Infinite Series Symmetry Reduction Solutions to the Perturbed Coupled Nonlinear Schr¨odinger Equation

Infinite Series Symmetry Reduction Solutions to the Perturbed Coupled Nonlinear Schr¨odinger Equation

Xue-Ping Cheng^a,b, Qing-Qing Yuan^c, Xiang-Qing Zhao^a, and Jin-Yu Li^a

a Department of Physics, Zhejiang Ocean University, Zhoushan 316004, P.R. China

b Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, P.R. China

c Department of Physics, Ningbo University, Ningbo 315211, P.R. China Reprint requests to X.-P. C.; E-mail:chengxp2005@126.com

Z. Naturforsch.67a,381 – 388 (2012) / DOI: 10.5560/ZNA.2012-0034 Received November 10, 2011 / revised March 16, 2012

By the approximate symmetry perturbation method, the perturbed coupled nonlinear Schr¨odinger equation (PCNLSE) is investigated. As a result, the approximate symmetries and infinite series sym- metry reduction solutions are obtained. Specially, we take the symmetry reduction for soliton solu- tions as an example, where the effects of perturbations on soliton solutions are briefly discussed.

Key words:Perturbed Coupled Nonlinear Schr¨odinger Equation; Approximate Symmetry Perturbation Method; Infinite Series Symmetry Reduction Solution.

PACS numbers:42.81.Dp; 02.30.Jr; 03.40.Kf

1. Introduction

The highly idealized nonlinear evolution equa- tions, such as the Korteweg–de Vries (KdV) equa- tion, the nonlinear Schr¨odinger (NLS) equation, the Kadomtsev–Petviashvili (KP) equation, etc., may have been solved analytically many times by kinds of well- known methods, including the inverse scattering trans- formation (IST), the Hirota bilinear method, the Dar- boux transformation, and so on. However, to obtain the analytical solutions of their correction systems that differ from the standard ones by some small additional terms (so-called perturbations), it still does not seem to be an easy task. One of the most powerful techniques in dealing with these systems is based on the IST [1].

This method requires that the unperturbed equations must be exactly solvable via the IST, which seriously restricts the range of application. Besides, the direct perturbation method has also been frequently used to find approximate analytical solutions to perturbed par- tial differential equations. Supplements for this method can be found in [2–6] and the references therein. An- other effective way to investigate perturbed nonlinear evolution equations is the approximate symmetry per- turbation method, which is an integration of the per- turbation method and the symmetry reduction method first proposed by Fushchich and Shtelen [7]. Recently, the approximate symmetry perturbation method has

been further improved by Lou and his colleagues, and has been widely applied to many equations [8–11].

In the present paper, we shall introduce the approx- imate symmetry perturbation method to the perturbed coupled system of two NLS equations:

ip_t+β1p_xx+ (κ₁|p|²+γ₁|q|²)p

=εP(p,q,p_x,q_x,p_t,q_t, . . .), (1a) iq_t+β₂q_xx+ (κ₂|q|²+γ2|p|²)q

=εQ(q,p,q_x,p_x,p_t,q_t, . . .). (1b) Equation (1) has been applied in various physical fields, such as plasma physics [12], nonlinear op- tics [13,14], condense matter physics [15,16], geo- physics fluid dynamics [17], etc. In the context of non- linear optics, pandqin (1) describe the envelopes of the waves with mutually orthogonal linear polariza- tion, x andt correspond to normalized distance and time, respectively. The parametersβ1,β2are the dis- persion coefficients,κ₁,κ₂the Landau constants which describe the self-modulation of the wave packets, and γ1, γ2 the wave–wave interaction coefficients which describe the cross-modulations of the wave packets.

εP and εQ are perturbations of arbitrary form, and ε is a small parameter characterizing the magnitude of these perturbations. Whenε=0, (1) reduces into the coupled nonlinear Schr¨odinger equation (CNLSE).

Manakov [18] was the first to show that the CNLSE is

c

2012 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

integrable in terms of the method of IST for the case whereβ1=β2=1,κ1=κ2=γ1=γ2. Later Zakharov and Schulman [19] have thoroughly studied the inte- grability of system (1) with ε=0, and they summa- rized that only when β₁=β₂, κ₁=κ₂=γ₁=γ₂ or β1=−β₂,κ1=κ2=−γ₁=−γ₂the CNLSE is inte- grable, i.e. it can be solved by IST. To put it in an- other way, with other more generalβ1,β2,κ1,κ2,γ1,γ2

values, system (1) is non-integrable, and the perturbed ones are included.

For the sake of simplicity and convenience, here we take account of the special case of (1) with coefficients β1 = β2 = κ1 = κ2 = γ1 = γ2 = 1 andεP(p,q,p_x,q_x, . . .) =−iεp,εQ(q,p,q_x,p_x, . . .) =

−iεq, corresponding to the action of linear damping in fiber systems [20]. Thereby, system (1) becomes

ip_t+p_xx+ (|p|²+|q|²)p+iεp=0, (2a) iq_t+q_xx+ (|p|²+|q|²)q+iεq=0. (2b) As we know, several perturbation methods have been employed to obtain the zero-order solutions of (2) [21], but a perturbation method for deriving the first order or much higher-order modifications is still quite rare.

2. Approximate Symmetry Perturbation Method to the Perturbed Coupled Nonlinear Schr¨odinger Equation

To search for the solutions of the PCNLSE (2), we would like to assume that the complex functionspand qare in exponential form:

p=ue^iv, q=re^is, (3) where u≡u(x,t), v≡v(x,t),r≡r(x,t), s≡s(x,t).

Substituting expression (3) into (2) and separating real and imaginary parts, (2) is then replaced by

−uv_t+u_xx−uv²_x+u³+ur²=0,

u_t+2u_xv_x+uv_xx+εu=0, (4a)

−rs_t+r_xx−rs²_x+ru²+r³=0,

r_t+2r_xs_x+rs_xx+εr=0. (4b) According to the approximate symmetry perturbation theory, the solutions of (4) are expressed as series:

u=

∞

∑

j=0

ε^ju_j, v=

∞

∑

j=0

ε^jv_j,

r=

∞

∑

ε^jr_j, s=

∞

∑

ε^js_j.

(5)

uj,vj,rj, andsjare functions with respect toxandt.

When one inserts expansion (5) into (4) and vanishes the coefficients of all different powers ofε, a system of partial differential equations is given:

−u0v0t+u0xx−u0v²_0x+u³₀+u0r₀²=0, (6a) u_0t+2u_0xv_0x+u₀v_0xx=0, (6b)

−r₀s_0t+r_0xx−r₀s²_0x+r₀u²₀+r³₀=0, (6c) r_0t+2r_0xs_0x+r₀s_0xx=0, (6d)

−u₀v_1t−u₁v_0t+u_1xx−u₁v²_0x−2u₀v_0xv_1x

+2u₀r₀r₁+u₁r₀²+3u²₀u₁=0, (7a) u_1t+u₀v_1xx+u₁v_0xx+2(u_1xv_0x+u_0xv_1x)

+u₀=0, (7b)

−r0s1t−r1s0t+r1xx−2r₀s0xs1x−r1s²_0x

+2r₀u₀u₁+r₁u²₀+3r₀²r₁=0, (7c) r_1t+r₀s_1xx+r₁s_0xx+2(r_0xs_1x+r_1xs_0x)

+r₀=0, (7d)

. . . ,

−

j

∑

i=0

u_ivj−i,t+uj,xx−

j

∑

i=0 j−i

∑

k=0

u_iv_k,xvj−i−k,x

+

j i=0

∑

j−i k=0

∑

u_i(r_krj−i−k+u_kuj−i−k) =0,

(8a)

uj,t+

j i=0

∑

uivj−i,xx+2

j i=0

∑

ui,xvj−i,x+u_j−1=0, (8b)

−

j

∑

i=0

r_isj−i,t+rj,xx−

j

∑

i=0 j−i

∑

k=0

r_is_k,xsj−i−k,x

+

j i=0

∑

j−i k=0

∑

r_i(u_kuj−i−k+r_krj−i−k) =0,

(8c)

rj,t+

j i=0

∑

risj−i,xx+2

j i=0

∑

ri,xsj−i,x+rj−1=0, (8d) withu−1=v−1=r−1=s−1=0. To find the exact solu- tions of (8), we first construct its Lie point symmetries and then give the corresponding symmetry reductions.

In what follows, we shall suppose that the Lie symme- try transformations satisfy

σuj=X u_j,x+Tu_j,t−U_j, σvj=X v_j,x+T v_j,t−V_j, (9a) σ_rj =X rj,x+Trj,t−R_j, σ_sj =X sj,x+T s_j,t−S_j,

(9b) whereX,T,U_j,V_j,R_j, andS_j are functions with re- spect tox,t,u_j,v_j,r_j, ands_j(j=0,1,2, . . .), that is, (8)

X.-P. Cheng et al.·Solutions to Perturbed Coupled Nonlinear Schr¨odinger Equation 383 is invariant under the transformations

{x,t,u_j,v_j,r_j,s_j} → {x+εX,t+εT,u_j

+εU_j,v_j+εV_j,r_j+εR_j,s_j+εS_j} (10) with an infinitesimal parameter ε. Under the nota- tion (9),σu_j,σv_j,σr_j, andσs_j are then the solutions of the symmetry equations, i.e., the linearized equations for (8) listed below:

−u₀σv₀,t−σ_u0v_0t+σu₀,xx−σu₀v²_0x−2u₀v_0xσv₀,x

+2u₀r0σr₀+σu₀r²₀+3u²₀σu₀=0, (11a) σu₀,t+u₀σv₀,xx+σv₀v0xx+2u_0xσv₀,x

+2σu₀,xv0x=0, (11b)

−r₀σ_s0,t−σ_r0s_0t+σ_r0,xx−2r₀s_0xσ_s0,x−σ_r0s²_0x +2r₀u₀σu₀+σr₀u²₀+3r²₀σr₀=0, (11c) σr₀,t+r₀σs₀,xx+σr₀s_0xx+2r_0xσs₀,x

+2σ_r0,xs_0x=0, (11d)

−u₀σv₁,t−u₁σv₀,t−σ_u0v_1t−σu₁v_0t+σu₁,xx

−2σ_u0v_0xv_1x−2u₀σv₀,xv_1x−2u₀v_0xσv₁,x

−2u₁v_0xσ_v0,x−σ_u1v²_0x+2r₀r₁σ_u0+σ_u1r₀² +6u₀u₁σ_u0+2u₀r₁σ_r0+2r₀u₀σ_r1+2r₀u₁σ_r0 +3u²₀σu₁=0,

(12a)

σu₁,t+u₀σv₁,xx+u₁σv₀,xx+σu₀v_1xx+σu₁v_0xx +2u_0xσv₁,x+2u_1xσv₀,x+2σ_u0,xv_1x+2σ_u1,xv_0x +σu₀=0,

(12b)

−r₀σs₁,t−r₁σs₀,t−σr₀s_1t−σ_r1s_0t+σr₁,xx

−2r₀s0xσs₁,x−2r₀σs₀,xs1x−2r₁s0xσs₀,x

−2σ_r0s_0xs_1x−σ_r1s²_0x+2u₀r₀σ_u1+2u₀u₁σ_r0 +2r₀u₁σu₀+2u₀r₁σu₀+6r₀r₁σr₀+σr₁u²₀ +3r₀²σr₁ =0,

(12c)

σr₁,t+r0σs₁,xx+r1σs₀,xx+σ_r0s1xx+σ_r1s0xx

+2r0xσ_s1,x+2r1xσ_s0,x+2σ_r0,xs1x+2σ_r1,xs0x

+σ_r0 =0,

(12d) . . . ,

−

j i=0

∑

(u_iσvj−i,t+σu_ivj−i,t) +σu_j,xx

−

j

∑

i=0 j−i

∑

k=0

·(σ_uiv_k,xvj−i−k,x+u_iσv_k,xvj−i−k,x

+u_iv_k,xσ_vj−i−k,x) +

j

∑

i=0 j−i

∑

k=0

[σ_ui(r_krj−i−k+u_kuj−i−k) +ui(σ_rkrj−i−k+r_kσr_j−i−k+σu_kuj−i−k+u_kσu_j−i−k)]

=0,

(13a)

σ_uj,t+

j i=0

∑

(u_iσ_vj−i,xx+σ_uivj−i,xx)

+2

j

∑

i=0

(u_i,xσ_vj−i,x+σ_ui,xvj−i,x) +σ_uj−1 =0,

(13b)

−

j

∑

i=0

(r_iσ_sj−i,t+σ_risj−i,t) +σ_rj,xx

−

j i=0

∑

j−i k=0

∑

(σ_ris_k,xsj−i−k,x+r_iσs_k,xsj−i−k,x

+risk,xσsj−i−k,x) +

j i=0

∑

j−i k=0

∑

[σ_ri(u_kuj−i−k+r_krj−i−k) +r_i(σ_ukuj−i−k+u_kσu_j−i−k+σ_rkrj−i−k+r_kσr_j−i−k)] =0,

(13c)

σrj,t+

j i=0

∑

(r_iσsj−i,xx+σrisj−i,xx)

+2

j i=0

∑

(r_i,xσsj−i,x+σ_ri,xsj−i,x) +σrj−1=0,

(13d)

andσu−1 =σv−1 =σr−1 =σs−1 =0. Substituting (9) into the linearized equation system (13) and eliminat- ing u_jt, v_jt,r_jt, and s_jt according to (8) yield thou- sands of determining equations by vanishing all co- efficients of the different partial derivatives of uj, v_j, r_j, and s_j for the functions X, T, U_j, V_j, R_j, and S_j. Solving these determining equations, we conclude

X=c₀

2x+c1t+x0, T =c0t+t0, (14a) U₀=−c₀u₀

2 , R₀=−c₀r₀

2 , V₀=S₀=c₁x

2 , (14b) U₁=c₀u₁

2 , R₁=c₀r₁

2 , V₁=c₀v₁, S₁=c₀s₁, (14c) . . . ,

U_j=(2j−1)c₀u_j

2 , V_j=jc₀v_j+c₁x

2 δj,0, (14d) R_j=(2j−1)c₀r_j

2 , S_j=jc0s_j+c1x

2 δj,0. (14e) c0,c1,x0, and t0are all arbitrary constants. Thus the Lie symmetries take the forms

σu₀ =c₀

2x+c₁t+x₀

u_0x+ (c₀t+t₀)u_0t+c₀u₀ 2 ,

(15a) σ_v0=c0

2x+c₁t+x₀

v_0x+ (c₀t+t₀)v_0t−c1x 2 ,

(15b)

σr₀=c0

2x+c1t+x₀

r0x+ (c₀t+t0)r_0t+c0r0

2 , (15c) σs0=c₀

2x+c₁t+x₀

s_0x+ (c₀t+t₀)s_0t−c₁x 2 ,

(15d) σ_u1=c₀

2x+c₁t+x₀

u_1x+ (c₀t+t₀)u_1t−c₀u₁ 2 ,

(16a) σv₁ =c₀

2x+c₁t+x₀

v_1x+ (c₀t+t₀)v_1t−c₀v₁, (16b) σ_r1=c₀

2x+c₁t+x₀

r_1x+ (c₀t+t₀)r_1t−c₀r₁ 2 ,

(16c) σs₁=c₀

2x+c₁t+x₀

s_1x+ (c₀t+t₀)s_1t−c₀s₁, (16d) . . . ,

σuj =c₀

2x+c₁t+x₀

u_j,x+ (c₀t+t₀)u_j,t

−(2j−1)c₀u_j

2 ,

(17a)

σv_j=c₀

2x+c₁t+x₀

v_j,x+ (c₀t+t₀)v_j,t

−jc₀v_j−c1x 2 δj,0,

(17b)

σ_rj =c0

2x+c₁t+x0

rj,x+ (c₀t+t₀)r_j,t

−(2j−1)c₀rj

2 ,

(17c)

σ_sj =c₀

2x+c₁t+x₀

sj,x+ (c₀t+t₀)sj,t

−jc₀s_j−c₁x 2 δj,0.

(17d) Note that the notationδj,isatisfiesδj,j=1 andδj,i= 0(j6=i). Subsequently, the similarity solutions to (8) can be obtained by solving the characteristic equations

dx X = dt

T = du₀ U₀ = dv₀

V₀ = dr₀ R₀ = ds₀

S₀ =. . .

= du_j U_j = dv_j

V_j = dr_j R_j = ds_j

S_j =. . . .

(18)

In the following paragraphs, two subcases are distin- guished concerning the solutions to (18).

Case 1. Whenc06=0, we choose the group invariant as

ξ=c²₀x−2c₀c₁t+2c₀x₀−4c₁t₀ c²₀√

c₀t+t₀ .

Then the similarity solutions for fieldsuj,vj,rj, andsj

are

u0=U0(ξ)(c₀t+t0)⁻¹², u1=U1(ξ)(c₀t+t₀)¹², . . . , u_j=U_j(ξ)(c₀t+t₀)^2j−12 , (19a) v₀=V₀(ξ) +f, v₁=V₁(ξ)(c₀t+t₀), . . . ,

v_j=V_j(ξ)(c₀t+t₀)^j+δ_j,0f, (19b) r₀=R₀(ξ)(c₀t+t₀)⁻¹², r₁=R₁(ξ)(c₀t+t₀)¹², . . . , rj=Rj(ξ)(c₀t+t₀)^2j−12 , (19c) s0=S0(ξ) +f, s1=S1(ξ)(c₀t+t₀), . . . ,

s_j=S_j(ξ)(c₀t+t0)^j+δ_j,0f. (19d) As follows from (19), the perturbation series solutions to (4) are described by the equations

u=

∞ j=0

∑

ε^jU_j(ξ)(c₀t+t₀)^2j−12 ,

v=f+

∞

∑

j=0

ε^jV_j(ξ)(c₀t+t₀)^j,

(20a)

r=

∞

∑

ε^jR_j(ξ)(c₀t+t₀)^2j−12 ,

s=f+

∞

∑

ε^jS_j(ξ)(c₀t+t₀)^j,

(20b)

where f =c1

c³₀

(c₁t₀−c₀x₀)ln(c₀t+t₀) +c²₀x

−c₀c₁t+2c₀x₀−4c₁t₀

andU−1=V−1=R−1=S−1=0. Inserting similarity solutions (19) into (8), we summarize the similarity re- duction equations

U_{0ξ ξ}−U₀V_0ξ² +1

2c0ξU0V_0ξ+U₀ U₀²+R²₀ +λU0=0,

(21a) 2U0V_{0ξ ξ}+4U_0ξV_0ξ−c₀ξU_0ξ−c0U0=0, (21b) R_{0ξ ξ}−R₀S²_0ξ+1

2c₀ξR₀S_0ξ+R₀ U₀²+R²₀ +λR₀=0,

(21c) 2R₀S_{0ξ ξ}+4R_0ξS_0ξ−c₀ξR_0ξ−c₀R₀=0, (21d)

X.-P. Cheng et al.·Solutions to Perturbed Coupled Nonlinear Schr¨odinger Equation 385

U_{1ξ ξ}−2U₀V_1ξV_0ξ−U₁V_0ξ² +1

2c₀ξ(U₀V_1ξ+U₁V_0ξ) +2U₀R₀R₁+U₁R²₀+3U₀²U₁−c₀U₀V₁

+λU₁=0,

(22a)

2(U₀V_{1ξ ξ}+U₁V_{0ξ ξ}) +4(U_0ξV_1ξ+U_1ξV_0ξ)

−c₀ξU_1ξ+c₀U₁+2U₀=0, (22b) R_{1ξ ξ}−2R₀S_1ξS_0ξ−R₁S²_0ξ+1

2c₀ξ(R₀S_1ξ+R₁S_0ξ) +2R₀U₀U₁+R₁U₀²+3R²₀R₁−c₀R₀S₁

+λR₁=0,

(22c)

2(R₀S_{1ξ ξ}+R₁S_{0ξ ξ}) +4(R_0ξS_1ξ+R_1ξS_0ξ)

−c₀ξR_1ξ+c₀R₁+2R₀=0, (22d) . . . ,

U_{j,ξ ξ}−

j

∑

i=0 j−i

∑

k=0

U_iV_k,ξV_j−i−k,ξ

+

j i=0

∑

j−i k=0

∑

U_i(R_kR_j−i−k+U_kU_j−i−k)

+1 2c0ξ

j i=0

∑

U_iV_j−i,ξ−c0 j i=0

∑

(j−i)U_iVj−i+λU_j=0, (23a)

2

j

∑

i=0

U_iV_{j−i,ξ ξ}+4

j

∑

i=0

U_i,ξV_j−i,ξ+c₀ξU_j,ξ + (2j−1)c₀U_j+2U_j−1=0,

(23b)

R_{j,ξ ξ}−

j

∑

i=0 j−i

∑

k=0

R_iS_k,ξS_j−i−k,ξ

+

j i=0

∑

j−i k=0

∑

Ri(R_kRj−i−k+U_kUj−i−k)

+1 2c₀ξ

j

∑

i=0

R_iS_j−i,ξ−c₀

j

∑

i=0

(j−i)R_iVj−i+λR_j=0, (23c)

2

j

∑

i=0

R_iS_{j−i,ξ ξ}+4

j

∑

i=0

R_i,ξS_j−i,ξ+c₀ξR_j,ξ + (2j−1)c₀R_j+2Rj−1=0.

(23d)

The constantλ isλ = ¹

c²₀(c₀c₁x₀−c²₁t₀).

Case 2. In the casec₀=0, the similarity solutions be- come

uj=Uj(ξ), vj=Vj(ξ) +gδj,0, rj=Rj(ξ), sj=Sj(ξ) +gδj,0

(24)

with the similarity variable ξ and function g being taken as

ξ =c1t²+2x₀t−2t₀x

2t₀ ,

g=−(c₁t+x₀)

6c₁t₀² (c²₁t²+2c₁x₀t−3c₁t₀x+x²₀). Hence, the perturbation series solutions to (4) are

u=

∞

∑

j=0

ε^jU_j(ξ), v=g+

∞

∑

j=0

ε^jV_j(ξ),

r=

∞

∑

ε^jRj(ξ), s=g+

∞

∑

ε^jSj(ξ).

(25)

And the similarity reduction equations related to simi- larity solutions (24) can be expressed by

U_{0ξ ξ}−U₀V_0ξ² +U₀(R²₀+U₀²) +κU₀=0, (26a) U₀V_{0ξ ξ}+2V_0ξU_0ξ =0, (26b) R_{0ξ ξ}−R0S²_0ξ+R₀(R²₀+U₀²) +κR0=0, (26c) R0S_{0ξ ξ}+2S_0ξR_0ξ =0, (26d) U_{1ξ ξ}−2U₀V_0ξV_1ξ−U₁V_0ξ² +2U₀R₀R₁

+3U₀²U1+U₁R²₀+κU1=0, (27a) U0V_{1ξ ξ}+U₁V_{0ξ ξ}+2(V_0ξU_1ξ+V_1ξU_0ξ)

+U₀=0, (27b)

R_{1ξ ξ}−2R₀S_0ξS_1ξ−R1S_0ξ² +2R₀U0U1

+3R²₀R₁+R₁U₀²+κR₁=0, (27c) R₀S_{1ξ ξ}+R₁S_{0ξ ξ}+2(S_0ξR_1ξ+S_1ξR_0ξ)

+R₀=0, (27d)

. . . ,

U_{j,ξ ξ}−

j i=0

∑

j−i k=0

∑

UiV_k,ξVj−i−k,ξ

+

j

∑

i=0

U_i(R_kRj−i−k+U_kUj−i−k) +κU_j=0,

(28a)

j

∑

i=0

U_iV_{j−i,ξ ξ}+2

j

∑

i=0

U_i,ξV_j−i,ξ+Uj−1=0, (28b)

R_{j,ξ ξ}−

j i=0

∑

j−i k=0

∑

R_iS_k,ξV_j−i−k,ξ

+

j i=0

∑

Ri(R_kRj−i−k+U_kUj−i−k) +κRj=0,

(28c)

j i=0

∑

R_iS_{j−i,ξ ξ}+2

j i=0

∑

R_i,ξS_j−i,ξ+Rj−1=0, (28d)

whereU−1=V−1=R−1=S−1=0, and the parameter κ is determined byκ=^2c1t0ξ+x02

4t₀² .

On observation of (23) and (28), it is not difficult to find that thejth similarity reduction equations con- tain U₀,U₁, . . . ,U_j, V₀,V₁, . . . ,V_j, R₀,R₁, . . . ,R_j, and S0,S1, . . . ,Sj. When the previous U0,U₁, . . . ,Uj−1, V₀,V₁, . . . ,Vj−1, R₀,R₁, . . . ,Rj−1, and S₀,S₁, . . . ,Sj−1

are known, the jth similarity reduction equations re- duce to the second-order linear ordinary differential equations ofU_j,V_j,R_j, and S_j. One can then solve them one after another.

3. Symmetry Perturbation Reduction for One-Soliton Solutions

For more detailed and further studying, let us take the simple casec₁=c₀=0 for example and consider the effects of perturbations on soliton solutions of the CNLSE with the help of Maple. Now the group vari- ableξ becomes

ξ=t−t₀ x0

x.

Calculating as above steps, it is easy to get the sim- ilarity reduction equations (for convenience, we do not write out them in detail). Solving of the reduc- tion equations leads to the solutions of different orders.

–4 –2 0 2 4

x –15–10

–5 0 5 10

15 t

0 0.5 1

|p|^2

–2 –4 2 0

4

x –15–10

–5 0 5 10

15 t

–1 –0.5

0

|q|^2

(a) (b)

Fig. 1. Evolution of intensity profile|p|²(a) and|q|²(b) of the one-soliton solutions of (2) with no perturbation, where the functionsu,v,r,sin (3) are respectively replaced byu0,v0,r0,s0shown in (29). The parameters are set atx0=t0=b1= k0=1.

Here for simplicity, we give the perturbation series so- lutions up to order 1, that is,

u0=U0(ξ) =k0tanh √

2x²₀ξ 4t₀² +b₁

, (29a)

r₀=R₀(ξ)

=

q− 4k₀²t₀²+x₀² tanh

^√_2x2 0ξ 4t₀² +b1

2t₀ ,

(29b)

s₀=v₀=S₀(ξ) =V₀(ξ) =−x²₀ξ

2t₀², (29c)

u₁=U₁(ξ) = k₁ 2 cosh²

^√_2x2 0ξ 4t₀² +b1

, (30a)

r1=R1(ξ) = k1

q− 4k²₀t₀²+x²₀ 4k₀t₀cosh^2√_2x2

0ξ

4t²₀ +b₁, (30b) s1=v1=S1(ξ) =V1(ξ)

=−x²₀ξ² 2t₀² +4√

2x²₀ξe

√ 2x2

0ξ 2t2

0

+2b1

−16t₀² x²₀



e

√ 2x2

0ξ 2t2

0

+2b₁

−1





, (30c)

. . . ,

wherek0,k1,b1are arbitrary constants.

In Figures1 and2, the evolution of intensity pro- files|p|² and |q|² of the one-soliton solutions of (2) without and with perturbations, respectively, are plot- ted. The parameters in (29) and (30) are fixed at

X.-P. Cheng et al.·Solutions to Perturbed Coupled Nonlinear Schr¨odinger Equation 387

–2 –4 2 0

4

x –15–10

–5 0 5 10

15 t

0 0.5 1

|p|^2

–2 –4 2 0

4

x –15–10

–5 0 5 10

15 t

–1.5 –1 –0.5

0

|q|^2

(a) (b)

Fig. 2. Evolution of intensity profile|p|²(a) and|q|²(b) of PCNLSE (2) withε=0.9. The zeroth-order and the first-order solutions are chosen as (29) and (30), where parameterk₁=2 and others are the same as those in Figure1.

0.2 0.4 0.6 0.8 1 1.2

|p|^2

–15 –10 –5 5 10

t

–1.4 –1.2 –1 –0.8 –0.6 –0.4 –0.2 0 0.2

|q|^2

–15 –10 t –5 5 10

(a) (b)

Fig. 3. Time evolution of amplitudes of the one-soliton solutions of PCNLSE (2) with dif- ferentεvalues forx= 0. ε=0, 0.5, 0.7, 0.8 from right to left.

x₀=t₀=k₀=b₁=1, k₁=2, ε=0.9. Some gen- eral behaviours of solitons in this case can also be found in [22]. Under the action of perturbations−iεp and −iεq, the solitons become deformed. For larger ε values, the distortion is more serious and the width of solitons is narrower and narrower. Another effect of the perturbations on solitons is that the perturba- tions have also changed the initial positions of soli- tons. It can be found obviously in Figure3, which illustrates the time evolution of amplitudes of solu- tions of (2) with differentε values, where the param- eter ε =0, 0.5, 0.7, 0.8, respectively, from right to left.

4. Summary and Discussion

In summary, approximate similarity reductions of the PCNLSE were studied in the frame of the approx-

imate symmetry reduction method, and the approx- imate symmetries and similarity symmetry solutions of different orders are given. From the results (23) and (28), it is not difficult to discover that the jth- order similarity reduction equations are linear vari- able coefficient ordinary differential equations ofU_j, V_j, R_j, and S_j, which depend on particular solutions of the previous similarity reduction equations from zero order to (j−1) order and Uj, Vj, Rj, and Sj

can be solved step by step. Compared with other per- turbation methods, the approximate symmetry pertur- bation method is much easier for treating the per- turbed partial differential equations. Besides, with the help of Maple, we have also analyzed the effects of perturbations on the one-soliton solutions, where we found that the perturbations −iεp and −iεq have not only changed the shapes of the solitons, but also the initial positions of the solitons. Also, if choos-

ing other solutions as initial approximate, we can an- alyze the effects of perturbations on them in the sim- ilar way. Moreover, extending the approximate sym- metry perturbation method to more other perturbed nonlinear evolution equations is worthy of further study.

Acknowledgement

The work is supported by the National Natural Sci- ence Foundation of China (No. 11047155) and the Nat- ural Science Foundation of Zhejiang Province of China (No. Y1100088 and No. Y6110074).

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