Damped parametric oscillator and exactly solvable complex Burgers equations

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1 Journal of Physics: Conference Series Damped parameric oscillaor and exacly solvable complex Burgers euaions To cie his aricle: Širin A Büyükašik and Okay K Pashaev 1 J. Phys.: Conf. Ser View he aricle online for updaes and enhancemens. Relaed conen - The Hiroa bilinear mehod for he coupled Burgers euaion and he high-order Boussines Burgers euaion Zuo Jin-Ming and Zhang Yao-Ming - Moving boundary problems for he Burgers euaion S De Lillo - Darboux Transformaions and Solion Soluions for Classical Boussines Burgers Euaion Xu Rui This conen was downloaded from IP address on 16/3/19 a 5:35

2 Damped parameric oscillaor and exacly solvable complex Burgers euaions Şirin A. Büyükaşık, Okay K. Pashaev Deparmen of Mahemaics, Izmir Insiue of Technology, 353 Urla, Izmir, Turkey Absrac. We obain exac soluions of a parameric Madelung fluid model wih dissipaion which is linearazible in he form of Schrödinger euaion wih ime variable coefficiens. The corresponding complex Burgers euaion is solved by a generalized Cole-Hopf ransformaion and he dynamics of he pole singulariies is described explicily. In paricular, we give exac soluions for variable parameric Madelung fluid and complex Burgers euaions relaed wih he Surm-Liouville problems for he classical Hermie, Laguerre and Legendre ype orhogonal polynomials. 1. Inroducion In 196, E. Schrödinger derived he fundamenal euaion of uanum mechanics and few monhs laer E. Madelung proposed a represenaion of he complex wave funcion in erms of modulus and phase, Ψ = ρexpis/ h). I convers he linear Schrödinger euaion ino a sysem of nonlinear hydrodynamic-like euaions, [1]. Originally, Madelung inroduced his euaions as an alernaive formulaion for he Schrödinger euaion, and since hen, he Madelung euaions provide a basis for numerous classical inerpreaions of uanum mechanics, []. Laer, Madelung fluid descripion of uanum sysems became a useful ool for describing he evoluion of classical/uanumlike sysems, and sudying he dispersionless or semiclassical limi of nonlinear parial differenial euaions of Schrödinger ype, [3]. As known, he Madelung fluid represenaion is fundamenal also in superconduciviy heory, [] and in he descripion of uanum fluids like superfluid He, [5], where he Madelung hydrodynamic variables have direc physical meaning, such as ρ = Ψ plays he role of he superfluid densiy, and v = argψ) is he superfluid velociy. From mahemaical poin of view, we emphasize a remarkable propery concerning inegrabiliy of he Madelung models. Since Madelung ransform of he linear Schrödinger euaion leads o nonlinear Madelung sysems, using an inverse Madelung ransform, clearly he nonlinear Madelung models are linearizable in he form of Schrödinger euaion. Then, he inverse Madelung ransform is a complex linearizaion ransform similar o he Cole-Hopf ransformaion, and nonlinear models admiing such ype of direc linearizaion are known as C-inegrable models. From his perspecive, Madelung fluid model and he complex Burgers euaion are nonlinear sysems belonging o he class of C-inegrable models. This allows us o describe heir nonlinear dynamics in erms of he corresponding linear Schrödinger euaion. Published under licence by Ld 1

3 Long ime ago he parameric Schrödinger model for harmonic oscillaor was proposed by A. Sakharov as descripive of uanum cosmological models, [6]. Moreover, in molecular physics, uanum chemisry, uanum opics and plasma physics many uanum-mechanical effecs are reaed by means of ime-dependen oscillaor, [7], [8]. Then, he Madelung fluid represenaion appears as a dual descripion of he same realisic sysems in erms of uanum hydrodynamic variables. Moivaed by his idea, in [9], we found exac soluions for nonlinear parameric Madelung model relaed wih he Caldirola-Kanai dissipaive oscillaor, [1],[11]. I was seen ha he probabiliy densiy ρ k,) sueezes and ends o Dirac-dela disribuion as, and he moving pole singulariies k l ), l = 1,,...,k of he complex velociy funcion V k,) are merging wih ime due o dissipaion, i.e. k l ) as. In his work, we inroduce generic parameric Madelung fluids wih dissipaion and complex Burgers euaions linearizable in he form of parameric Schrödinger euaion for harmonic oscillaor relaed wih he classical orhogonal polynomials. Indeed, uanizaion of he singular Surm-Liouville problems provides a rich se of exacly solvable harmonic oscillaor models wih ime dependen parameers, known as uanum Surm-Liouville problems, [1]. Then, exac soluions of he corresponding Madelung models are obained using generalized Cole- Hopf ransformaion. Precisely, we discuss Madelung fluid and complex Burgers euaions of Hermie, Laguerre and Legendre ype. For each ype, exac soluions are obained, dynamics of he zeros of he probabiliy densiy, and dynamics of he pole singulariies of he complex velociy funcion are described explicily. Compuer plos are given o illusrae some ypical behavior of he soluions.. Quanum Surm-Liouville problems and he Madelung represenaion In his secion we ouline some basic resuls discussed in [1] and [9]. Consider he second order differenial euaion [ d µ) d ] x) + [λr)]x) =, a < < b, 1) d d where λ is a specral parameer, µ) > on a,b), µ) C 1 a,b), and λr) > on a,b), r) Ca,b). If singulariies are allowed a he end poins of he fundamenal domain a,b), and boundary condiions are imposed like x) mus be coninuous or bounded or become infinie of an order less han he prescribed, one has he well known singular Surm-Liouville problem, [13]. On he oher hand, E.1) can be wrien in he form of damped parameric oscillaor ẍ + µ)/µ))ẋ + λr)/µ))x =, a,b), where Γ) = µ)/µ) is he damping coefficien and λr)/µ) is he freuency. Then, self-adjoin uanizaion leads o ime-dependen uanum Hamilonian, and Schrödinger euaions for harmonic oscillaor relaed wih he classical orhogonal polynomials, also known as uanum Surm-Liouville problems, see [1]. For generaliy, we shall use noaion λr) µ)ω ), and consider he IVP i h Ψ = h Ψ µ) + µ)ω ) Ψ, ) Ψ, ) = ψ), < <. 3) I is known ha, if x) is soluion of he IVP for he classical parameric oscillaor ẍ + µ) µ)ẋ + ω )x =, x ) = x, ẋ ) =, )

4 and if he iniial funcion Ψ, ) is given as hen he ime-evolved sae is exp i ϕ k ) = N k e Ω H k Ω ), k =,1,,..., 5) µ)ẋ) hx) Ω g)r ) Ψ k,) = N k R) exp i ) ) k + 1 ) ) arcanω g)) exp Ω ) ) R ) H k Ω R), 6) where g) = hx ) dξ µξ)x ξ), g ) = ; R) = x x ) + Ω x)g)) )1, 7) and he corresponding probabiliy densiy ρ k,) = Ψ k,) becomes ρ k,) = 1 k k! π ) ) Ω R) exp Ω R) Hk Ω R)), k =,1,,... 8) Now, wriing he wave funcion in polar form Ψ,) = ) ī 1 ρ,) exp h S,) = exp ln ρ,) + ī ) h S,), 9) where ρ, ), S, ) is real-valued funcion and seing v, ) = 1/µ)) S/), Schrödinger euaion ) ransforms o a sysem of parameric Madelung fluid euaions wih dissipaion v + µ) µ) v + v v = 1 [ h µ) µ) ρ + [ρv] =. 1 ) ρ ρ ] + µ)ω ), Here, ρ,) is he probabiliy densiy, v,) is he velociy field and Γ) = µ)/µ) is he fricion coefficien, which for nonconsan µ) reflecs he dissipaive naure of he sysem. Wih real-valued iniial condiions v, ) = ṽ), ρ, ) = ρ), i has formal soluion 1) v,) = i h ) Ψ,) µ) ln, ρ,) = Ψ,), 11) Ψ, ) where Ψ,) is soluion of he Schrödinger euaion ) wih iniial condiion Ψ, ) = i ) ρ) exp h µ ) ṽξ)dξ. 1) Clearly, he explici form of he soluions v,) and ρ,) depends on he properies of he iniial funcions ṽ) and ρ). In his work, we shall consider iniial funcions ṽ), ρ), so ha in 1) one has Ψ, ) L R). Specifically, for all paricular problems he iniial condiions will be aken such ha Ψ, ) = ϕ k ), where ϕ k ) is given by 5). 3

5 Anoher represenaion of he wave funcion in he form i ) Ψ,) = exp h µ) V ξ,)dξ, 13) where V, ) is a complex velociy, ransforms Schrödinger euaion ) o a nonlinear complex Burgers euaion wih ime dependen coefficiens. Then, he corresponding general IVP + µ) µ) V + V + ω ) = i h V µ), 1) V, ) = Ṽ ), has formal soluion given by he generalized complex Cole-Hopf ransformaion V,) = i h ln Ψ,)), 15) µ) where Ψ,) is soluion of he Schrödinger euaion ) wih he iniial condiion ī ) Ψ, ) = exp h µ ) Ṽ ξ)dξ. 3. The Hermie ype Madelung fluid and complex Burgers euaion Here, we inroduce nonlinear Madelung model relaed wih he Surm-Liouville problem for he classical Hermie polynomials. For his, we chose exponenially decreasing mass µ) = e, and consan freuency ω ) = n,, ). Then, we obain he sysem of Hermie ype Madelung fluid euaions v v + v v = e ρ + [ρv] =. [ h e 1 ) ] ρ ρ + ne, Using ransformaion v,) = i he ) Ψ,) ln, Ψ, ) ρ,) = Ψ,), 17) sysem 16) is linearizable in he form of Hermie ype Schrödinger euaion 16) i h Ψ = he Ψ + ne Ψ, 18) where µ) = e and ω ) = n,, ), n =,1,,... I follows ha sysem 16) wih general iniial condiions v, ) = ṽ), ρ, ) = ρ), has formal soluion given by 17), where Ψ, ) is soluion of he Schrödinger euaion 18) wih iniial condiion Ψ, ) = ρ) exp i/ h)e is Ψ, ) = ϕ k ), is soluion akes he form ṽξ)dξ). In paricular, when he iniial sae for E.18) i h ) ) Ḣn) Ψ n,k,) = N k R n ) exp e exp i ) H n ) Ω g n)r n ) exp i k + 1 ) ) arcanω g n )) exp Ω ) R n) ) H k Ω R n ), k =,1,,..., 19)

6 . Ρ ÈVÈ^ a) - b) Figure 1: Hermie ype model. a) Probabiliy densiy ρ,, ). b) Plo of V,, ). where Hn ), n =, 1,,... are he Hermie polynomials saisfying H n H n + nhn =, Hn ) 6=, H n ) =, and he auxiliary funcions are gn ) = h Hn ) Z eξ dξ, g ) = ; Hn ξ) Rn ) = Hn ) Hn ) + [Ω gn )Hn )]!1. ) Therefore, sysem 16) wih specific iniial condiions vk, ) =, has exac soluions vn, ) = and ρn,k, ) = p p ρk, ) = Nk exp Ω ) Hk Ω ), Nk k =, 1,,...! H n ) h Ω e gn )Rn ), k =, 1,,..., Hn ) Rn ) exp p Ω Rn ) Hk p Ω Rn ), 1) where gn ) and Rn ) are defined by ). Nex, we inroduce he Hermie ype complex Burgers euaion for complex velociy field V, ) ih V V + V + n = e, n =, 1,,... ) By he generalized complex Cole-Hopf ransformaion V, ) = ih e ln Ψ, )) 3) i linearizes in he form of Schro dinger euaion 18). Then, he IVP for ) wih iniial condiion V, ) = Ve ) has formal soluion given by 3), where Ψ, ) is soluion of he IVP R for he Schro dinger euaion 18) wih iniial condiion Ψ, ) = exp i/h )e Ve ξ)dξ. 5

7 a) Figure : Hermie ype model. Moving poles of V n,k,), n =,k =. As an example, we consider he IVP wih specific iniial condiions V + V + n = i h V e, n =,1,,..., [ V k, ) = i he Ω H k ] Ω ) H k, k =,1,,... Ω ) ) I has explici soluion ] [ [Ḣn ) V n,k,) = H n ) hω e g n )Rn ) + i he Ω Rn ) H k ] Ω R n )) H k. 5) Ω R n )) Noe ha, since soluion Ψ n,k,) of he Hermie uanum oscillaor given by 19) has zeros a poins where H k Ω R n ) ) =, hen due o Cole-Hopf ransformaion 3), hese zeros become pole singulariies of he complex Burgers soluion V n,k,) given by 5). If τ l) k, l = 1,,...,k, denoe he zeros of he Hermie polynomial H kξ), hen he moion of he zeros of Ψ n,k,) and poles of V n,k,) for fixed n =,1,... and k = 1,,... is described by l) k ) = τ l) k Ω R n ) = τl) k H n ) + [Ω g n )H n )] Ω Hn ) )1, l = 1,,...,k. 6) In Fig.1 we show he behavior of he probabiliy densiy ρ n,k,) and V n,k,) for n =,k =. In Fig., we illusrae he moion of he zeros/poles for he case k =, n =. Clearly, k is he number of he moving zeros/poles and n is he number of oscillaions. A imes close o =, he singulariies show oscillaory moion. Then, when ±, one has R n ) and l) ), showing ha he zeros/poles go away from he origin =. Moreover, he k disance beween differen zeros/poles also increases wih ime, ha is for fixed k and i j, i) k ) j) k ) as ±.. The Laguerre ype Madelung fluid and complex Burgers euaion Now, we inroduce Madelung sysem relaed o he Surm-Liouville problem for he Laguerre polynomials. We sar by choosing ime variable coefficiens of he form µ) = e and 6

8 ω ) = n/,, ). This leads o Laguerre ype parameric Madelung fluid sysem v + 1 )v + v v [ = h e 1 ) ] ρ e ρ + ne, ρ + [ρv] =, 7) wih variable fricion Γ) = 1 )/,, ). Using ransformaion v,) = i he ) Ψ,) ln, ρ,) = Ψ,), 8) Ψ, ) i can be linearized in he form of Laguerre ype Schrödinger euaion i h Ψ = h e ne Ψ + Ψ, 9) where µ) = e, and ω ) = n/,, ), n =,1,,... Then, sysem 7) wih general iniial condiions v, ) = ṽ), ρ, ) = ρ), has formal soluion given by 8), where Ψ,) is soluion of 9) wih iniial condiion Ψ, ) = ρ) exp i/ h) e ṽξ)dξ ). The nonsaionary Schrödinger euaion 9) wih iniial condiion Ψ, ) = ϕ k ), k =, 1,,..., has exac soluion i Ψ n,k,) = N k R n ) exp e L ) ) n ) exp i h L n ) Ωg n )R n) ) exp i k + 1 ) ) arcanω g n )) exp Ω ) R n ) ) H k Ω R n ), k =,1,,..., 3) where L n ), n =,1,,... are he Laguerre polynomials saisfying L n + 1 ) and he auxiliary funcions are g n ) = hl n ) L n + n L n =, L n ), Ln ) =, e ξ dξ ξl nξ), g ) = ; R n ) = Therefore, sysem 7) wih specific iniial condiions v k, ) =, has exac soluions v n,) = L n ) L n ) + [Ω g n )L n )] ρ k, ) = Nk exp Ω ) ) Hk Ω ), k =,1,,..., ) Ln ) L n ) e g n )Rn) hω, k =,1,,... )1. 31) and ) ) ρ n,k,) = Nk R n ) exp Ω R n ) Hk Ω R n )), 3) 7

9 1.6 Ρ. ÈVÈ^ a) b) Figure 3: Laguerre ype model. a) Probabiliy densiy ρ,3, ). b) Plo of V,3, ). where gn ) and Rn ) are given by 31). Nex model which we give is he Laguerre ype complex Burgers euaion n h e V 1 + )V + V + =i, n =, 1,,... 33) By he complex Cole-Hopf ransformaion V, ) = ih e ln Ψ, )) 3) i is linearizable in he form of Schro dinger euaion 9). Then, E.33) wih general iniial condiion V, ) = Ve ) has soluion given by 3), where Ψ, ) is soluion of 9) wih iniial R e daa Ψ, ) = exp i/h ) e V ξ)dξ. The special IVP n h e V 1 + )V + V + = i, n =, 1,,..., " # Hk Ω ) h e d ih e ln ϕk )) = Ω, Vk, ) = i d Hk Ω ) 35) k =, 1,,... has exac soluions of he form # " # g )e h e H Ω R )) L n ) n n k h Ω Rn ) + i Ω Rn ). Vn,k, ) = Ln ) Hk Ω Rn )) " 36) Since soluion of he Schro dinger euaion, Ψn,k, ) given by 3) has zeros a poins where Hk Ω Rn ) =, hen hese zeros are pole singulariies of he soluion Vn,k, ) given by 36). Explicily, he moion of he zeros/poles for fixed n =, 1,,... and k = 1,,... is described by l) k ) l) l) τ τ = k = k Ω Rn ) Ω Ln ) + [Ω gn )Ln )] Ln )!1, l = 1,,..., k. 37) In Fig.3, we illusrae he behavior of ρn,k, ) and Vn,k, ) for n =, k = 3, h = Ω = 1). We observe ha, a zeros of L ) here are finie ime singulariies of Vn,k, ), and also since k = 3, here are hree moving zeros of ρ,3, ), which are also he moving poles of V,3, ). 8

10 5. Legendre ype Madelung fluid and complex Burgers euaion Las models which we inroduce are relaed wih he Legendre polynomials. For his, we consider ime variable coefficiens µ) = 1 ) and ω ) = nn + 1)/ 1 ), 1,1). Tha gives a sysem of parameric Legendre ype Madelung hydrodynamic euaions v 1 ) v + v v = 1 ρ + [ρv] =, [ 1 ) h 1 ) 1 ) ρ ρ + ] nn + 1), wih fricion coefficien Γ) = / 1 ), 1,1). Under ransformaion 38) v,) = i h ) Ψ,) 1 ) ln, ρ,) = Ψ,), 39) Ψ, ) i convers o he Legendre ype Schrödinger euaion i h Ψ = h 1 ) nn + 1) Ψ + Ψ, ) where 1,1), µ) = 1 ), and ω ) = nn + 1)/ 1 ), n =,1,,... Then, sysem 38) wih general iniial condiions v, ) = ṽ), ρ, ) = ρ), has formal soluion given by 39), where Ψ,) is soluion of ) wih iniial condiion Ψ, ) = ρ)exp i/ h)1 ) ṽξ)dξ ). E. ) wih iniial condiions Ψ, ) = ϕ k ) has exac soluions Ψ n,k,) = N k R n ) exp exp i k + 1 ) i 1 ) P n ) h P n ) ) ) ) exp i ) Ω g n)r n ) arcanω g n )) exp Ω ) R n ) ) H k Ω R n ), k =,1,,..., 1) where P n ), n =,1,,... are he classical Legendre polynomials saisfying P n and he auxiliary funcions are g n ) = hp n ) 1 P ) nn + 1) n + 1 ) P n =, P n ), P n ) = dξ 1 ξ )Pnξ), g ) = ; R n ) = P n ) P n ) + [Ω g n )P n )] )1. ) Therefore, sysem 38) wih specific iniial condiions v k, ) =, ρ k, ) = ϕ k ), k =, 1,,..., has exac soluions P n ) v n,) = P n ) g n )R hω n ) ) 1, k =,1,,... 3) ) and ) ) ρ n,k,) = Nk R n ) exp Ω R n ) Hk Ω R n )). ) 9

11 .6 3 Ρ. ÈVÈ^ a) b) Figure : Legendre ype model. a) Probabiliy densiy ρ,, ). b)plo of V,, ). Finally, we wrie he Legendre ype complex Burgers euaion nn + 1) h V + V + = i, n =, 1,,... 1 ) 1 ) 1 ) 5) By he generalized Cole-Hopf ransform V, ) = ih ln Ψ, )), 1 6) i is linearizable in he form of Schro dinger euaion ). Then, he complex Burgers euaion 5) wih general iniial condiion V, ) = Ve ), has formal by 6), soluion given where R e Ψ, ) is soluion of ) wih iniial condiion Ψ, ) = exp i/h )1 ) V ξ)dξ. In paricular, he complex Burgers euaion 5) wih specific iniial daa " # Hk Ω ) ih Ω, k =, 1,,..., Vk, ) = 1 ) Hk Ω ) has exac soluion # " # g ) h H Ω R )) P n ) n n k h Ω R ) + i Ω Rn ). 7) Vn,k, ) = Pn ) 1 ) n 1 ) Hk Ω Rn )) " The moion of he zeros of Ψn,k, ) given by 1), and poles of soluion Vn,k, ) given by 7), for fixed n =, 1,,... and k = 1,,... is described by l) k ) l) l) τ τ = k = k Ω Rn ) Ω Pn ) + [Ω gn )Pn )] Pn )!1, l = 1,,..., k. 8) In Fig., we give he plo of ρn,k, ), and Vn,k, ) for n =, k =. One can observe also, he four zeros and poles respecively, which show oscillaory moion on he ime inerval -1,1). 1

12 Acknowledgmens This work is suppored by TÜBİTAK, Projec No: 11T679. References [1] Madelung E 196 Z. Phys., 3 [] Holland P R 1993 The Quanum Theory of Moion, Cambridge Universiy Press) [3] Zakharov V E 199 NATO Adv. Sci. Ins. Ser. B, Phys. 3, N.M. Ercolani e al, eds, NY Plenum) [] Feynman R 1965 The Feynman Lecures on Physics, Quanum Mechanics, Addison-Wesley) [5] Feynman R 1998 Saisical Mechanics, Wesview Press, nd Ed) [6] Sakharov A 1965 Zh. Eksp. Teor. Fiz [ 1966 Sov. Phys. JETP 1] [7] Lewis H R, Riesenfeld W B 1969 J. Mah. Phys. 1 8, 158 [8] Pedrosa I A, Serra G P, Guides I 1997 Phys. Rev. A 56 5, 3 [9] Büyükaşık Ş A, Pashaev O K 1 J. Mah. Phys [1] Caldirola P 191 Nouovo Cimeno 18 [11] Kanai E 198 Prog.Theo.Physics 3 [1] Büyükaşık Ş A, Pashaev O K, Ulaş-Tigrak E 9 J. Mah. Phys 5 71 [13] Couran R, Hilber D Mehods of Mahemaical Physics Vol.I, NY J. Wiley and Sons) 11