B¨acklund Transformation and N-Soliton Solutions for the Cylindrical Nonlinear Schr¨odinger Equation from the Diverging Quasi-Plane Envelope Waves

B¨acklund Transformation and N-Soliton Solutions for the Cylindrical Nonlinear Schr¨odinger Equation from the Diverging Quasi-Plane Envelope Waves

Pan Wang, Bo Tian, Wen-Jun Liu, and Yan Jiang

State Key Laboratory of Information Photonics and Optical Communications, and School of Science, P.O. Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China

Reprint requests to B. T.; E-mail:tian.bupt@yahoo.com.cn

Z. Naturforsch.67a,441 – 450 (2012) / DOI: 10.5560/ZNA.2012-0037 Received November 2, 2011 / revised March 21, 2012

This paper investigates a cylindrical nonlinear Schr¨odinger (cNLS) equation, which describes the cylindrically diverging quasi-plane envelope waves in a nonlinear medium. With the Hirota method and symbolic computation, bilinear form andN-soliton solutions in the form of anNth-order poly- nomial inNexponentials are obtained for the cNLS equation. By means of the properties of double Wronskian, theN-soliton solutions in terms of the double Wronskian is testified through the direct substitution into the bilinear form. Based on the bilinear form and exchange formulae, the bilinear B¨acklund transformation is also given. Those solutions are graphically depicted to understand the soliton dynamics of the cylindrically diverging quasi-plane envelope waves. Soliton properties are discussed and physical quantities are also analyzed. Dispersion parameter has the effect that it may extend (or shorten) the periodic time of soliton interaction and change the direction of soliton propa- gation. Amplitudes of solitons are related to the cubic nonlinearity parameter.

Key words:Cylindrical Nonlinear Schr¨odinger Equation; Soliton Solutions; Double Wronskian;

Hirota Method; B¨acklund Transformation; Symbolic Computation.

PACS numbers:02.30.Ik; 02.30.Jr; 02.70.Wz; 05.45.Yv

1. Introduction

With the development of nonlinear science, solitons have been an attractive topic in the research of optics, plasmas, fluids, condensed matter, astrophysics, and particle physics [1–8]. Progress on the discrete and cavity solitons [2], incoherent and vector solitons [3], and temporal, spatial, and spatiotemporal optical soli- tons [4,5] has been made.

Optical solitons have their potential applications in the long distance communication and all-optical ultra- fast switching devices [6,7]. Optical solitons exist in the optical fibers on the basis of the balance between the group velocity dispersion and self-phase modula- tion [6,7]. Dynamics of nonlinear pulse propagation in a monomode fiber in the absence of optical losses can be described by the nonlinear Schr¨odinger (NLS) equation [6,7],

iE_z−1

2k⁰⁰E_{τ τ}+σ|E|²E=0, (1)

whereE(z,τ)represents the complex envelope ampli- tude, τ andzdenote the time and distance along the direction of propagation, respectively,k⁰⁰is the second derivative of the axial wave numberkwith respect to the angular frequencyω₀and describes the group ve- locity dispersion, andσ=n₂ω0/cA_effis the self-phase modulation parameter withn₂,c, andA_effas the Kerr coefficient, speed of light, and effective core area of the fiber, respectively. Equation (1) admits the bright and dark soliton-type pulses respectively propagating in the anomalous and normal dispersion regimes [6,7].

However, (1) is an ideal model [6,7,9]. Effects not accounted for by (1) have been discussed mainly in two aspects [10–20]: (i) attention has been paid to the study of generalized higher-order NLS equa- tions containing the terms accounting for such ef- fects as the third-order dispersion, delayed nonlinear response, and self-steepening [10,11]; (ii) nonlinear optical fibers with the inhomogeneous dispersion and nonlinearity have been considered, including the pulse

c

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

compression [12,13], soliton control [14,15], stimula- tion of modulation instability [16,17], dispersion man- agement [18], and soliton amplification [19]. Dynam- ics of the optical solitons is affected by the presence of inhomogeneities in the media [20].

In this paper, we will consider an inhomogeneous cylindrical NLS (cNLS) equation [21–24],

iq_t+α(t)q_xx+β(t)q|q|²= q

2t, (2)

which describes the cylindrically diverging quasi-plane envelope waves in a nonlinear medium. Equation (2) plays a role in the theory of light wave envelopes in dispersive media with nonlinear refractive index [22].

Hereby, q(x,t) is the complex envelope,t represents the ‘distance’ along the direction of propagation, and xdenotes the ‘time’ in the group velocity frame. The dispersion parameterα(t)and cubic nonlinear param- eter β(t)are both real functions. Lax pair for (2) has been derived through the Painlev´e analysis [21], and the one-soliton solution has also been given via the B¨acklund transformation (BT) corresponding to the Lax pair [21].

Physically meaningful, our work will be focused on t >0 for (2). However, to our knowledge, the inte- grable properties such as the N-soliton solutions and bilinear B¨acklund transformation (BT) for (2) have not been obtained yet. The paper will proceed as follows:

in Section2, with the Hirota method and symbolic computation [25–36], we will obtain theN-soliton so- lutions in the form of an Nth-order polynomial in N exponentials.N-soliton solutions in terms of the dou- ble Wronskian will be given by means of the Wron- skian technique, and will be verified through the di- rect substitution into the bilinear form in Section3. In Section4, based on the bilinear form, bilinear BT will be constructed, which can be used to construct theN- soliton solutions from the vacuum solution in an itera- tive manner. Section5will be our discussions. Conclu- sions will be given in Section6.

2. Bilinear Form andN-Soliton Solutions in Terms of the Exponential Polynomials for (2)

Based on the Hirota method [38], once the nonlin- ear evolution equations (NLEEs) are transformed into the bilinear forms, analytic solutions can be derived by the perturbation technique without employing the in- verse scattering method [25–31]. In this section, bilin-

ear form andN-soliton solutions for (2) will be derived with the help of symbolic computation.

Equation (2) can be transformed into the bilinear form

iD_t+α(t)D²_x− 1 2t

g·f =0, (3a) D²_xf·f =γ|g|², (3b) with the transformation q=g/f and the coefficient condition β(t) =γ α(t). Hereby,g(x,t)is a complex differentiable function, f(x,t)is a real one,γis a real constant, andD_xandD_tare the bilinear derivative op- erators [38] defined by

D^m_xDⁿ_t(ν·υ) = ∂

∂x− ∂

∂x⁰ m

∂

∂t− ∂

∂t⁰ n

·ν(x,t)υ(x⁰,t⁰)|_x0=x,t⁰=t. Let us expandg(x,t)and f(x,t)as

g(x,t) =εg₁(x,t) +ε³g₃(x,t) +ε⁵g₅(x,t) +. . . , (4a) f(x,t) =1+ε²f₂(x,t) +ε⁴f₄(x,t)

+ε⁶f₆(x,t) +. . . , (4b)

where ε is a formal expansion parameter, and the coefficients fi(x,t) (i=2,4,6, . . .) and gj(x,t) (j= 1,3,5, . . .)are the differentiable functions to be deter- mined.

Without loss of generality, we setε =1. Truncat- ing expression (4) with g_m=0 (m=3,5,7, . . .) and fn=0 (n=4,6,8, . . .), and substituting it into bilinear form (3), we get

g=g₁=e^ξ, f =1+f₂

=1+ γ

2(w+w^∗)²e^ξ+ξ∗, (5a) ξ =k(t) +wx+ζ, k(t)

=i Z n

[2tw²α(t)−1]/(2t)o

dt, (5b)

wherewandζ are the complex constants with^∗denot- ing the complex conjugate. Thus, the one-soliton solu- tions for (2) can be expressed as

q=g f

= g1

1+f₂= e^iIm(ξ) 2 e^∆ sech

1

2(ξ+ξ^∗+2∆)

, (6)

P. Wang et al.·Cylindrical NLS Equation from Diverging Quasi-Plane Envelope Waves 443 where

e^2∆= γ 2(w+w^∗)²,

with Im(ξ)denoting the imaginary part ofξ. We take

g=g₁+g₃, f =1+f₂+f₄, (7) whereg1=e^ξ1+e^ξ2 andξj=kj(t) +wjx+ζj,kj(t) (j=1,2)are the differentiable functions to be deter- mined, w_j and ζ_j are both complex constants. Sub- stituting expression (7) into bilinear form (3), we getk_j(t) =i^R[2tw²_jα(t)−1]/(2t)dt,(j=1,2). After some calculations, we obtain

f₂=M₁₁e^ξ1+ξ1∗+M₁₂e^ξ1+ξ2∗+M₂₁e^ξ2+ξ1∗ +M₂₂e^ξ2+ξ2∗, (8a)

f4=N12N₁₂^∗M11M12M21M22e^{ξ1+ξ2+ξ1∗+ξ2∗}, (8b) g₃=N₁₂M₁₁M₂₁e^{ξ1+ξ2+ξ1∗}

+N₁₂M₁₂M₂₂e^{ξ1+ξ2+ξ2∗}, (8c) where

M11= γ

2(w₁+w^∗₁)², M12= γ 2(w₁+w^∗₂)², M₂₁= γ

2(w₂+w^∗₁)², M₂₂= γ 2(w₂+w^∗₂)², N₁₂=2(w₁−w2)²

γ .

The two-soliton solutions for (2) is obtained as fol- lows:

q=g

f = g₁+g₃

1+f₂+f₄= e^ξ1+e^ξ2+N₁₂M₁₁M₂₁e^{ξ1+ξ2+ξ1∗}+N₁₂M₁₂M₂₂e^{ξ1+ξ2+ξ2∗}

W , (9)

where

W =1+M₁₁e^ξ1+ξ1∗+M₁₂e^ξ1+ξ2∗+M₂₁e^ξ2+ξ1∗ +M22e^ξ2+ξ2∗

+N₁₂N₁₂^∗M₁₁M₁₂M₂₁M₂₂e^{ξ1+ξ2+ξ1∗+ξ2∗}. In general, N-soliton solutions for (2) can be pre- sented in the following form [37,38]:

q=g

f , (10a)

f=

∑

µ=0,1 0exp

"

2N

∑

l<j

η(l,j)µ_lµ_j+

2N

∑

l=1

µ_lθ_l

#

, (10b)

g=

∑

µ=0,1 00exp

"

2N

∑

l<j

η(l,j)µ_lµj+

2N

∑

l=1

µlθl

#

, (10c)

g^∗=

∑

µ=0,1 000exp

"

2N

∑

l<j

η(l,j)µ_lµj+

2N

∑

l=1

µlθl

#

, (10d)

where

θ_l=k_l(t) +w_lx+ζ_l, θl+N=θ_l^∗, k_l+N(t) =k^∗_l(t) (l=1,2, . . . ,N), wl+N=w^∗_l, ζl+N=ζ_l^∗ (l=1,2, . . . ,N), k_l(t) =i

Z n

2tw²_lα(t)−1 /(2t)o

dt,

η(l,j) =ln γ

2(w_l+w_j)²

(l=1,2, . . . ,Nand j=N+1, . . . ,2Nor l=N+1, . . . ,2Nand j=1,2, . . . ,N), η(l,j) =ln

2(w_l−w_j)² γ

(l=1,2, . . . ,Nand j=1,2, . . . ,Nor l=N+1, . . . ,2Nand j=N+1, . . . ,2N), with w_l, w_j, ζl, and ζj as the complex constants.

Hereby,∑^2N_l<j indicates the summation over all possi- ble pairs taken form 2N elements with the condition l< j.∑µ=0,10 ,∑µ=0,100 , and ∑µ=0,1000 indicate the sum- mations over all possible combinations of µ_l =0,1 (l=1,2, . . .2N) and require that

N l=1

∑

0µl=

N l=1

∑

0µl+N,

N l=1

∑

00µl=1+

N l=1

∑

00µl+N,

1+

N

∑

l=1 000µ_l=

N

∑

l=1 000µl+N.

3.N-Soliton Solutions in Terms of the Double Wronskian for (2)

In the following, we will give the N-soliton so- lutions in terms of the double Wronskian [39–41]

for (2). We postulate thatgandf have the form f=W^N−1,N−1(φ,ψ), g=2√

√2

γ W^N,N−2(φ,ψ), g^∗=2√

√2

γ W^N−2,N(φ,ψ),

(11)

where the double Wronskian is defined as W^N,M(φ,ψ) =det(φ,∂_xφ, . . . ,∂_x^N−1φ;

ψ,∂xψ, . . . ,∂_x^M−1ψ),

with φ = (φ₁,φ2, . . . ,φN+M)^T and ψ = (ψ₁,ψ2, . . . , ψN+M)^T. Hereby,φ_jandψ_jare postulated to be

φ_j=a_je^{−ρjx+2iρ2j}

Rα(x)dx−ⁱ

4lnt, ψ_j=b_je^{ρjx−2iρ2j}

Rα(x)dx+ⁱ₄lnt

, φN+j=−ψ^∗_j, ψN+j=φ^∗_j,

(12)

witha_j,b_j, andρj(j=1,2, . . . ,N)as all the complex constants. It can be verified thatφ_j andψ_j satisfy the following conditions:

(φ_j)_x=−ρ_jφ_j, (ψ_j)_x=ρ_jψ_j, (φ_N+j)_x=ρ^∗_jφ_N+j, (ψ_N+j)_x=−ρ^∗_jψN+j,(φ_j)_t=2iα(t)(φ_j)_xx− i

4tφ_j, (ψ_j)_t=−2iα(t)(ψ_j)_xx+ i

4tψ_j, (φ_N+j)_t=2iα(t)(φ_N+j)_xx− i

4tφN+j, (ψ_N+j)_t=−2iα(t)(ψ_N+j)_xx+ i

4tψN+j.

Via the abbreviated notation [39–41] for the Wron- skian and its derivatives, (11) becomes

f=

N[−1; N[−1

, g=2√

√2 γ

bN; N[−2 , g^∗=2√

√2 γ

N[−2; Nb .

(13)

From (12) and (13), we have f_x=

N[−2, N; N[−1 +

N[−1; N[−2, N ,

(14a)

g_x=2√

√2 γ

N[−1, N+1; N[−2

+

bN; N[−3, N−1

,

(14b)

f_xx=

N[−3,N−1,N; N[−1

+

N[−2,N+1; N[−1

+2

N[−2, N; N[−2, N

+

N[−1; N[−3, N−1, N

+

N[−1; N[−2, N+1 ,

(14c)

g_xx=2√

√2 γ

N[−2, N, N+1; N[−2

+

N[−1, N+2; N[−2

+2

N[−1, N+1; N[−3, N−1

+

bN; N[−4, N−2, N−1

+

bN; N[−3, N

,

(14d)

f_t=2iα(t)

−

N[−3, N−1, N; N[−1

+

N[−2, N+1; N[−1

+

N[−1; N[−3, N−1, N

−

N[−1; N[−2, N+1

,

(14e)

gt=4i√ 2α(t)

√γ

−

N[−2, N, N+1; N[−2

+

N[−1, N+2; N[−2

+

bN; N[−4, N−2, N−1

−

bN; N[−3, N

−

√2i

√γt

bN; N[−2 .

(14f)

The following identical equations will be employed to verify the final result:

N[−1; N[−1

"

−

N

∑

j=1

ρ_j−ρ^∗_j

#2

bN; N[−2

=

"

−

N

∑

ρj−ρ^∗_j

#

N[−1; N[−1

(15a)

·

"

−

N

∑

ρj−ρ^∗_j

#

bN; N[−2 ,

P. Wang et al.·Cylindrical NLS Equation from Diverging Quasi-Plane Envelope Waves 445

bN; N[−2

"

−

N

∑

ρi−ρ_i^∗

#2

N[−1; N[−1

=

"

−

N

∑

j=1

ρj−ρ^∗_j

#

bN; N[−2

(15b)

·

"

−

N i=1

∑

ρj−ρ^∗_j

#

N[−1; N[−1 .

Substituting (13) – (15) into bilinear form (3a), we ob- tain

ig_tf−ig f_t+α(t)g_xxf−2α(t)g_xf_x+α(t)g f_xx−1 2tg_xf

=8√ 2α(t)

√ γ

N[−1; N[−1

N[−2, N, N+1; N[−2 +

N[−1; N[−1

bN; N[−3, N

+

bN; N[−2

N[−2, N+1; N[−1 +

bN; N[−2

N[−1; N[−3, N−1, N

−

N[−1, N+1; N[−2

N[−2, N; N[−1 −

bN; N[−3, N−1

N[−1; N[−2, N

=4√ 2α(t)

√γ

N[−2, 0, N−1, N, N+1; N[−2, 0, N−1 0, N[−2, N−1, N, N+1; 0, N[−2, N−1

+

N[−1, 0, N; N[−3, 0, N−2, N−1, N 0, N[−1, N; 0, N[−3, N−2, N−1, N

!

=0.

With the help of (13) – (15), we can simplify bilinear form (3b) as 2f f_xx−2f_x²−γ|g|²

=8

N[−1; N[−1

N[−2, N; N[−2, N −8

N[−2, N; N[−1

N[−1; N[−2, N

−8

bN; N[−2

N[−2; Nb

=4(−1)^−5N2+2

N[−2, 0, N−1, N; N[−2, 0, N−1, N 0, N[−2, N−1, N; 0, N[−2, N−1, N

=0.

Therefore,N-soliton solutions for (2) in terms of the double Wronskian can be obtained as

q=2√

√2 γ

W^N,N−2(φ,ψ)

W^N−1,N−1(φ,ψ). (16)

4. Bilinear B¨acklund Transformation for (2)

The BT provides a way of constructing new solu- tions from the known ones for the NLEEs [42]. Based on bilinear form (3), a bilinear BT will be presented in this section. In order to derive the bilinear BT for (2) by means of exchange formulae [42], we have

Q≡f² h

iD_t+α(t)D²_x− 1 2t i

g⁰·f⁰

−f⁰² h

iD_t+α(t)D²_x−1 2t i

g·f

= (f f⁰)h

iD_t(g⁰·f+f⁰·g)−α(t)D²_x(g⁰·f−f⁰·g) + (g f⁰−g⁰f)/(2t)−µ(g⁰f+f⁰g)i

+ (g⁰f+f⁰g)h

−iD_t(f⁰·f)−2λ α(t)D_x(f⁰·f) +α(t)γ(g⁰g^∗−gg^0∗)/2+µf⁰fi

+ (g⁰f−f⁰g)

·h

α(t)D²_x(f·f⁰)−γ α(t)(gg^0∗+g⁰g^∗)/2+2λ²α(t)f⁰fi . (17)

Decoupling (17), we can get the bilinear BT as D_x(g⁰·f+f⁰·g) =λ(g⁰f−f⁰g), (18a) iD_t(g⁰·f+f⁰·g)−α(t)D²_x(g⁰·f−f⁰·g)

+ (g f⁰−g⁰f)/(2t)−µ(g⁰f+f⁰g) =0, (18b) iD_t(f⁰·f) +2λ α(t)D_x(f⁰·f)

−γ α(t)(g⁰g^∗−gg^0∗)/2−µf⁰f=0, (18c) D²_x(f·f⁰)−γ(gg^0∗+g⁰g^∗)/2+2λ²f⁰f =0, (18d) whereλ,µare both the complex constants to be deter- mined. Choosing the trivial solutionq=0, i.e.,g=0 and f =1, and solving BT (18), we can obtaing⁰and

f⁰as follows:

g⁰=χ1e^iκx+i

R−1+2κ2tα(t)

2t dt,

f⁰=m₁e

√2κx−2√

2κ^2Rα(t)dt

+m₂e⁻

√ 2κx+2√

2κ^2Rα(t)dt,

(19)

with λ =iκ,µ =0, |χ₁|²=16κ²m1m2. Hereby, κ, m₁, andm₂are the real constants andχ1is a complex one. Thus, based on BT (18), the one-soliton solutions for (2) can be given as

q=g⁰ f⁰

= χ₁e^iκx+i

R−1+2κ2tα(t)

2t dt

m1e

√2κx−2√

2κ^2Rα(t)dt+m₂e⁻

√ 2κx+2√

2κ^2Rα(t)dt. (20)

5. Discussions

Our analysis begins with the dynamics of cylin- drically diverging quasi-plane envelope waves evolv- ing under one-soliton solutions (6). Characterizing the inhomogeneous features of propagating cylindrically diverging quasi-plane envelope waves, some physical

1

3.5 x 0.5

2.5 t 0 1.1 q

1

3 5 x 0

1 5 x

0.3 1 q

t 2 t 3

(a) (b)

Fig. 1 (colour online). One soliton via solutions (6). Parameters are w = 1+1.1i,α(t) =0.52,γ=4, andζ=1.

quantities such as the amplitude, width, velocity, and energy are, respectively, given as

A= 1

2|e^∆|,W = 2

w+w^∗,v=−iα(t)(w−w^∗), E=

Z +∞

−∞

|q|²dx=2|w+w^∗|

|γ| .

It should be noticed that (6) includes two main vary- ing parameters, i.e., the dispersion parameterα(t)and cubic nonlinearity parameterγ. Through the choice of values of parameters, it is possible to explain the var- ious soliton adjustments. Profile and evolution of one soliton are presented in Figure1. Soliton amplitude is in inverse proportion to the value ofγ, and direct pro- portion to the real part ofw. Velocities of solitons are related toα(t)and the imaginary part ofw. With the choice of α(t)as ±0.4t, 0.2t², and 0.4sin(0.8t), the propagation for the soliton along the distancetare de- picted in Figures2and3. Main feature of the soliton dynamics presented in Figure3b is the phenomenon of periodic propagation.

Seen in Figures4–9are the evolution and interac- tion of two solitons via (9). The situation ofα(t)se- lected as constant is presented in Figures4–7, while the cases ofα(t)in the profiles of polynomial and tri- angle functions are given in Figures8and9. With the parameters given in Figure4a, the two solitons propa- gate parallel and do not interfere with each other. Once the values of the parametersζ1andζ2decrease, the pe- riodic interaction occurs, as seen in Figure4b. The pe- riodic time gets shorter as the value ofα(t)increases, via the comparison between Figures4b and5b. If the value of γ increases, amplitudes of the solitons de- crease (Fig.5a). Under the condition that the real parts of w1 andw2 have the same signs, we note that the signs of the imaginary parts ofw₁andw₂have certain effect on the interaction between the two solitons from Figures6and7. If the imaginary parts of w₁andw₂

P. Wang et al.·Cylindrical NLS Equation from Diverging Quasi-Plane Envelope Waves 447

4

3 x 1

t 3 0 1.1 q

4

3 x

0 4

3 x 1

t 3 0 1.1 q

4

3 x 0

(a) (b)

Fig. 2 (colour online). One soliton via solutions (6). Parameters are (a) w= 1.2−i,α(t) =−0.4t,γ=3, andζ=2;

(b)w=1.2−i,α(t) =0.4t,γ=3, and ζ=2.

3 1

x 1

4 t 0 1.1 q

3 1

x

0 3

1 x 1

4 t 0 1.1 q

3 1

x 0

(a) (b)

Fig. 3 (colour online). One soliton via solutions (6). Parameters are (a) w= 1.2− i, α(t) = 0.2t², γ = 3, and ζ = 2; (b) w =1.2−0.92i, α(t) = 0.4sin(0.8t),γ=3, andζ=2.

3

6 x

2 t 8 0 2.5 q

3

6 x

0 3

6 x 2

8 t 0 2.5 q

3 6

x 0

(a) (b)

Fig. 4 (colour online). Two solitons via solutions (9). Parameters are (a) w₁= 1.1,w2=1.8,α(t) =1,γ=4,ζ1=−3, andζ2=3; (b) the same parameters as (a) except forζ1=1 andζ2=1.

2 4

x 1

t 4 0 2.5 q

2 4

x 2 0

4 x 1

t 4 0 2.5 q

2 4

x 0

(a) (b)

Fig. 5 (colour online). Interaction be- tween two solitons via solutions (9). Pa- rameters are (a) w₁ =1.1, w₂= 1.8, α(t) =1,γ=6,ζ1=1, andζ2=1; (b) w1=1.1,w2=1.8,α(t) =1.61,γ=4, ζ1=1, andζ2=1.

have the opposite signs, the places of interaction are different. If the imaginary parts ofw₁andw₂have the same sign, the difference between Figures 6and7 is that the amplitudes of the solitons do not increase at the moment of interaction. Propagation and interaction of the two solitons are depicted in Figure8whenα(t)is a linear polynomial. Figure9gives the periodic prop-

agation of solitons whenα(t)is chosen as the triangle function. If the imaginary parts ofw₁andw₂are both positive, the two solitons travel along the same direc- tion without interaction (Fig.9a); while in Figure9b, the two solitons travel along the same direction and in- teraction occurs as the imaginary parts ofw₁andw₂ are both negative. The phenomenon that the two soli-

3

5 x

0.8 2.8 t 0 2.5

q 3

5 x

0 3

5 x 0.8

t 2.8 0 2.5

q 3

5 x 0

(a) (b)

Fig. 6 (colour online). Interaction be- tween two solitons via solutions (9). Pa- rameters are (a)w₁=1.2−0.2i,w₂= 1.8+0.4i,α(t) =1,γ=4, ζ1=−1, andζ2=1; (b)w1=1.2+0.2i,w2= 1.8−0.4i,α(t) =1,γ=4, ζ1=−1, andζ2=1.

3 5

x

1

3 t

0 1.1

q

3 5

x

0 3

5 x 1

t 3 0 1.1

q 3

5 x 0

(a) (b)

Fig. 7 (colour online). Interaction be- tween two solitons via solutions (9). Pa- rameters are (a)w1=1.2+0.1i,w2= 1+1.4i, α(t) =1, γ=4, ζ1= −1, andζ2=1; (b)w1=1.2−0.1i,w2= 1−1.4i,α(t) =1,γ=4,ζ1=−1, and ζ2=1.

7

7 x

0.5

3.5 t 0

q 1.1 7

7 x

7

7 x

0.5

3.5 t 0

q 1.1 7

7 x

7

7 x

0.5

3.5 t 0

q 1.1 7

7 x

(a) (b) (c)

Fig. 8 (colour online). Interaction between two solitons via solutions (9). Parameters are (a)w1=1+i,w2=−1.22+i, α(t) =t,γ=4,ζ1=0, andζ2=0; (b) the same parameters as (a) except forw2=−1.22−i; (c) the same parameters as (b) except forw₁=1−i.

3

14 x

2

7 t 0

1.1 q

3

14 x 3

14 x

2

7 t 0

1.1 q

3

14 x

3

14 x

2

7 t 0

1.1 q

3

14 x

(a) (b) (c)

Fig. 9 (colour online). Interaction between two solitons via solution (9). Parameters are (a)w1=1+1.01i, w2=1+i, α(t) =sin(t),γ=4,ζ1=0, andζ2=0; (b) the same parameters as (a) except forw1=1−1.01i,w2=1−i; (c) the same parameters as (a) except forw2=1−i.

P. Wang et al.·Cylindrical NLS Equation from Diverging Quasi-Plane Envelope Waves 449 tons intertwine and periodically propagate is shown in

Figure9c.

As seen from Figures 1–9, the values of the soli- ton amplitude |q| are bounded. From the figures, we can also observe that the finite regions with the enve- lope waves existing centralize almost all of the ampli- tudes and energies, and the values of the soliton am- plitude tend to zero when x→ ±∞, while the values oftare arbitrary real numbers. In general, to deal with the classical solutions of differential equations, the no- tion of weak derivatives is introduced in the Sobolev spaces [43]. Here, to search for the analytic soliton so- lutions and integrability, we have taken advantage of the Wronskian technique and Hirota method, and do not need to introduce the notion of weak derivatives in the Sobolev spaces.

In order to understand the interaction dynamics be- tween the two solitons, we use the asymptotic analysis to investigate two-soliton solutions (9) as follows:

(i) Before the interaction (t→ −∞):

q→S⁻₁ = e^iIm(ξ1) 2 e^∆1 sech

1

2 ξ1+ξ₁^∗+2∆₁

(ξ₁+ξ₁^∗∼0,ξ2+ξ₂^∗→ −∞),

(21a)

q→S⁻₂ = e^iIm(ξ2) 2 e^∆2 sech

1

2 ξ₂+ξ₂^∗+2∆₄

(ξ₂+ξ₂^∗∼0,ξ₁+ξ₁^∗→+∞),

(21b)

with

e^2∆1 = γ 2(w₁+w^∗₁)²,

e^2∆4 = γ(w₁−w₂)²(w^∗₁−w^∗₂)² 2(w₂+w^∗₂)²(w₂+w^∗₁)²(w₁+w^∗₂)². (ii) After the interaction (t→+∞):

q→S⁺₁ = e^iIm(ξ1) 2 e^∆1 sech

1

2 ξ₁+ξ₁^∗+2∆₃

(ξ₁+ξ₁^∗∼0,ξ₂+ξ₂^∗→+∞),

(22a)

q→S⁺₂ = e^iIm(ξ2) 2 e^∆2 sech

1

2 ξ2+ξ₂^∗+2∆₂

(ξ₂+ξ₂^∗∼0,ξ1+ξ₁^∗→ −∞),

(22b)

with

e^2∆2 = γ 2(w₂+w^∗₂)²,

e^2∆3 = γ(w₁−w₂)²(w^∗₁−w^∗₂)² 2(w₁+w^∗₁)²(w₁+w^∗₂)²(w₂+w^∗₁)².

Consequently, from asymptotic expressions (21) and (22), we conclude that the interaction between the two solitons forqis elastic.

6. Conclusions

In this paper, based on symbolic computation, we have studied some analytic properties of equation (2) which describes the cylindrically diverging quasi-plane envelope waves in a nonlinear medium. Via the Hirota method, one-soliton solutions (6), two-soliton solu- tions (9), and N-soliton solutions (10) for (2) in the form of an Nth-order polynomial in N exponentials have been obtained. Moreover, we have givenN-soliton solutions (16) in terms of the double Wronskian for (2) and verified it through the direct substitution into the bilinear form (3). On the other hand, bilinear BT (18) and the corresponding one-soliton solutions (20) have been constructed based on bilinear form (3). Finally, to understand the soliton dynamics of the cylindrically diverging quasi-plane envelope waves in a nonlinear medium, (6) and (9) have been graphically discussed in Figures 1–9. The dispersion parameter α(t) has the effect that it may extend (or shorten) the periodic time of soliton interaction and change the direction of soliton propagation. The soliton amplitude is in inverse proportion to the cubic nonlinearity parameter √

γ.

Through the asymptotic analysis, we have proved the interaction between the two solitons to be elastic.

Acknowledgements

We express our sincere thanks to Dr. X. L¨u for his valuable discussions and other members of our discussion group for their valuable suggestions. This work has been supported by the Open Fund of State Key Laboratory of Information Photonics and Op- tical Communications (Beijing University of Posts and Telecommunications), by the Fundamental Re- search Funds for the Central Universities of China under Grant No. 2011BUPTYB02, by the Supported Project (No. SKLSDE-2010ZX-07) and Open Fund (No. SKLSDE-2011KF-03) of the State Key Labo- ratory of Software Development Environment, Bei- jing University of Aeronautics and Astronautics, by the National High Technology Research and Devel- opment Program of China (863 Program) under Grant No. 2009AA043303, and by the Specialized Research Fund for the Doctoral Program of Higher Education (No. 200800130006), Chinese Ministry of Education.

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