Remrk on boundry vlue problems rising in Ginzburg-Lndu theory ANITA KIRICHUKA Dugvpils University Vienibs Street 13, LV-541 Dugvpils LATVIA nit.kiricuk@du.lv FELIX SADYRBAEV University of Ltvi Institute of Mthemtics nd Computer Science Rin bulvris 9, LV-1459, Rig LATVIA felix@ltnet.lv Abstrct: The eqution x (x x 3 ) (i) is considered together with the boundry conditions x (), x (1) (ii), x (), x (T ) (iii). The exct number of solutions for the boundry vlue problems (BVP) (i), (ii) nd (i), (iii) is given. The problem of finding the initil vlues x x() of solutions to the problem (i), (iii) is solved lso. Key Words: Boundry vlue problem, Jcobin elliptic functions, cubic nonlinerity, phse trjectory, multiplicity of solutions 1 Introduction The cubic complex Ginzburg - Lndu eqution t A A + (1 + ib) A + (1 + ic) A A, ppers in numerous descriptions of remrkble physicl phenomen. It suffices to mention the superconductivity theory [6]. A survey of multiple pplictions cn be found in [] nd [11]. Sttionry solutions of rel Ginzburg - Lndu eqution [] t A A + A A A generte vrious problems for ordinry differentil equtions. It ws mentioned in the work [9] tht there re lcking in the literture the results on boundry vlue problems tht rise in Ginzburg - Lndu theory of superconductivity. Nmely, the boundry vlue problem x (x x 3 ), (1) x (), x (1) () ws mentioned nd the problem of finding the initil vlues x of solutions of the problem (1), x (), x (T ), (3) tht is interpreted s eigenvlue problem. Our intent is to fill this gp. First, we nlyze the problem nd provide the exct number of solutions depending on the prmeter. Despite of the fct tht the phse portrit of the eqution is very well known we re unwre of the proof tht exct number of solutions to the Neumnn problem depends entirely on the properties of solutions of the linerized eqution round the zero equilibrium point. To prove this, monotonicity of period for closed trjectories lying in G (the region of phse plne between two heteroclinic trjectories) must be proved first. Second, using the theory of Jcobin elliptic functions, we give explicit expressions for solutions of the problem. Avoiding cumbersome formuls provided by symbolic computtion softwre. Generlly it is known tht solutions of the eqution cn be expressed in terms of the Jcobin elliptic functions. However, it is nor esy nor convenient to get the respective formuls explicitly. Only few sources provide the relted informtion nd generlly re useless in specific contexts like the Neumnn boundry vlue problem. Stndrd softwre provides generl expressions tht re not esy to use in order to get explicit nlyticl formuls for specific cses under considertion. Formuls for solutions of the qudrtic eqution re quite different for solutions tht behve differently in different subsets of phse plne. For instnce, no solutions of the Neumnn BVP cn exist (if speking bout the trjectories of solutions) outside the region G. We mde ll the necessry computtions for the Neumnn problem by ourselves nd the results look quite stisfctory. The explicit formul for solutions of the respective Cuchy problem ws obtined. In order to study the problem numericlly pproximtions of the initil vlues for possible solutions re needed. We derived the eqution for finding the initil vlues x of solutions to the Neumnn problem. E-ISSN: 4-88 9 Volume 17, 18
These formuls involve severl Jcobin elliptic functions nd cn be used to effectively find the initil vlues of solutions. The initil conditions of the Neumnn problem (1), () cn be found esily now s zeros of some one rgument function. The dvntges of the proposed pproch re demonstrted considering the exmple in the finl section. Phse portrit First consider eqution (1). The phse portrit is wellknown. There re three criticl points of eqution (1) t x 1,3 ±1, x. The origin is center nd x 1,3 ±1 both re sddle points. Two heteroclinic trjectories connect the two sddle points, Fig. 1. nd therefore the respective C 1. Then ny trjectory locted in the region G stisfies the reltion (4), where C < 1. Since ny trjectory in the region G is closed it is convenient to consider the respective solutions in polr coordintes. Introduce polr coordintes by x(t) r(t) sin φ(t), x (t) r(t) cos φ(t) (6) Eqution (1) written in polr coordintes turns to system: { φ (t) sin φ(t) r (t) sin 4 φ(t) + cos φ(t), r (t) 1 r(t) sin φ(t) ( 1 + r (t) sin φ(t) ). (7) Consider ny solution of eqution (1) with the initil conditions (x(t ), x (t )) G. Let initil conditions be written s φ(t ) φ, r(t ) r, (φ, r ) G, r >. (8) Lemm 1 The ngulr function φ(t) of ny solution of (7), (8) is monotoniclly incresing. Proof: Consider the first eqution of system (7) multiplied by r Figure 1: The phse portrit of eqution (1), shded is the region G Any solution of (1) stisfies the energy reltion x (t) x (t) + 1 x4 (t) + C, (4) where C is n rbitrry constnt. It is cler from the nlysis of the phse portrit tht solutions (trjectories) of the Neumnn problem cn exist only in the region (we denote it G) between the heteroclinic trjectories. 3 Monotonicity result Denote open region bounded by the two heteroclinic trjectories connecting sddle points by G. Consider trjectories (closed curves) tht fill the region G. The heteroclinic solution t infinity stisfies x ( ) x ( ) + 1 x4 ( ) + C (x ( ) 1) + 1 + C (5) r φ (t) r sin φ(t) r 4 (t) sin 4 φ(t) + r cos φ(t). (9) Returning to (x, y) coordintes r φ (t) x (t) x 4 (t) + y (t) x (t)(1 x (t)) + y (t) >. (1) Since x (t) < 1 for ny solution of (1) locted in the domin G, the ngulr function φ(t) is incresing. Corollry Let x(t) be solution of eqution (1) with the initil conditions in G. Then between ny two consecutive zeros of x(t) there is exctly one point of extremum. 4 Existence nd multiplicity theorem We cn prove the following result considering the boundry vlue problem (1), (3). Eqution (1) hs n integrl x (t) x (t) + 1 x4 (t) + C, (11) where C is n rbitrry constnt. Solutions x(t; x ) of the Cuchy problem (1), (1) x() x, x (), < x < 1 (1) E-ISSN: 4-88 91 Volume 17, 18
stisfy the reltion (11) where C x 1 x4. Denote solution of the Cuchy problem (1), (1) by x(t; x ). The series of trnsformtions x (t; x ) x (t; x ) + 1 x4 (t; x ) + x 1 x4, dx dt ± x (t; x ) + 1 x4 (t; x ) + x 1 x4, (13) (notice tht x (t; x ) < ) nd therefore dx dt x (t; x ) + 1 x4 (t; x ) + x 1 x4 (14) dx leds to x x x x + 1 x4 +x 1 x4 dx x + 1 x4 +x 1 x4 t dt t dx x + 1 x4 1 x4 +x x 1 dx (x 1 x4 ) (x 1 x4 ) ξ x x 1 T x (1 ξ ) 1 x (1 ξ4 ) dt T x, (15) (16) where T x is the time needed to move on phse plne from (x, ) to the verticl xis x ( qurter of period). It follows then tht T x is incresing function of x. Thus the following lemm ws proved. Lemm 3 The function T x monotoniclly increses (from π, this will be shown below) to + s x chnges from zero to 1. The exct number of solutions for problem (1), (3) is given by Theorem 4. Theorem 4 Let i be positive integer such tht i π (i + 1)π < T <, (17) where T is the right end point of the intervl in (3). The Neumnn problem (1), (3) hs exctly i nontrivil solutions such tht x() x, x (), 1 < x < 1. Proof: Consider solutions of the Cuchy problem (1), x() x, x (), where < x < 1. Solutions for x smll enough behve like solutions of the eqution of vritions y y round the trivil solution. The solution of the linerized eqution is y(t) x cos t. (18) Due to the ssumption i π (i + 1)π < T < solutions y(t) long with solutions x(t; x ) (for smll enough x ) hve exctly i extrem in the intervl (, T ) nd t T is not n extremum point. These extrem due to Lemm 3 move monotoniclly to the right s x increses. Solutions x(t; x ) with < x < 1 nd close enough to 1 hve not extrem in (, 1] since the respective trjectories re close to the upper heteroclinic (nd the period of heteroclinic solution is infinite). Therefore there re exctly i solutions of the problem (1), (3). The dditionl i solutions re obtined considering solutions with x ( 1, ) due to symmetry rguments. Hence the proof. The phse plne nlysis ws used to study multiple solutions of BVP in [8], [1], [13]. The review of some spects of multiple solutions of BVP is in [3]. The relted results re in [4], [5]. 5 Eigenvlue problem Let us ddress the eigenvlue problem posed in [9]. Consider the Cuchy problem (1), (1): x (x x 3 ), x() x, x (), < x < 1. Let nd T (in (3)) be given. We wish to find x such tht the respective solutions x(t; x ) of the bove problem stisfy the boundry condition x (T ), i.e. x(t; x ) solve the Neumnn problem (1), (3). The following ssertion provides the explicit formul for solution of (1), (1). Lemm 5 The function x(t,, x ) x cd where k, is solution of the Cuchy problem (1), (1). x x ( x ) t; k, (19) Proof: Consider eqution (16) where the qurter of E-ISSN: 4-88 9 Volume 17, 18
period T x is given by formul T x 1 (1 ξ ) 1 x (1 ξ4 ) 1 x (1 ξ )(1 x x ξ ) ( x ) (1 ξ )(1 x x ξ ) ( x ) (1 ξ )(1 k ξ ) ( x )K(k), where < k x x we hve K(k) 1 () < 1. Replcing ξ sin φ(t) π (1 ξ )(1 k ξ ) dφ 1 k sin φ(t). Inverse function of function F (φ(t), k) t F (φ(t), k) φ(t) (1) t () 1 k sin ψ(t) is Jcobin mplitude φ(t) m(t, k) [14], [1], but sin φ(t) sin m(t, k) sn(t, k). For solutions of the problem (1), (3) ( x ) t π φ(t) 1 k sin ψ(t) π φ(t) 1 k sin ψ(t) 1 k sin ψ(t) K(k) φ(t) 1 k sin ψ(t), (3) F (φ(t), k) φ(t) 1 k sin ψ(t) (4) ( x K(k) ) t, ( x φ(t) m K(k) ) t, k, (5) [ sin φ(t) sin m sn [ K(k) K(k) ( x ) ( x ) t, k Note tht the x(t) x ξ x sin φ(t) x(t) x sn K(k) ( x ) ] t, k ]. (6) t, k. (7) It cn be shown by using the reduction formul sn(k u) cd u ([1]) x(t) x cd ( x ) t, k. (8) Denote f(t,, x ) x t(t,, x ). This derivtive cn be computed nd the following formul is vlid. One obtins using [1, p.] (nd Wolfrm Mthemtic) tht ( ) ( x ) f(t,, x ) x cd t t; k ( ) ( x ) ( x nd ) t; k sd We hve rrived t the sttement. x (k 1) ( ) ( x ) t; k. (9) Lemm 6 For given nd T the eigenvlue problem (1), (3) cn be solved by solving the below eqution with respect to x f(t,, x ). (3) Theorem 7 A solution to the Neumnn problem (1), (3) is given by (19) where x is solution of (3). Applictions of the theory of Jcobin elliptic functions to vrious problems for ordinry differentil equtions cn be found lso in [], [7]. 6 Results for n-th order eqution Mny of the fcts tht were estblished previously for the cubic cse re vlid lso for cses of higher degree polynomils in the right side of equtions. Consider equtions of the type x (x x 3 ), x [(x x 3 ) + (x 5 x 7 )], x [(x x 3 ) + (x 5 x 7 ) + (x 9 x 11 )],... x [(x x 3 ) + (x 5 x 7 ) + (x 9 x 11 ) +... + +(x n 1 x n+1 )]. (31) All equtions hve only three criticl points t x 1,3 ±1, x. The origin is center nd x 1,3 ±1 both re sddle points. Two heteroclinic trjectories connect the two sddle points. The phse portrits of these equtions re similr to tht for the cubic cse s depicted in Fig. 1. The nlogues of Theorem 4 re vlid lso for the equtions of type (31). We formulte the following theorem. E-ISSN: 4-88 93 Volume 17, 18
Theorem 8 Let i be positive integer such tht i π (i + 1)π < T <, (3) where T is the right end point of the intervl in (3). The Neumnn problem where eqution is of type s in eqution (31) with condition (3) hs exctly i nontrivil solutions such tht x() x, x (), 1 < x < 1. Proof: The proof is similr to tht for Theorem 4 becuse solutions of the Cuchy problem (1), x() x, x (), where < x < 1 for x smll enough behve like solutions of the eqution of vritions y y round the trivil solution. The solution of the linerized eqution is s given in (18). Due to the ssumption i π < T < solutions (i + 1)π y(t) long with solutions x(t; x ) (for smll enough x ) hve exctly i extrem in the intervl (, T ) nd t T is not n extremum point. At the sme time solutions with x tending to the criticl point t x 1 hve not points of extrem since the periods of such solutions tend to infinity s x goes to 1. The extrem of solutions x(t; x ) leve the intervl (, T ) pssing through the point T s x goes to 1. Therefore there re exctly i solutions of the problem nd the dditionl i solutions re obtined considering solutions with x ( 1, ) due to symmetry rguments. The vlue T x defined in formul () seems to be monotoniclly incresing for ll equtions (31) s computtions show. We believe tht this cn be proved similrly to the proof of Lemm 3. Figure : The grph of f(1, 11, x ) The grph of f(1, 11, x ) is depicted in Fig.. There re three zeros of (34) nd, respectively, three initil vlues x t x.59, x.913 nd x.998. In Fig. 3 nd Fig. 4 respectively the grphs of solutions x(t) of the problem (33), () nd its derivtives x (t) re depicted. Figure 3: Grphs x(t) for solutions of the problem (33), (), x.59 (solid), x.913 (dshing tiny), x.998 (dshing lrge) 7 Exmple Consider eqution (1) with 11: x 11(x x 3 ). (33) Let the initil conditions be x() x, x (), < x < 1. Then the number of solutions stisfying the boundry conditions (), x(), x (1), is three nd, symmetriclly, for initil conditions x() x, x (), 1 < x < there re lso three solutions to the problem (33), (), totlly six nontrivil solutions. By Theorem 4, this is the cse for T 1 nd i 3. Indeed, 3 π 11 < T 1 < 4 π 11 ) in the inequlity (17). Consider eqution (3) with 11 nd T1, then f(1, 11, x ) 11x (k 1) 1 1 x ( ) ( ) nd 11 1 1 x ; k sd 11 1 1 x ; k. (34) Figure 4: Grphs x (t) for solutions of the problem (33), (), x.59 (solid), x.913 (dshing tiny), x.998 (dshing lrge) For incresing T the number of zeros x of eqution (3) increses lso nd the number of solutions of the problem (33), () increses consequently. The exmples for T nd T 3 re given in Fig. 5 nd Fig. 6 respectively. E-ISSN: 4-88 94 Volume 17, 18
Figure 5: The grph of f(, 11, x ), T, the number of solutions is six Figure 6: The grph of f(3, 11, x ), T 3, there re ten solutions, becuse 1 π < 3 < 11 π π 11 11 8 Conclusion The number of solutions of the boundry vlue problem (1), (3) depends entirely on the coefficient (for given T ) nd is known precisely (Theorem 4). The initil vlues x for solutions of boundry vlue problem (1), (3) cn be found by solving the eqution (3) tht is composed of certin Jcobin elliptic functions. Then the solutions of the Neumnn problem (1), (3) re known nlyticlly using the formul (19). References: [1] J. V. Armitge, W. F. Eberlein, Elliptic Functions, Cmbridge University Press, 6. [] I. Arnson, The World of the Complex Ginzburg-Lndu Eqution, Rev. Mod. Phys. Vol. 74, 99,. Avilble t https://doi.org/1.113/revmodphys.74.99. [3] M. Dobkevich, F. Sdyrbev, N. Sveikte, nd I. Yermchenko, On Types of Solutions of the Second Order Nonliner Boundry Vlue Problems, Abstrct nd Applied Anlysis, Vol. 14, Spec. Is. (13), Article ID 594931, 9 pges. [4] M. Dobkevich, F.Sdyrbev, Types nd Multiplicity of Solutions to Sturm - Liouville Boundry Vlue Problem, Mthemticl Modelling nd Anlysis, Vol., 15 - Issue 1,1-8. [5] M. Dobkevich, F.Sdyrbev, On Different Type Solutions of Boundry Vlue Problems, Mthemticl Modelling nd Anlysis, Vol. 1, 16 - Issue 5, 659-667. [6] V. L. Ginzburg, Nobel Lecture: On superconductivity nd superfluidity (wht I hve nd hve not mnged to do) s well s on the physicl minimum t the beginning of the XXI century, Rev. Mod. Phys., Vol. 76, No. 3, 4, pp. 981 998. [7] K. Johnnessen, A Nonliner Differentil Eqution Relted to the Jcobi Elliptic Functions, Interntionl Journl of Differentil Equtions, Vol. 1, Article ID 41569, 9 pges, http://dx.doi.org/1.1155/1/41569. [8] A. Kirichuk, F. Sdyrbev, Multiple Positive Solutions for the Dirichlet Boundry Vlue Problems by Phse Plne Anlysis, Abstrct nd Applied Anlysis, Vol. 15(15), http://www.hindwi.com/journls//15/ 3185/. [9] N. B. Konyukhov, A. A. Shein, On n Auxiliry Nonliner Boundry Vlue Problem in the Ginzburg Lndu Theory of Superconductivity nd its Multiple Solution, RUDN Journl of Mthemtics, Informtion Sciences nd Physics, Vol. 3, pp. 5, 16. Avilble t http://journls.rudn.ru/miph/rticle/view/13385. [1] L. M. Milne-Thomson, Hndbook of Mthemticl Functions, Chpter 16. Jcobin Elliptic Functions nd Thet Functions, Dover Publictions, New York, NY, USA, 197, Edited by: M. Abrmowitz nd I. A. Stegun. [11] Wim vn Srloos nd P. C. Hohenberg, Fronts, pulses, sources nd sinks in generlized complex Ginzburg-Lndu equtions, Physic D (North- Hollnd), Vol. 56 (199), pp. 33 367. [1] Kun-Ju Hung, Yi-Jung Lee, Tzung-Shin Yeh, Clssifiction of bifurction curves of positive solutions for nonpositone problem with qurtic polynomil, Communictions on Pure nd Applied Anlysis, Vol. 16, 15(4): 1497-1514, doi: 1.3934/cp.16.15.1497 [13] S. Ogorodnikov, F. Sdyrbev, Multiple solutions of nonliner boundry vlue problems with oscilltory solutions, Mthemticl modelling nd nlysis, Vol. 11, N 4, pp. 413 46, 6. [14] E. T. Whittker nd G. N. Wtson, A Course of Modern Anlysis, Cmbridge University Press, (194, 1996), ISBN -51-5887-3. E-ISSN: 4-88 95 Volume 17, 18