Two-component nonlinear wave of the NLS equation
aa r X i v : . [ n li n . S I] D ec Two-component nonlinear wave of the NLS equation
G. T. Adamashvili
Technical University of Georgia,Kostava str.77, Tbilisi, 0179, Georgia.email: guram − adamashvili @ ymail.com. Using the generalized perturbation reduction method the scalar nonlinear Schr¨odinger equationis transformed to the coupled nonlinear Schr¨odinger equations for auxiliary functions. A solutionin the form of a two-component vector nonlinear pulse is obtained. The components of the pulseoscillate with the sum and difference of the frequencies and wave numbers. Explicit analyticalexpressions for the shape and parameters of the two-component nonlinear pulse are presented.
Keywords:
Two-component nonlinear waves, Single and coupled nonlinear Schrodinger equations,Generalized perturbation reduction method.
PACS numbers: 05.45.Yv, 02.30.Jr, 52.35.Mw
I. INTRODUCTION
The nonlinear solitary wave solutions (soliton, breather, vector breather and etc.) of various nonlinear partialdifferential equations plays a fundamental role in the theory of nonlinear waves and applied mathematics. Theseequations include the Sine-Gordon equation, the Maxwell-Bloch equations, different versions of the modified Benjamin-Bona-Mahony equations, the Maxwell-Liouville equations, the system of the magnetic Bloch equations and the acousticwave equation, the wave equation in dispersive and Kerr-type media, the system of wave equation and materialequations for multi-photon resonant excitations, among others. To obtain the solitary wave solutions to these nonlinearevolution equations, many methods were developed, such as the inverse scattering transform, different asymptoticalapproaches, Lie group method, the factorization technique, Exp-function method and so on [1-9]. Special interestexpress the single (scalar) and coupled nonlinear Schr¨odinger equations. They are being one of the basic equationsfor studying the one-component and two-component solitary waves of stable profile of any nature in different physicalfields of research: in optics, acoustics, magnetics, fluid dynamics, quantum electronics, particle physics, plasma physicsand etc. By means of various mathematical approaches we can establish a connection between two different nonlinearequations. In particular, to apply the generalized perturbation reduction method we can transform various nonlinearpartial differential equations to the coupled nonlinear Schr¨odinger equations for auxiliary functions. As a result,the two-component waves, which are the bound state of two small-amplitude scalar one-component breathers withthe identical polarization, have been obtained. The first breather oscillates with the sum, and the second with thedifference of the frequencies and wave numbers. This two-component pulse has a very special form and they are metin different fields of research: in optics, acoustics, plasma physics, hydrodynamics, particle physics, and etc. In theoptical and acoustic self-induced transparency, such wave is called the vector 0 π pulse [10-24].It is necessary to separately consideration the nonlinear partial differential equations for real and complex functions.We consider the real function U ( z, t ) of the space coordinate z and time t which can be presented to the form U = U (+) + U ( − ) , (1)where the complex conjugate functions U (+) and U ( − ) are satisfied the scalar nonlinear Schr¨odinger equation ± i ∂U ( ± ) ∂t + β ∂ U ( ± ) ∂z + a | U ( ± ) | U ( ± ) = 0 , (2) U (+) = U ( − ) ∗ , a and β are real constants. Eq.(2) is completely integrable by the inverse scattering transform and has N -soliton solutions [1-4].One-soliton solution of this equation has the form U ( ± ) ( z, t ) = K e ± iφ ( z,t ) coshφ ( z, t ) , (3)where φ ( z, t ) = 12 [ V s β z − ( V s β − aK ) t ] , φ ( z, t ) = K r a β ( z − V s t ) ,K is the amplitude of the nonlinear Schr¨odinger equation soliton, V s is the velocity of the nonlinear wave.Depending from the width of the nonlinear pulse we can consider different nonlinear solitary wave solutions. In thepresent work we investigate one-component scalar breather and two-component vector breather solutions of Eqs.(1)and (2).The rest of this paper is organized as follows: Section II is devoted to the small amplitude one-component breathersolution. In Section III, we will investigate the two-component (vector) breather solution of the Eqs.(1) and (2) withthe generalized perturbation reduction method. Finally, in Section IV, we will discuss the obtained results. II. THE ONE-COMPONENT BREATHER SOLUTION
When the width of the pulse
T >> ω − for the real function U ( z, t ) Eq.(1) we can using the method of slowlyvarying shape. In this case we represent the functions U ( ± ) in the form U ( ± ) = ˆ u ± Z ± , (4)where ˆ u ± is the slowly varying complex envelope function, Z ± = e ± i ( kz − ωt ) is the fast oscillating function, ω and k are the frequency and the wave number of the carrier wave. For the reality of U , we set ˆ u +1 = ˆ u ∗− .The complex function ˆ u ± vary sufficiently slowly in space and time compared to the carrier wave part Z ± , i.e. | ∂ ˆ u ± ∂t | << ω | ˆ u ± | , | ∂ ˆ u ± ∂z | << k | ˆ u ± | are valid.Substituting Eq.(4) into Eq.(2), we obtain the nonlinear Schr¨odinger equation for envelope function ˆ u ± in the form ± i ( ∂ ˆ u ± ∂t + 2 βk ∂ ˆ u ± ∂z ) + β ∂ ˆ u ± ∂z = − a | ˆ u ± | ˆ u ± (5)and dispersion relation for nonlinear pulse ω = βk . (6)The equation (5) is transformed to the scalar nonlinear Schr¨odinger equation in the following form ± i ∂ ˆ u ± ∂t + β ∂ ˆ u ± ∂y + a | ˆ u ± | ˆ u ± = 0 , (7)where y = z − βk t, t = t. The one-soliton solution of the Eq.(7) is given by the Eq.(3) if in this equation replace z by the variable y .Substituting this solution to the Eqs.(1) and (4) we obtain U = 2 K sin( kz − ωt + φ ( y, t ) − π )cosh φ ( y, t ) . (8)This is small amplitude single-component breather for the function U .We have to note that Eq.(8) is not breather of the nonlinear Schr¨odinger equation (7). The breather of nonlinearSchr¨odinger equation is unstable solution to relatively infinitesimal perturbations of the initial data. The real function U can be solution of the another nonlinear equation, for instance, the Sin-Gordon equation. In this case, we obtainwell known result when one soliton solution of nonlinear Schr¨odinger equation is connected with the small amplitudebreather solution of the Sine-Gordon equation[1]. III. THE TWO-COMPONENT VECTOR BREATHER SOLUTION
The single-component breather Eq.(8) is not the only possible nonlinear wave for the function U = X l = ± ˆ u l Z l , (9)when envelope of this function ˆ u l satisfied scalar nonlinear Schr¨odinger equation Eq.(5).Indeed, for more wider pulses for which the condition ω >> Ω l,n >> T − (10)is fulfilled, we can consider also the two-component nonlinear solitary waves, where parameters Ω l,n will be determinedbelow.For the study of the two-component nonlinear solitary wave solution of Eqs.(5) and (9) we apply the general-ized perturbative reduction method [10-18] by means of which we can transform Eq.(5) into the coupled nonlinearSchr¨odinger equations for auxiliary functions. In this method the function ˆ u l ( z, t ) can be represented as:ˆ u l ( z, t ) = ∞ X α =1 + ∞ X n = −∞ ε α Y l,n f ( α ) l,n ( ζ l,n , τ ) , (11)where Y l,n = e in ( Q l,n z − Ω l,n t ) , ζ l,n = εQ l,n ( z − v gl,n t ) ,τ = ε t, v g ; l,n = d Ω l,n dQ l,n ,ε is a small parameter. Such an expansion allows us to separate from ˆ u l the even more slowly changing auxiliaryfunction f ( α ) l,n . It is assumed that the quantities Ω l,n , Q l,n , and f ( α ) l,n satisfy the conditions: ω ≫ Ω l,n , k ≫ Q l,n , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f ( α ) l,n ∂t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Ω l,n (cid:12)(cid:12)(cid:12) f ( α ) l,n (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f ( α ) l,n ∂z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Q l,n (cid:12)(cid:12)(cid:12) f ( α ) l,n (cid:12)(cid:12)(cid:12) . for any value of indexes l and n .Substituting Eq.(11) into the left-hand side of the Eq.(5), we obtain wave equation ∞ X α =1 + ∞ X n = −∞ ε α Y l,n [ W l,n + iεJ l,n ∂∂ζ + ilε ∂∂τ + βε Q ∂ ∂ζ ] f ( α ) l,n + a | ˆ u l | ˆ u l = 0 , (12)where W l,n = ln Ω l,n − lβknQ l,n − n βQ l,n ,J l,n = − lQ l,n v g ; l,n + 2 lβkQ l,n + 2 βnQ l,n . Following the standard procedure characterized for any perturbative expansions while equating to each other theterms of the same order to ε , from the Eq.(12), we obtain the chain of the equations. As a result, in the first andsecond order of ε we determine the connection between the parameters Ω l,n and Q l,n which is given byΩ l,n = βQ l,n (2 k + nlQ l,n ) (13)and the equation v g ; l,n = 2 β ( k + nlQ l,n ) . (14)In the third order of ε we obtain the coupled nonlinear Schr¨odinger equations in the following form i ( ∂λ ± ∂t + v ± ∂λ ± ∂z ) + β ∂ λ ± ∂z + a ( | λ ± | + 2 | λ ∓ | ) λ ± = 0 , (15)where λ ± = εf (1)+1 , ± ,v ± = v g ;+1 , ± = 2 β ( k ± Q +1 , ± ) . The solution of Eq.(15) we seek in the form of [10-18] λ ± = K ± b T Sech ( t − zV T ) e i ( k ± z − ω ± t ) , (16)where K ± , k ± and ω ± are the real constants, V is the velocity of the nonlinear wave. We assume that k ± << Q +1 , ± , ω ± << Ω +1 , ± . (17)The connections between K ± , k ± and ω ± are given by K = K − , k ± = V − v ± β ,ω + = ω − + v − − v β . (18)Substituting Eq.(16) into (15), when the complex envelope functions ˆ u ± satisfied the scalar nonlinear Schr¨odingerequation (7), we obtain the two-component pulse of the function U : U ( z, t ) = 2 bT Sech ( t − zV T ) { K + cos[( k + Q +1 , +1 + k + ) z − ( ω + Ω +1 , +1 + ω + ) t ]+ K − cos[( k − Q +1 , − + k − ) z − ( ω − Ω +1 , − + ω − ) t ] } , (19)where T is the width of the two-component nonlinear pulse, T − = V v + k + + k β − ω + β , b = 3 V a β K . (20)This pulse oscillating with the sum ω + Ω +1 , +1 and difference ω − Ω +1 , − of the frequencies and the wave numbers k + Q +1 , +1 and k − Q +1 , − (taking into account Eq.(17)). IV. CONCLUSION
In the present paper, we investigate the wide class of phenomena for the nonlinear waves of different nature (optical,acoustic, elastic, magnetic and etc.) in various media and different fields of research which can be describe by meansof the scalar nonlinear Schr¨odinger equation. We study separately nonlinear waves with different width and is shownthat can be formed different nonlinear waves: soliton Eq.(3), and for more wider pulse small amplitude breatherEq.(8), which coincide with the small amplitude breather of Sine-Gordon equation.For more wider width of pulses for which the inequalities (10) is valid, situation became different. Using thegeneralized perturbation reduction method Eq.(11), the Eq.(5) for the complex functions ˆ u l , transform to the couplednonlinear Schr¨odinger equations (15) for the auxiliary functions λ ± . As a result, the two-component nonlinear pulseoscillating with the sum and difference of the frequencies and wave numbers Eq.(19), can be formed. The dispersionrelation and the connection between parameters Q +1 , ± and Ω +1 , ± are determined from Eqs.(6) and (13). Theparameters of the two-component vector breather of Eqs.(1) and (2) are determined from Eqs.(14), (18) and (20). [1] A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, 1985).[2] S. P. Novikov, S. V. Manakov L. P. Pitaevski and V. E. Zakharov , Theory of Solitons: The Inverse Scattering Method ,(Academy of Science of the USSR, Moscow, USSR. 1984).[3] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris
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