Evolution equations for pulse propagation in nonlinear media

Ev olution equations for pulse propagation in nonlinea r media Debabrata P al, Amita v a Ch oudhuri and B T alukdar a Dep artment of Physics, Visva-Bhar ati University, Santiniketan 7312 35, India e-mail : binoy1 23@bsnl.in Abstract W e sh o w that the complex mo dified K dV (cmKdV) equation and generalized nonlinear Sc hr¨ odinger (GNLS) equation b elong to the Ablo witz, Kaup, New ell and Segur or so-called AKNS hierarc hy . Both equations do not follo w from the action p rinciple and are nonintegrable. By in- tro ducing some auxiliary fields we obtain the v ariatio nal principle f or them and s tudy th eir canonical stru ctures. W e make use of a coupled amplitude-phase metho d to solv e the equations analytica lly and de- riv e cond itions u nder w hic h they can supp ort brigh t and dark solitary w a ve solutions. P A CS n umbers : 47.20 .Ky , 4 2.81.Dp, 02.30.Jr 1. In tro duction In a pioneering w ork Zakharov and Shabat [1] solv ed the nonlinear Sch r¨ odinger (NLS) equation b y the use of in v erse sp ectral metho d. Subsequen tly , Ablo witz, Kaup, New ell and Segur (AKNS) [2 ] sough t a generalization of the method and in tro duced the so-called AKNS hierarc h y which in v olve s a fa mily of in- tegrable equations asso ciated with the Zakharov -Shabat eigen v alue problem. It is now fa ir ly w ell know n that man y ph ysically imp ortant integrable pa r t ial differen tia l equations b elong to the AKNS hierarch y . F or example, b esides the NLS equations, this hierarch y yields the KdV-, mKdV-, Sine-Gordon-, Harry-Dym equations as well a s the constrain t KP hierarc h y [3]. There ar e some relative ly recen t attempts to disclose the p ossible connections b etw een the tw o-comp onent Camassa-Holm equation and AKNS hierarc h y extended b y a negative flow [4 , 5]. Th us realization of ph ysically imp ortan t nonlin- ear ev olutio n equations in the frame of AKNS mo del is still an in teresting curiosit y . 1 In the pr esen t w ork we shall prov ide, in Sec. 2, a deriv ation for tw o non- in tgrable par t ia l differen tial equations by using the integrable AKNS mo del for whic h there exists a w ell defined sp ectral pro blem. The equations of our in terest are the complex mo dified KdV- (cmKdVI) [6] and g eneralized third- order nonlinear Sc hro dinger (GNLS) equations [7]. T he cmKdVI equation describes the nonlinear steady-state propagation of lo w er- h ybrid w a v es in a uniform plasma and more inte restingly , its solutions rev eal a close connec- tion b et w een classical soliton and en v elop e solitons. On the o ther hand, the GNLS equation is used t o mo del propag a tion of ultra short pulses in optical fib ers [8] a nd is traditionally obtained fr o m Maxw ell’s equations with sp ecial atten tion to nonlinear susceptibilities of the associat ed optical medium [9]. An awkw ard a nalytical constrain t for b oth cmKdVI- and G NLS eq uations is that these do not follow from the action principle to hav e a Lagrangian represen tation which plays a role in many applicativ e con texts [1 0]. F or an arbitrary differential equation one can a lw ays find the v ariat io nal principle b y taking recourse to the use of auxiliary field v aria bles [11]. W e follow this route, in Sec. 3, to construct expressions for Lag r a ngian densities of the cmKdV and GNLS equations and use them to study their canonical structures. In Sec. 4 w e mak e use of a coupled a mplitude-phase f o rm ulation [12] to solv e t hese equations analytically a nd deriv e conditio ns under whic h cmKdVI- and GNLS equations can supp ort brigh t and dark solitary w av e solutions. Finally , in Sec. 5, w e mak e some concluding remarks with a view to summarize our o utlo ok on the presen t work. 2. Deriv ation of cmKdV I- and GNLS equations us ing the A KNS hierarc h y The AKNS hierarc h y with the Zakharov-Shabat e igen v alue problem is giv en b y [2] i  u t − v t  − ω (2 L z s )  u v  = 0 . (1) Here u = u ( x, t ) and v = v ( x, t ) represe n t tw o (1 + 1) dimensional complex fields. The op erator ω (2 L z s ) giv es the dispersion relation o f the linearization equation in the u comp onent with L z s , an integro differen tia l op erator written as [13] L z s = 1 2 i  ∂ ∂ x − 2 u R x −∞ dy v 2 u R x −∞ dy u − 2 v R x −∞ dy v − ∂ ∂ x + 2 v R x −∞ dy u  . (2) The subscript zs of L merely indicates that the AKNS hierarc h y is the family of integrable equations b elonging to the Zakharov -Shabat eigen v a lue prob- 2 lem. The NLS hierarc hy can b e obtained b y choosing ω (2 L z s ) = (2 L z s ) n , n = 2 , 4 , 6 ... and v = − u ∗ . In particular, fo r n = 2, (1) giv es the NLS equation. F or n = 4, we get the forth-order equation in the NLS hierarch y and so on. The c ho ice for the disp ersion relation amounts to demanding that ω (2 L z s ) is an en tir e function of the argumen t. T o derive the noninte grable equations o f our in terest w e b egin b y intro- ducing ω ( 2 L z s ) = ( − 2 L z s ) n (3) where n = 1 , 2 , 3 ... . Note that f or ev en v alues of n the relation in (3) coincides with the dispersion relation used for the NLS hierarc h y . F or n = 3 ω ( 2 L z s ) = − i  a 11 a 12 a 21 a 22  (4) with a 11 = ∂ 3 ∂ x 3 − 2 u 2 x Z x −∞ dy v − 6 u x v − 4 uv ∂ ∂ x + 4 u 2 v Z x −∞ dy v + 2 u x Z x −∞ dy v y − 2 u Z x −∞ dy v 2 y , (5 a ) a 12 = 2 u 2 x Z x −∞ dy u + 2 uu x − 4 u 2 v Z x −∞ dy u + 2 u x Z x −∞ dy v y + 2 u Z x −∞ dy u 2 y , (5 b ) a 21 = − 2 u x Z x −∞ dy v y − 2 v v x − 2 v Z x −∞ dy v 2 y + 4 u v 2 Z x −∞ dy v − 2 v 2 x Z x −∞ dy v (5 c ) and a 22 = − ∂ 3 ∂ x 3 + 2 v 2 x Z x −∞ dy u + 6 v x u + 4 u v ∂ ∂ x − 4 uv 2 Z x −∞ dy u − 2 v x Z x −∞ dy u y +2 v Z x −∞ dy u 2 y . (5 d ) F rom (1), (3), (4) a nd (5) we g et cmKdVI equation u t + 2( | u | 2 u ) x + u 3 x = 0 (6) for v = − u ∗ . Similarly , for n = 5 w e g et the fifth-o r der cmKdVI equation u t − (8 | u | 2 u 2 x + 4 | u | 4 u + 2 u 2 u ∗ 2 x + 6 u 2 x u ∗ ) x − 10 u ∗ x u 2 x + 14 u x u 2 x u ∗ − 12 uu x u ∗ 2 x − 10 u 2 u ∗ 2 u x − u 5 x = 0 . (7) 3 The family of higher-order equations obtained in this wa y do es not f orm a hierarc h y in the same sense as used in the case of integrable equations. W e ha v e just seen that for n = 3 the disp ersion relation in (3) g enerates cmKdV I equation. Let us no w define ω ( 2 L z s ) = 3 X n =2 ( − 2 L z s ) n . (8) W ritten explicitly (8) b ecomes ω ( 2 L z s ) = −  b 11 + ia 11 b 12 + ia 12 b 21 + ia 21 b 22 + ia 22  (9) with b 11 = − ∂ 2 ∂ x 2 + 4 u v + 2 u x Z x −∞ dy v − 2 u Z x −∞ dy v y , (10 a ) b 12 = − 2 u x Z x −∞ dy u − 2 u Z x −∞ dy u y , (10 b ) b 21 = − 2 v x Z x −∞ dy v − 2 v Z x −∞ dy v y (10 c ) and b 22 = − ∂ 2 ∂ x 2 + 4 u v + 2 v x Z x −∞ dy u − 2 v Z x −∞ dy u y . (10 d ) F rom (1), (9) and (10) w e get GNLS equation iu t + 2 i | u | 2 u x + 2 i ( | u | 2 ) x u + u 2 x + 2 | u | 2 u + iu 3 x = 0 (11) for v = − u ∗ . 3. V ariational principle and canonical formul ation The first step tow ards a canonical form ulation o f any system is to assure the existenc e of a Lagr a ngian. Both cmKdVI and GNLS equations in (6) and (11) inv alidate the Helmholtz condition [14] suc h that these equations are not Euler-Lagrange express ions. One standard metho d for find ing v ariational principle [11] for suc h equations is to introduce a set of auxiliary v ariables v { v 1 , ......., v i } a nd consider the Lagrangian densit y L L ( u, v ) = v ∆[ u ] (12) 4 for the problem where ∆[ u ] = 0 defines an ar bitr ary system of differential equations with ∆ = ∆(∆ 1 , ....., ∆ i ). F or the cmKdVI equation in (6 ) w e consider an additional complex field v = v ( x, t ) and in tro duce the Lagra ngian densit y L = v  u t + 4 u u ∗ u x + 2 u 2 u ∗ x + u 3 x  − v ∗  u ∗ t + 4 u u ∗ u ∗ x + 2 u ∗ 2 u x + u ∗ 3 x  (13) to write the action principle as δ Z L dx dt = 0 . (14) Clearly , the Euler-Lagra ng e equations for v and v ∗ giv e the cmKdV I and its complex conjugate equations. The Euler-Lagrange equations for u and u ∗ yield the equations for v and v ∗ . The set of four equations th us obtained can b e written in the matr ix for m     u u ∗ v v ∗     t =     − ∂ 3 x − 4 | u | 2 ∂ x − 2 u 2 ∂ x 0 0 − 2 u ∗ 2 ∂ x − ∂ 3 x − 4 | u | 2 ∂ x 0 0 0 0 − ∂ 3 x − 4 | u | 2 ∂ x 2 u ∗ 2 ∂ x 0 0 2 u 2 ∂ x − ∂ 3 x − 4 | u | 2 ∂ x     ×     u u ∗ v v ∗     . (15) The canonical momen tum densities are π = ∂ L ∂ u t = v and π ∗ = ∂ L ∂ u ∗ t = − v ∗ (16) corresp onding to the Lagra ngian density in (13 ) whic h via Legendre trans- formation leads to the Hamilto nian densit y H = v ∗  4 uu ∗ u ∗ x + 2 u ∗ 2 u x + u ∗ 3 x  − v  4 uu ∗ u x + 2 u 2 u ∗ x + u 3 x  (17) with the Hamiltonian written as H = Z H dx . (18) The equations in (15) can b e written in the Hamiltonian form ˙ ξ = δ H δ η = { η ( x ) , H ( y ) } (19) 5 where ξ and η stand for appropriate field v a r ia bles. It is easy to verify that (19) is endow ed with P oisson structures { u ( x ) , u ( y ) } = δ ( x − y ) , { u ( x ) , u ∗ ( y ) } = 0 , { u ( x ) , v ( y ) } = 0 , { u ( x ) , v ∗ ( y ) } = 0 , (20 a ) { u ∗ ( x ) , u ( y ) } = 0 , { u ∗ ( x ) , u ∗ ( y ) } = δ ( x − y ) , { u ( x ) , v ( y ) } = 0 , { u ( x ) , v ∗ ( y ) } = 0 , (20 b ) { v ( x ) , u ( y ) } = 0 , { v ( x ) , u ∗ ( y ) } = 0 , { v ( x ) , v ( y ) } = δ ( x − y ) , { v ( x ) , v ∗ ( y ) } = 0 (20 c ) and { v ∗ ( x ) , u ( y ) } = 0 , { v ∗ ( x ) , u ∗ ( y ) } = 0 , { v ∗ ( x ) , v ( y ) } = 0 , { v ∗ ( x ) , v ∗ ( y ) } = δ ( x − y ) . (20 d ) The Lagra ngian and Hamiltonian densities for the GNLS equation in (11) can b e written as those in (1 3) and (17). The Lag rangian system of equations is giv en by i     u u ∗ v v ∗     t =     c 11 c 12 c 13 c 14 c 21 c 22 c 23 c 24 c 31 c 32 c 33 c 34 c 41 c 42 c 43 c 44         u u ∗ v v ∗     (21) with c 11 = c 44 = − i∂ 3 x − ∂ 2 x − 4 i | u | 2 ∂ x , (22 a ) c 12 = − (2 u 2 + 2 iu 2 ∂ x ) , (22 b ) c 21 = − c ∗ 12 (22 c ) c 22 = c 33 = − i∂ 3 x + ∂ 2 x − 4 i | u | 2 ∂ x , (22 d ) c 13 = c 14 = c 23 = c 24 = 0 , (22 e ) c 31 = 2 u ∗ v , (22 f ) c 32 = 2 uv , (22 g ) c 34 = 2 u ∗ 2 − 2 iu ∗ 2 ∂ x , (22 h ) c 41 = − c ∗ 32 , (22 i ) c 42 = − c ∗ 31 (22 j ) and c 43 = − c ∗ 34 (22 k ) 6 The set of equations in (21 ) can b e written in the Hamiltonian form as giv en in (19) with Poiss on structures similar to tho se in (20) 4. Solitary w a ve solutions T o o btain the solitary w a v e solutio ns of (6) and (11) w e tak e recourse to the use of coupled amplitude-phase formulation [1 2] and write u ( x, t ) in the form u ( x, t ) = P ( χ ) exp [ i ( k x − ω t )] , χ = x + β t , (23) with χ the tra v elling cordinate containing the g roup v elo cit y β of the w a v e pac k et. Here the function P is real. F rom (6) and (23 ) w e obtain β P χ + 6 P 2 P χ + P χχχ − 3 k 2 P χ + i (2 k P 3 − ω P + 3 kP χχ − k 3 P ) = 0 . (24) Equating the real a nd imaginary parts of ( 2 4) seperately to zero w e ha v e P χχχ + 6 P 2 P χ + ( β − 3 k 2 ) P χ = 0 (25) and P χχ + 2 3 P 3 − ω + k 3 3 k P = 0 . (26) The third order equation in (25) can b e integrated to write P χχ + 2 P 3 + ( β − 3 k 2 ) P = 0 . (27) Both equations (26) and (27) can b e solve d analytically . The solution of (26 ) will dep end explicitly on ω and k while the solution of (27) will ha v e similar dep endence on β and k . In particular, P ( χ )’s will b e giv en b y P ( χ ) = (3 k 2 − β ) 1 2 sech [(3 k 2 − β ) 1 2 χ ] (26 ′ ) or P ( χ ) = ( ω + k 3 k ) 1 2 sech [( ω + k 3 3 k ) 1 2 χ ] . (27 ′ ) Here equations (26 ′ ) and ( 27 ′ ) refer to solutions of (26 ) and ( 27) when the first in tegra ls of these equations are ta k en as zero. Clearly , the compatibilit y condition of thes e solutions implies that (26 ′ ) m ust satisfy (27) and (27 ′ ) m ust satisfy (26). This viewp oint yields the necessary and sufficien t condition to get a r elat io n among β , ω and k and w e ha v e β = 2 k 2 − ω k . (28) 7 Since the first integral of (26) or (27 ) gives the energy E of the w av e, w e infer from (26 ′ ) and (27 ′ ) that the zero energy solution of the cmKdV equation represen ts a bright solitory w a v e solution pro vided the ve lo city β satisfies the constrain t in (28). The c hoice − ( β − 2 k 2 ) 2 4 and − ω + k 3 4 k for the first in tegrals of (26) a nd (2 7) leads to the solutio ns P ( χ ) =  β − 3 k 2 2  1 2 tanh "  β − 3 k 2 2  1 2 ( x + β t ) # (26 ′′ ) and P ( χ ) =  ω + k 3 2 k  1 2 tanh "  ω + k 3 2 k  1 2 ( x + β t ) # (27 ′′ ) The compatibilit y condition of (26 ′′ ) and (2 7 ′′ ) giv es the r elat io n β = 4 k 2 + ω k . (29) It is of interes t to not e that for β as given in (29 ) , t he expressions fo r the first integral for (26) a nd (27) b ecome identically equal. Thus w e infer tha t for negativ e v a lues of E the cmKdVI equation supp orts dark solitory w av e solution. W e hav e carried out a similar analysis for the GNLS equation in (11) and found that the zero energy solutions a re given by P ( χ ) =  k 2 − k 3 − ω 1 − ω  1 2 sech "  k 2 − k 3 − ω 1 − ω  1 2 ( x + β t ) # (30) or P ( χ ) =  3 k 2 − 2 k − β  1 2 sech h  3 k 2 − 2 k − β  1 2 ( x + β t ) i (31) sub j ect to the consistancy condition β = k ( 3 k − 2) + k 2 ( k − 1) 1 − ω + ω 1 − ω . (32) It is eviden t from (30) or (3 1), as with t he cmKdVI equation, the zero en- ergy solution of the GNLS equation also represen ts a brigh t solitory w a v e. Exp ectedly , the negativ e energy solution P ( χ ) =  ω + k 3 − k 2 2(1 − k )  1 2 tanh "  ω + k 3 − k 2 2(1 − 3 k )  1 2 ( x + β t ) # (33) 8 or P ( χ ) =  2 k − β − 3 k 2 2  1 2 tanh "  2 k − β − 3 k 2 2  1 2 ( x + β t ) # (34) with β = ω 1 − k − 2 k (1 − k ) (35) corresp onds to a dark solitory w av e. 5. Conclusions W e b egan by noting tha t a large n um b er of ph ysically imp o rtan t partial differen tia l equation b elongs to the AKNS hierarc h y . T his motiv ated us to examine if the nonintegrable cmKdVI and GNLS equations could b e embed- ded into the same hierarc h y . W e pro vided deriv ation of t hese equations in the frame o f the AKNS mo del. The equations are nonLagrangian ev en in the p oten tial represen tation. But we fo und that one can o btain the v ariational principle for them b y ta king recourse to the use of some auxiliary fields. This pro vided a natural basis t o study their canonical structures. W e solv ed these equations by using a coupled amplitude-phase method and explicitly demon- strated t he zero-energy solutions corresp ond to bright solitary w a v es while the negativ e-energy solutions represen t dark solitary w a v es. References [1] Zakharov V E a nd Shabat A B 197 2 JETP 34 62 [2] Ablo witz M J, Kaup D J, New ell A C a nd Segur H 19 74 Stud. 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