlinear integrable partial differential equation (PDE)for models with weak nonlinear effects [1].It has been central for almost fourty years in a large variety of areas in sciences and it appears in many phys-ical contexts,see for instance [2,3,4,5,6].From the integrability of this PDE it follows the existence of in-finitely many symmetries and conservation laws,and the possibility of solving its Cauchy problem,once the initial data are prescribed.In correspondence with its symme-tries one finds an infinite number of exact solutions,the solitons,which,up to a phase,emerge unperturbed from the interaction among themselves.Many physical and biological applications involve lat-tice systems.In literature,one may find a few discrete forms of the NLS equation.The most relevant one is the discrete NLS (DNLS)equationi ∂t f n +12h 2(f n +1−2f n +f n −1)==σSome attempts to extend this approach to discrete equations have been proposed[25,26,27,28,29,30,31, 32,33].In[30,31,33]one canfind a multiscale expansion technique on the lattice which,starting from dispersive integrable Z2-lattice equations,provides other Z2-lattice equations.To do so one had to introduce a slow-varying condition on the amplitudes by requiring(∆n)p+1f n=0,(4) p being a positive integer and∆n f n=f n+1−f n.As a consequence,the resulting reduced equation turned out to be non-integrable even if the ancestor equation was in-tegrable.However,as shown in[27],if p=∞the reduced equations become formally continuous and their integra-bility may be preserved by the discrete reductive pertur-bation procedure.In this way the multiscale expansions easilyfit with both difference-difference and differential-difference equations.The illustrative example considered in[27]has been the lattice potential KdV equation,a dis-persive nonlinear Z2-lattice equation obtained from the superposition formula for the soliton solutions of the KdV equation.There one performed the multiscale expansion of the weak plane wave solutions of the discrete dispersive linear system.A proper representation of the discrete shift operators in terms of differential ones,provides the integrable NLS equation(1)as the lowest order secu-larity condition from the multiscale expansion.Further examples have been considered in[29].They confirmed a discrete analog of the Zakharov-Kuznetsov’s claim[20]:“if a nonlinear dispersive discrete equation is integrable then its lowest order multiscale reduction is an integrable NLS equation”.In the present letter we consider the multiscale per-turbation analysis of Eq.(2).Multiscale analysis of the DNLS equation(2)has been already considered in[26], giving a differential-difference system which does not ful-fill any integrability criterium.However this result did not prove the non-integrability of the DNLS equation, as similar results have been obtained in the case of the integrable equation(3)[26,32].Here,extending to lat-tice equations the approach used in[20,22,23,24]and computing the higher order terms in the reductive per-turbation expansion we are able to provide an analytical evidence of the non-integrability of Eq.(2).In fact,even if its lowest order multiscale reduction is an integrable KdV-type equation,the higher orders reductions exhibit non-integrable behaviors.DISCRETE MULTISCALE ANALYSIS OF THEDNLS EQUATIONBy the position f n(t)=[νn(t)]1/2exp[iφn(t)]the DNLS equation(2)may be written as the following system of real differential-difference equations:∂tνn+1α+n sinβ+n+h2−1γ+n cosβ+n+i!,δκ=∞i=1(−1)i−1where[x]denotes the integer part of x.Let us introduce the multiscale expansions(12-13),to-gether with Eq.(10),into Eqs.(5-6)and require that these equations be satified at all orders inǫ.At the lowest non-trivial orderǫ2onefindsν(1)=−σ∂t1φ(1).From now on all results will be presented just for the functionsφ(i).At orderǫ3we get ∂2t1−c2δ2κ φ(1)=0,where c=± ζσ1/2 /h.As c has to be real thenσ=1;moreover we chooseζ=h so that c remainsfinite as h→0. Therefore the resulting equation at this order is satisfied byφ(1)=φ(1)(ξ,{t m}m≥2)withξ=κ−ct1.At orderǫ5,the no-secular term condition implies ∂2t1−c2δ2κ φ(2)=0,i.e.φ(2)=φ(2)(ξ,{t m}m≥2).The evolution equation forφ(1)w.r.t.the slow-time t2reads∂t2φ(1)=K2 φ(1) ,(14)K2 φ(1) =a ∂3ξφ(1)−3a f(ξ)−∂2ξφ(1)24∂3ξφ(2)+3 64 ∂2ξφ(1)2+ch216∂ξφ(1)∂3ξφ(1)−c(h4−30h2−15)4Eq.(21)is a necessary condition for the integrability of Eq.(2).In this case only nine out of the fourteen coef-ficients of g(t2)are independent.Thus we havefive in-tegrability conditions,whose explicit expressions will be published elsewhere in a more detailed paper[34].The obtained constraints on the polynomial g(t2)are not sat-isfied by the coefficients computed in Eq.(19).Then, this incompatibility implies that the DNLS equation(2) cannot be integrable.CONCLUDING REMARKSBy performing a discrete multiscale analysis,we have proven that the DNLS equation(2)is not integrable.Al-though its lowest order reduction gives a KdV equation, the higher order terms do not satisfy the required integra-bility conditions.It is remarkable to note that a similar analysis performed on the integrable discrete NLS equa-tion(3)provides integrable reductions[34]. 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