Lax pairs and Fourier analysis: The case of sine- Gordon and Klein-Gordon equations

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1 Jounal of Physics: Confeence Seies Lax pais and Fouie analysis: The case of sine- Godon and Klein-Godon equations To cite this aticle: Pavle Saksida 22 J. Phys.: Conf. Se View the aticle online fo updates and enhancements. Related content - Dispesive estimates fo the Schödinge and Klein-Godon equations Elena A Kopylova - Scatteing fo massive scala fields on Coulomb potentials and Schwazschild metics J Dimock and B S Kay - Asymptotic iteation method fo solution of the Katze potential in D-dimensional Klein-Godon equation Dewanta Aya Nugaha, A Supami, C Cai et al. Recent citations - On nonlinea Fouie tansfom: towads the nonlinea supeposition Pavle Saksida - Complex nonlinea Fouie tansfom and its invese Pavle Saksida This content was downloaded fom IP addess on 9/9/28 at 6:

2 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 Lax pais and Fouie analysis: The case of sine-godon and Klein-Godon equations Pavle Saksida Faculty of Mathematics and Physics, Univesity of Ljubljana Abstact. In this pape we constuct a new Lax pai fo the Klein-Godon equation. The stuctue algeba of this Lax pai is the algeba T A 2 of uppe tiangula Toeplitz block matices with su2 blocks. Fo the suitable choice of the values of the spectal paamete, the integals of motion, obtained fom the holonomy of the spatial pat of the Lax pai, have simple expessions in tems of the Fouie data. We compae these integals to the coesponding integals of the sine-godon system.. Intoduction The invese scatteing method is often efeed to as the non-linea vesion of the stategy of solving the patial diffeential equations by means of the Fouie tansfom. Often this compaison is given in somewhat vague tems and it is not even meant to be taken too liteally. On the othe hand, thee exists a substantial body of deep and impotant esults on non-linea vesions of Fouie tansfom, applied to the study of integable systems see e. g. [4], [5], [6], [7]. In this note we will descibe a case of a simple and concete connection between the zeo-cuvatue condition and Fouie analysis. We shall conside the peiodic sine-godon equation and its lineaization at the vacuum solution, the peiodic Klein-Godon equation. These equations ae equations of motion of two Hamiltonian systems H sg, {, }, M and H kg, {, }, M. The phase space of both systems is and the Poisson backet is given by M = {qx, px; qx, px : S R} {F, G} = 2π δf δg δqx δpx δf δg dx. δpx δqx The Hamiltonian H sg of the sine-godon system is given by H sg qx, px = 2π and the one fo the Klein-Godon system is H kg qx, px = 2 p2 x + 2 q2 xx cos qx dx 2π 2 p2 x + 2 q2 xx + q 2 x dx. Published unde licence by Ltd

3 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 The sine-godon system is a well-known integable system. In paticula, the sine-godon equation is equivalent to the zeo-cuvatue condition fo the Lax pai Lz, Az given in 6 below. Hee z denotes the spectal paamete. The main esults pesented in this pape ae the following: i By means of a petubation method we constuct a new Lax pai with the spectal paamete Lz, Az fo the Klein-Godon system. The stuctue goup of this Lax pai is the Lie goup, coesponding to the Lie algeba T A 2 of the uppe-tiangula block Toeplitz matices with su2 blocks. Ou Lax pai is given in fomula. ii The integals of the Klein-Godon system, yielded by the Lax pai Lz, Az, have simple expessions in tems of the Fouie coefficients of the elements fom the phase space M. Moe concetely, let Hqx, px; z be the holonomy of L qx,px z evaluated along the loop x x,. Let F be an Ad-invaiant function on the Lie algeba T A 2. Then, as usual, the function F z : M R given by F z qx, px = F Hqx, px; z is a conseved quantity of the Klein-Godon system fo evey value of the spectal paamete z. Now, let us develop qx and px into the Fouie seies qx = α n e inx, px = β n e inx. Fo evey intege n Z let the value z n of the spectal paamete be given by Then fom we get the integals z n = n + n 2 +. F zn qx, px = iω n α n + β n 2, whee ω n = n These esults ae poved in sections 2 and 3. In section 4 we descibe a elationship between the integals F zn of the Klein-Godon system and the coesponding integals of the sine-godon system in some detail. Sections 2 and 3 povide a shote and moe easily eadable pesentation of the esults that appeaed in []. The discussion in section 4 is new. 2. Jet sine-godon system and its Lax pai The key ingedient in the poof of the above claims is the constuction of the so-called jet sine-godon system. Let qx, t; s : S [, T ] ɛ, ɛ R be a path of solutions of the peiodic sine-godon equation q tt q xx = sin q qx + 2π, t = qx, t, q x x + 2π, t = q x x, t. 3 2

4 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 Suppose that we have qx, t;, that is, the path stats at the vacuum solution. Conside the powe seies development qx, t; s = k n= s k k! qn x, t + Ok +. Then the coefficients q n x, t ae a solution of the system of patial diffeential equations q n tt q n xx = q n + f n q,..., q n 2, n =,..., k, 4 whee the functions f n ae given by sin sx + s2 2! x sk k! x k = k x n f n x,..., x n 2 s n + Ok +. n= The system 4 will be called the k-jet sine-godon system. We will show that this system is equivalent to a zeo-cuvatue condition fo a cetain Lax pai with spectal paamete. Let J km = {[q x, p x,..., q k x, p k x]} be the space of k-jets of paths s qx; s, px; s, based at, in the phase space M. The equations 4 can be ewitten in the fom of the system q n p n t = p n q n xx q n + f n q,..., q n 2, n =,..., k 5 whose solutions ae paths t j k q, pt = [q x, t, p x, t,..., q k x, t, p k x, t] in the jet space J km. Recall that the equation of motion 3 fo the sine-godon system H sg, {, }, M is equivalent to the zeo-cuvatue condition [ ] L q,p z t A q,p z x + L q,p z, A q,p z = fo the Lax pai z 2 L q,p z = i z 2 + sin q 2 cos q 2 z 2 + sin q 2 z 2 cos q 2 z 2 + A q,p z = i cos q 2 z2 sin q 2 z2 sin q 2 z2 + cos q 2 p + 4 p, 4 + q x4 qx 4 This Lax pai is slightly diffeent fom the usual one, found fo instance in [3]. It can be deived fom the Lax pai fo the Maxwell-Bloch equation, given in [8] and [9]. Conside now a path t j k q, pt = [q x, t, p x, t,..., q k x, t, p k x, t] in the jet space J k M. Choose a map t; s qs, ps = qx, t; s, px, t; s M, qx, t;, px, t;, d n such that ds n s= qx, t; s, px, t; s = q n x, t, p n x, t fo n =,..., k. Fo evey s fomulae 6 assign the Lax pai L qs,ps z, A qs,ps z to the path t qx, t; s, px, t; s in M. Let us denote. 6 L n z = dn ds n s= L qs,ps z, A n z = dn ds n s= A qs,ps z, n =,..., k. 3

5 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 We now aange the above matices into two uppe tiangula block-toeplitz matices with su2 blocks: L L k! Lk A A L k! Ak A L = k! Lk k 2! Lk L, A = k! Ak k 2! Ak A. 7 The above constuction associates a pai L, A to evey path t j k qx, t, px, t in the jet space J k M. It is easily seen that uppe tiangula block-toeplitz matices with su2 blocks fom a Lie algeba. We shall denote this Lie algeba by T A k. The poof of the following poposition is athe staightfowad. Poposition A path t j k q, p in the jet space J k M is a solution of the system 5 if and only if the associated Lax pai L j k q,pz, A j k q,pz satisfies the zeo-cuvatue condition [ ] L j k q,pz t A j k q,pz x + L j k q,pz, A j k q,pz =. When a system of diffeential equations has a zeo-cuvatue fomulation, one expects it to have a lage numbe of conseved quantities. These stem fom the Ad-invaiant functions of the stuctue goup. In ou case the stuctue goup is the Lie goup T G k of the Lie algeba T A k. The goupt G k is obtained fom the algeba T A k by exponentiation. Theefoe it is clea that the nontivial Ad-invaiant functions on T G k do not come fom the specta of the elements. The following poposition is poved in []. Poposition 2 Fo evey n =,..., k the function given by whee is an Ad-invaiant function. ϕ n g = n! ϕ n : T G k R n tg j g n j, j= g g g 2 g k g g = g g k..... T G k, 8 g The essential ingedient of the poof is the fact that the matix Lie algeba T A k is isomophic to the algeba J k su2 whose elements ae paths in su2 of the fom j k α = α + sα sk k! αk, α n su2 and whose backet is given by the tuncated polynomial multiplication combined with the backet on su2, k [j k α, j k s k n n β] = [α l, β n l ]. k! l n= l= 4

6 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 Poposition 2 enables us to compute the conseved quantities of the jet sine-godon system quite easily. Let j k q, p J k M be an abitay element in the jet space. The holonomy H j k q,pz of L j k q,pz is given by H j k q,pz = Φ2π; z, whee Φx; z is the solution of the initial poblem Φz x = L j k q,pz Φz, Φ; z = Id. Let the blocks of the uppe tiangula Toeplitz block matix Φx; z be denoted by Φ n x; z, n =,..., k. In tems of blocks the above initial poblem is given by the system n n Φ n x = L Φ n + L j Φ n j, Φ ; z = Id, Φ n ; z =, n. 9 j j= This is a system of non-homogeneous linea equations. They all have the same homogeneous pat Φ x = L Φ with the constant coefficient matix L. The non-homogeneity of n-th equation depends only on the solutions of the pevious n equations. Theefoe, we can successively compute the blocks Φ n x and, by evaluating at x = 2π, we obtain all the blocks H n z of the holonomy H j k q,pz. Fom poposition 2 it then folows that fo evey value z of the spectal paamete the functions F z : J k M R, given by F z j k q, p = ϕ n H j k q,pz = n j= n th j z H n j z, n =,..., k, j ae integals of the jet sine-godon system. Even though the jet sine-godon system has many conseved quantities, it is not difficult to see that it is not integable. Namely, the conseved quantities do not fom a complete system. 3. Lax pai and integals of the Klein-Godon system Let us now conside the jet sine-godon system 5 on J 2 M, that is, on the space of 2-jets. The equation of motion of this system is given simply by two uncoupled Klein-Godon equations q tt q xx = q q 2 tt q 2 xx = q 2. The 2-jet sine-godon system is the Hamiltonian system K, {, } p, M M, whee {, } p is the usual poduct Poisson backet on the Catesian poduct phase space M M, and the Hamiltonian K is given by K[q, p, q 2, p 2 ] = H kg q, p + H kg q 2, p 2. A path t [q x, t, p x, t, q 2 x, t, p 2 x, t] in J 2 M = M M is a solution of the 2-jet system if and only if q i x, t, p i x, t ae solutions of the Klein-Godon system fo i =, 2. In paticula, the path t [q x, t, p x, t,, ] is a solution of the 2-jet system pecisely when q x, t, p x, t is a solution of the Klein-Godon system. The above emaks lead to the fomulation of the following theoem. Theoem Let t qx, t, px, t be a path in the phase space M. Let us associate to this path the su2 matix functions L =, L = i z 2 i z2 i z2 8z q + p 4 i z2 8z q p 4, L 2 i z 2 = 6z q2, i z2 6z q2 5

7 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 A i +z 2 =, A i z2 i +z2 = 8z q + qx 4 i z2 8z q, A 2 i +z 2 qx = 6z q2. 4 i +z2 6z q2 Let the block Toeplitz matices Lx, t; z = L qx,t,px,t z and Ax, t; z = A qx,t,px,t z be given by Lx, t; z = L L 2 L2 L L L, Ax, t; z = A A 2 A2 A A A. Then the path t qx, t, px, t solves the Klein-Godon equation if and only if the associated Lax pai L, A defined above satisfies the zeo-cuvatue condition L t A x + [L, A] =. Poof: Let t; s Qx, t; s, P x, t; s M be a map such that Qx, t;, P x, t; =, and such that is satisfies the conditions d ds s=qx, t; s, P x, t; s = qx, t, px, t, d 2 ds 2 s=qx, t; s, px, t; s =,, whee qx, t, px, t is the path in M consideed in the statement of the theoem. Matices L i and A i fo i =,, 2 ae obtained by taking deivatives L i x, t; z = di ds i s=l Qx,t;s,P x,t;s z, A i x, t; z = di ds i s=a Qx,t;s,P x,t;s z, whee Lz, Az is the Lax pai of the sine-godon system, evaluated at qx, t; s, px, t; s. It now follows fom poposition that L, A given above is a Lax pai fo the 2-jet sine-godon system, esticted the subspace N J 2 M and given by N = {[qx, px,, ]} J 2 M. But we have seen that this estiction is pecisely the Klein-Godon system. If we estict the Lax pai to the functions, constant with espect to x, we get a Lax pai fo the hamonic oscillato. This Lax pai is indeed new and clealy diffeent fom the usual one, found e.g. in []. Let now F: J 2 M = M M R be an integal of the 2-jet system. Then the map given by F : M R F q, p = F[q, p,, ] is an integal of the Klein-Godon system. But at the end of section 2 we have seen that the conseved quantities of jet sine-godon systems ae explicitly computable. We shall now pefom the calculation fo the esticted 2-jet system and this will give us the conseved quantities of the Klein-Godon system. The expession gives F z qx, px = F z [qx, px,, ] = 2 th z H z + 2 th z H 2 z, 2 6

8 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 whee H i = Φ i 2π and Φ i x ae given by the system 9. In the case of the esticted 2-jet system we have Φ x = L Φ Φ x = L Φ + L Φ Φ 2 x = L Φ 2 + 2L Φ + L 2 Φ, whee the matices L i, i =,, 2 ae given in theoem. The integation gives us Φ x = exp xl Φ x = Φ x Φ 2 x = 2Φ x x + Φ x x Φ ξ L x ξ Φ ξ dξ Φ ξ L ξ Φ ξ Φ ξl 2 ξ ξ Φ ξ dξ. Φ η L η Φ η dηdξ The fist equation above gives Φ e i κx 2 x = κx i e 2 and the second Φ x = Φ x 4 x 4 px + iωqxeiκx dx px + iωqxe iκx dx, 3 whee κ = κz = z2 2z and ω = +z2 2z. Evaluation of Φ x at x = 2π will yield the fist tem in the expession 2. Fo the second tem we have 2π th H 2 = 2 t 2π + t Ad Φ x L x x Ad Φ x L2 x dx. Ad Φ ξ L ξ dξ dx The function L 2 takes values in the taceless Lie algeba su2, theefoe the second tem in the above expession is equal to zeo. If on the fist tem we pefom integation by pats and take into account the elation tab = tba, we get th H 2 = th H, and finally, F z qx, px = 4 t H qx,pxz H qx,px. z 4 Now, the expession 3 fo Φ x suggests setting the paamete κ = z2 2z to intege values. Then H = Φ 2π will be expessed in tems of the Fouie coefficients of the functions qx and px. Plugging the expession 3 into 4 and choosing the values of the spectal paamete z so that κz will be integal, give us the following theoem. 7

9 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 Theoem 2 Let the elements qx, px M of the phase space be given by the Fouie seies qx = α n e inx, px = β n e inx. If fo evey n Z we choose the value of the spectal paamete to be z n = n + n and set ω n = n 2 +, then the coesponding values of the integals of motion of the Klein- Godon system ae given by the fomula F zn qx, px = 2 iω nα n + β n 2, n Z. 4. On integals of the sine-godon system We now take a shot look at the application of the scheme, pesented above, to the evaluation of the integals of the peiodic sine-godon system. Not supisingly, it tuns out that the expessions of the sine-godon integals involve infinitely many Fouie coefficients of qx and px. Theefoe these expessions ae not easily manageable. Nevetheless, they give some insight by poviding the explicit compaison with the integals of the Klein-Godon system. In paticula, let the phase space M be eplaced by the space M P = {qx, px ; qx, px, q x x, p x x peiodic with peiod P }. We shall find an appoximation of the sine-godon integals on M P when P appoaches to zeo. Let Φx; z be the solution of the initial poblem whose quality inceases Φ x x; z = L qx,px z Φx; z, Φ; z = Id, whee L q,p z is given by 6. Denote Hz = Φ2π; z. Then, fo evey value of the spectal paamete z, the function G z qx, px = thz : M R is an integal of the sine-godon system. We shall conside lines of the fom s sqx, spx in M and the expansion of G z sqx, spx with espect to the petubation paamete s. We can think of a line s sqx, spx as of a line of initial conditions of the path of solutions s qx, t; s, px, t; s of the sine-godon system with initial conditions If we deive the equation qx, ; s, px, ; s = sqx, spx. Φ x x; s; z = L sqx,spx Φx; s; z consecutively with espect to s and evaluate at s =, we obtain Φ n x = L Φ n + n j= n L j Φ n j, Φ ; z = Id, Φ n ; z =, n, j 8

10 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 whee we denoted Φ n = dn ds n s= Φs and L n = dn ds n s= L sq,sp. Note that the above equation is the same as 9. The usual integation of a non-homogeneous odinay diffeential equation with constant coefficients gives us Φ n x = n l= n x Φ Φ ξ L l ξφ n l ξ dξ, l whee, as befoe, Φ x = exp xl = e iκx 2 e iκx. 2 Recusive application of the above fomula to itself eventually gives us the expession [ Φ n x = Φ x x L n ξ dξ + n L k ξ L k2 ξ 2 dξ 2 dξ k, k 2 k +k 2 =n 2 x n + L k ξ L k2 ξ 2... k,..., k L kl ξ kl dξ kl... dξ l k +...+k l =n l x + n! L ξ L ξ 2... L ξ n dξ n... dξ ], 6 whee nx and n = k, k 2,..., k l L k x = Φ x L l x Φ x = Ad Φ x Ll x, n! k!k 2!... k l!, is the multinomial symbol. Integals ae taken ove the simplices fo k + k k l = n l x = {ξ, ξ 2,..., ξ l ; x ξ... ξ l }. The integals G z, evaluated at sqx, spx M, ae given by G z sqx, spx = t H sqx,spx z = n= s 2n 2n! tφ2n 2π. Note that taces of Φ n fo odd n ae equal to zeo. By means of the expession 6 the above sum, evaluated at s =, can be eaanged in the fom G z qx, px = L 2 qx, px + 2 m=2 whee we denoted k = k 2π. The tems ae given by L m sqx,spx ξ,..., ξ m = n=m s n [ n! t k +...+k m=n 2m L 2m qx, px, 7 n k,..., k m L k ξ L k2 ξ 2 L ] km ξ m. The leading tem of the development 7 is given by the integations ove the 2-simplex. Evaluation of L 2 gives L 2 q,p ξ, ξ 2 = t L qξ,pξ I L qξ2,pξ 2 I. 9

11 7th Intenational Confeence on Quantum Theoy and Symmeties QTS7 Jounal of Physics: Confeence Seies doi:.88/ /343//29 Since this is a symmetic function of ξ, ξ 2, we get L 2 q,p ξ, ξ 2 = 2π t L qx,px I dx Fo the values of the spectal paamete to z n, given by 5, this tem is equal to L 2 q,p = 2 2 iω nγ n + β n 2 2π 2, + cosqx dx whee, as befoe, ω n = n 2 +, while γ n and β n ae the coefficients in sinqx = γ n e inx, px = β n e inx. The tems belonging to highe dimensional simplices, when evaluated at sq, sp, involve powes of s, highe than 2. Theefoe we clealy have d 2 ds 2 s= G zn sq, sp = d2 ds 2 s= L 2 sq,sp = F zn q, p, 2 whee F zn ae the integals of the Klein-Godon system, constucted in the pevious section. Conside now the functionals G zn : M P R, whee z n = 2πn P + 2πn P 2 + which ae the integals of the sine-godon system on M P defined above. The quality of the appoximation 2 L 2 q, p fo the integals G zn q, p is bette than the appoximation by the Klein-Godon integals F zn and it inceases as P appoaches zeo. The eason fo this is the fact that the volume of the simplex m P is equal to P m m!. Theefoe the absolute values of the highe-ode tems in 7, which ae integals ove m P fo m 4, decease with P faste that the leading tem. Refeences [] Babelon O, Benad D and Talon M 23 Intoduction to Classical Integable Systems Cambidge Univesity Pess: Cambidge [2] Dassios G 27 What non-linea methods offeed to linea poblems? The Fokas tansom method Intenat. J. Non-Linea Mech [3] Faddeev L D and Takhtajan L A 27 Hamiltonian Methods in the Theoy of Solitons Spinge: Belin [4] Fokas A S 22 Integable Nonlinea Evolution Equations on the Half-Line Comm. Math. Phys [5] Fokas A S and Gelfand I M 994 Integability of Linea and Nonlinea Evolution Equations and the Associated Nonlinea Fouie Tansfom Lett. Math. Phys [6] Fokas A S and Sung L Y 25 Genealized Fouie Tansfoms, Thei Nonlineaization and the Imaging of the Bain Notices Ame. Math. Soc., [7] Pelloni B 26 Linea and nonlinea genealized Fouie tansfoms Philos. Tans. R. Soc. Lond. Se. A Math. Phys. Eng. Sci [8] Saksida P 25 Maxwell-Bloch equations, C Neumann system and Kaluza-Klein theoy J. Phys. A: Math. Gen [9] Saksida P 26 Lattices of Neumann oscillatos and Maxwell-Bloch equations Nonlineaity [] Saksida P 2 On zeo-cuvatue condition and Fouie analysis J. Phys. A: Math. Gen