# Soliton and Breather Solutions for the Mixed Nonlinear Schrödinger Equation viaN-Fold Darboux Transformation

Authors: Hui-Qin Hao; Jian-Wen Zhang; Rui Guo
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Under investigation in this paper is the mixed nonlinear Schrödinger equation, which describes the propagation of the subpicosecond or femtosecond optical pulse in a monomodal optical fiber. The Darboux transformation is constructed andN-times iterative potential formula is presented. Two-soliton and breather solutions are derived on the vanishing and two types of nonvanishing backgrounds: the continuous wave(cw) background and constant background, respectively. The dynamic features of the solitons and breathers are discussed via analytic solutions and graphical illustration.


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## Body

## 1. Introduction

Investigations on the nonlinear Schrödinger equation have attracted certain interest in nonlinear optics [1–6]. In this paper, we mainly investigate the following mixed nonlinear Schrödinger equation (MNLS) [7]:
(1)iuz+12utt+σ14|u|2u-σ3iδ16(|u|2u)t+Γu=0,
where u denotes the slowly varying complex pulse envelope, z and t are the partial derivatives with respect to the longitudinal distance and retarded time, σ=±1 corresponds to the focusing and defocusing nonlinear Schrödinger (NLS) equation, δ is a real parameter describing the measure of the derivative cubic nonlinearity, and Γ denotes the loss or gain coefficient. Equation (1) usually governs the subpicosecond or femtosecond optical pulse propagation in a monomodal optical fiber.As a complete integrability model, (1) is equivalent to the following Wadati-Konno-Ichikawa (WKI) spectral system [7, 8]:(2a)Ψt=UΨ=(2i(λ2-1)3δi2λui2λu*2i(λ2-1)3δ)Ψ,(2b)Ψz=VΨ=(-4i9δ2(λ2-1)2+i8λ2|u|2+iΓ2-iλ(λ2-1)3δu+3i32δλ|u|2u-λ4ut-iλ(λ2-1)3δu*+3i32δλ|u|2u*+λ4ut*4i9δ2(λ2-1)2-i8λ2|u|2-iΓ2)Ψ,where Ψ=(ψ1,ψ2)′ (′ denotes the transpose of a matrix). One can deduce (1) directly from the zero curvature equation Uz-Vt+UV-VU=0.During the past decades, [8–10] have derived some soliton solutions via the dressing method, the Hirota bilinear method and the technique of determinant calculation, respectively. However, to our knowledge, the soliton and breather solutions of (1) have not been generated through N-fold Darboux transformation and the interaction characters between two breathers have not been analysed. The aim of this paper is mainly to construct N-fold Darboux transformation (DT) in Section 2, derive two-soliton, one- and two-breather solutions via the obtained DT, and discuss dynamic features of those solitons and breathers in Section 3. Finally, our conclusions will be addressed in Section 4.


## 2.N-Fold Darboux Transformation

Taking the transformation as(3)Ψ′=TΨ=(ABCD)Ψ=((-1)N+∑n=1N‍a2nλ2n∑n=1N‍b2n-1λ2n-1∑n=1N‍c2n-1λ2n-1(-1)N+∑n=1N‍d2nλ2n)Ψ,
we require that Ψ′ must satisfy(4a)Ψt′=U′Ψ′=(Tt+TU)T-1Ψ′,(4b)Ψz′=V′Ψ′=(Tz+TV)T-1Ψ′,where U′, V′ have the same form as U, V. From the rules of the Darboux transformation [11–14], we know that T|λ=λ1Ψ=0, a2n, b2n-1, c2n-1, and d2n can be calculated by the linear algebraic system:
(5)∑n=1N‍(a2nλj+αjb2n-1)λj2n-1=(-1)N,∑n=1N‍(c2n-1+αjd2nλj)λj2n-1=(-1)N+1αj,
with
(6)αj=ϕ2(λj)-γjψ2(λj)ϕ1(λj)-γjψ1(λj),1≤j≤2N,
and ϕ=(ϕ1,ϕ2)′, ψ=(ψ1,ψ2)′ as two basic solutions of the spectral system (2a)-(2b) and λj and γj(λk≠λj, γk≠γj, as k≠j) as parameters suitably chosen such that the determinant of coefficients for (5) is nonzero. Thus, a2n, b2n-1, c2n-1, and d2n are uniquely determined by (5). Equations (3) and (5) show that det⁡T(λ) is a 2N polynomial of λ, and
(7)det⁡T(λj)=A(λj)D(λj)-B(λj)C(λj),cA(λj)=-αjB(λj),C(λj)=-αjD(λj).
Therefore, one can derive that det⁡T(λ)=∏j=12N‍(λ-λj)=0, so λj(j=1,2,…) are 2N roots for the determinant of matrix T(λ).Meanwhile, defining(8)(Tt+TU)T*=(f11(λ)f12(λ)f21(λ)f22(λ)),(Tz+TV)T*=(g11(λ)g12(λ)g21(λ)g22(λ)),
one can deduce that f11(λ) and f22(λ) are (4N+2), f12(λ) and f21(λ) are 4N+1, g11(λ) and g22(λ) are (4N+4), g12(λ) and g21(λ) are 4N+3th-order polynomials of λ, and(9a)αjt=i2λju*-4i(λj2-1)3δαj-i2λjuαj2,(9b)αjz=c(λj)-2a(λj)αj-b(λj)αj2,where
(10)a(λj)=-4i9δ2(λj2-1)2+i8λj2|u|2,b(λj)=-iλj(λj2-1)3δu+3i32δλj|u|2u-λj4ux,c(λj)=-iλj(λj2-1)3δu*+3i32δλj|u|2u*+λj4ux*.
By virtue of the above conclusions, one can deduce that all λj(1⩽j⩽2N) are roots of fkj(λ), gkj(λ)(k,j=1,2). Noting (7), one can arrive at
(11)det⁡M∣fkj(λ),det⁡M∣gkj(λ)(k,j=1,2).
According to TT*=det⁡|T|E, one can transform (8) into
(12)Tt+TU=P(λ)T,Tz+TV=Q(λ)T,
where(13a)P(λ)=(p11(2)λ2+p11(1)λ+p11(0)p12(1)λ+p12(0)p21(1)λ+p21(0)p22(2)λ2+p22(1)λ+p22(0)),(13b)Q(λ)=(∑j=04‍q11(j)λj∑j=03‍q12(j)λj∑j=03‍q21(j)λj∑j=04‍q22(j)λj).Comparing the coefficients of λ2N+4, λ2N+3, λ2N+2, λ2N+1, and λ2N in (12), one can obtain(14a)p11(2)=-p11(0)=-p22(2)=p22(0)=2i3δ,p11(1)=p12(0)=p21(0)=p22(1)=0,(14b)p12(1)=i2(ua2Nd2N-83δb2N-1d2N)=i2u′,p21(1)=i2(u*d2Na2N+83δc2N-1a2N)=i2u′*,(14c)q22(4)=q22(0)=-q11(4)=-q11(0)=4i9δ2,(14d)q11(3)=q11(1)=q12(2)=q12(0)=q21(2)=q21(0)=q22(3)=q22(1)=0,(14e)q12(3)=-i3δ(ua2Nd2N-83δb2N-1d2N)=-i3δu′,(14f)q21(3)=-i3δ(u*d2Na2N+83δc2N-1a2N)=-i3δu′*,(14g)q11(2)=8i9δ2-i3δb2N-1a2Nu*+i8uu*+q12(3)c2N-1a2N=8i9δ2+i8u′u′*,(14h)q22(2)=-8i9δ2+i3δc2N-1a2Nu*-i8uu*-q21(3)b2N-1a2N=-8i9δ2-i8u′u′*,(14i)q12(1)=1d2N(3iδ32-8i9δ2b2N-1+8i9δ2b2N-3-q11(2)b2N-1mmmmm-q12(3)d2N-2+iu3δa2N-iu3δa2N-2-i|u|28b2N-1mmmmm+3iδ32a2N|u|2u-ut4a2N8i9δ2),(14j)q21(1)=1a2N(3iδ328i9δ2c2N-1-8i9δ2c2N-3-q22(2)c2N-1-q21(3)a2N-2mmmmm+iu*3δd2N-iu*3δd2N-2+i|u|28c2N-1mmmmm+3iδ32d2N|u|2u*+ut*4d2N8i9δ2).Substituting u′=u(a2N/d2N)-(8/3δ)(b2N-1/d2N), u′*=u*(d2N/a2N)+(8/3δ)(c2N-1/a2N) into (14i) and (14j), one can calculate that
(15)q12(1)=iu′3δ+3iδ32|u′|2u′-14ux′,q21(1)=iu′*3δ+3iδ32|u′|2u′*-14ux′*.
Summarying the above conclusions, one can arrive at the following theorm.Theorem 1.
The solutionq for (1) is mapped into the new solution q′ under the Darboux transformation:
(16)u′=ua2Nd2N-83δb2N-1d2N,u′*=u*d2Na2N+83δc2N-1a2N.


## 3. Soliton and Breather Solutions for (1)

In this section, we will construct soliton solutions for (1). Choosing ϕ(λ)=(ϕ1(λ),ϕ2(λ))′ and ψ(λ)=(-ψ2*(λ*),ψ1*(λ*))′ as two solutions of the spectral system (2a)-(2b), one can derive that λ2j=λ2j-1* and γ2j=γ2j-1*, so α2j-1=[ϕ2(λ2j-1)-γ2j-1ψ2(λ2j-1)]/[ϕ1(λ2j-1)-γ2j-1ψ1(λ2j-1)].ForN=1, from (5), one can obtain that
(17)a2=λ1|ϕ2|2+λ1*|ϕ1|2|λ1|2λ1|ϕ1|2+λ1*|ϕ2|2,b1=(λ1*2-λ12)ϕ1ϕ2*|λ1|2λ1|ϕ1|2+λ1*|ϕ2|2,
and d2=a2*, c1=-b1*. Then from (16), one can obtain
(18)u(1)=u(Δ*Δ)2+83δΔ2(λ12-λ1*2)ϕ1ϕ2*Δ*,
with Δ=λ1|ϕ1|2+λ1*|ϕ2|2.ForN=2, from d4=a4*, c3=-b3*, and (5), one can obtain that(19a)∑n=12‍(a2nλ2j-1+α2j-1b2n-1)λ2j-12n-1=1,(19b)∑n=12‍(a2nλ2j-1*α2j-1*-b2n-1)λ2j-1*(2n-1)=-α2j-1,with j=1,2, so (19a)-(19b) result:
(20)a4=Ωa4Ω,b3=Ωb3Ω,
where
(21)Ω=|λ12λ1α1λ14α1λ13λ32λ3α3λ34α3λ33λ1*2α1*-λ1*λ1*4α1*-λ1*3λ1*2α3*-λ3*λ3*4α3*-λ3*3|.Ωa4 and Ωb3 are produced from Ω by replacing its third and fourth columns with (1,1,-α1*,-α3*), respectively. Then the two-soliton solutions for (1) can be derived as
(22)u(2)=ua4d4-83δb3d4.Then we we mainly discuss soliton solutions from three different seeds.Case 1.
Considering the vanishing background ofu=0, we can generate elastic interactions and bound states between two solitons for (1) as shown in Figure 1.Evolution of the two-soliton solutions of (1). Parameters are (a) λ1=2+2i, λ2=3+2i, δ=2.5, Γ=0.1 and (b) λ1=1+2i, λ2=1.0001-2i, δ=2.5, Γ=0.1.
(a)
(b)Figure1(a) depicts the interactions of two solitons on the vanishing background. One can find that main features of the interaction are that the shapes, amplitudes, and pulse widths all remain invariant except for slightly visible phase shifts, so the interaction is elastic. Figure 1(b) shows that when suitable parameters are chosen, two bound solitons with the same amplitude propagate in parallel without any effect on each other even if the propagation distance grows long enough. From Figure 1, we can conclude that the parallel bound solitons will form when |λ1|→|λ2|, while if the value of ||λ1|-|λ2|| increases, the elastic interactions between two solitons will happen.Case 2.
In the case of the cw background asu=ucexp⁡i(κt+ωz), we can derive the nonlinear dispersion relation ω=(4uc2-8κ2+3uc2κδ+16Γ)/16 for (1). By the method of separation of variables and the superposition principle, we derive
(23)ϕ1=(c1eθ1+c2eθ2)exp⁡i(κt+ωz),ϕ2=c3eθ1+c4eθ2,
where
(24)θ1=t2(-iκ-13δB2-A2)+z2(-iω-13δB2-A2Θ),θ2=t2(-iκ+13δB2-A2)+z2(-iω+13δB2-A2Θ),
with
(25)A=4(λ12-1),B=3iucδλ1,Θ=9δ2uc2-32(λ12-1)48δ.
For simplicity, setting c4=c1 and through direct computations, we can generate that
(26)c3=θ1t+iκ-γ(i/2)λ1ucc1,c2=(i/2)λ1ucθ1t+iκ-γc4,
with γ=2i(λ12-1)/3δ. Substituting (27) into (18) and taking Re(λ1)=Im⁡(λ1), κ=0, we can obtain Akhmediev-breather solutions on the nonvanishing background for (1) as shown in Figure 2(a). One can observe that the main feature is propagation of the Akhmediev breather which is periodic in the space coordinate and aperiodic in the time coordinate. Therefore, it is considered as a modulation instability (MI) process in which a cw beam becomes unstable [15]. MI was predicted to occur in optical fibers and was experimentally observed [16].
Iterating the DT again, we can obtain the two-breather solution for (1) as shown in Figures 2(b), 3(a), and 3(b). One can observe that the interactions between those breathers are also elastic. Through adjusting the value of λ1 and λ2, we can control the directions of those breathers as shown in those figures.Evolution of the breather solutions of (1). Parameters are (a) uc=1, κ=0, δ=3, λ1=2.05+2.05i, Γ=0 and (b) uc=1, κ=0, λ1=2.05+2.05i, λ2=2+2i, δ=3, Γ=0.
(a)
(b)Evolution of the breather solutions of (1). Parameters are (a) uc=1, κ=0, λ1=2.05+1.05i, λ2=2+2i, δ=3, Γ=0 and (b) uc=1, κ=0, λ1=1.5+i, λ2=3+1.1i, δ=3, Γ=0.
(a)
(b)Case 3.
Considering the constant background ofu=κ1+iκ2, (1) gives the relation κ12+κ22=-4Γ. Defining λ=Ac+iκc, and for simplicity setting κc=8(Ac2-1)-9Γδ2, we can obtain
(27)ϕ1=c1eϑ+c2e-ϑ,ϕ2=c3eϑ+c4e-ϑ,
where
(28)ϑ=2t(ξ1+iη1)+z(ξ1+iη1)8(λ2-1)+9Γδ236δ2,c3=ϑt-(2i/3δ)(λ2-1)(i/2)λ(κ1+iκ2)c1,c2=(i/2)λ(κ1+iκ2)-ϑt-(2i/3δ)(λ2-1)c4,
with
(29)ξ1+iη1=β(16+β),β=9ΓAc2.
Owing to β which is always negative, we discuss two different cases according to the sign of (16+β).(A)
When(16+β)>0, the value of β(16+β) will be negative; then we can arrive at ξ1=0, η1=-β(16+β). Substituting these conclusions and (27) into (18), we can derive the Ma-breather solution for (1) as shown in Figure 4(a).(B)
When(16+β)<0, the value of β(16+β) will be positive; then we can arrive at ξ1=β(16+β), η1=0. Substituting these conclusions and (27) into (18), we can derive the Akhmediev-breather solution for (1) as shown in Figure 4(b).
One can observe from Figure4(a) that the breathers time periodically propagate on the constant backgrounds; that is, they are the Ma-breathers. In addition, as κ1+iκ2 approaches zero, the Ma-breather will become the one-soliton solution. Since κ12+κ22=-4Γ and Γ denotes the gain or loss coefficient in in (1), we can conclude that the absence of the gain or loss term can become breathers into one-soliton solutions.
From Figure4(b), one can find that the Akhmediev breather is periodic in the space coordinate and aperiodic in the time coordinate. Generally, the time-aperiodic solution can be regarded as a homoclinic or separatrix trajectory in the infinite-dimension phase space of the solutions for (1) with periodic boundary conditions in space. Through numerical simulation, one can gain the facts that are in Figure 4(b) as follows:(1)
the periods are in inverse proportion to the value of16+β, so the group velocities of the Akhmediev breathers are dependent on parameters β;(2)
parametersAc can affect the amplitudes.Evolution of the breather solutions of (1). Parameters are (a) Ac=0.5, κ1=1, κ2=1, δ=1 and (b) Ac=0.05, κ1=1, κ2=1, δ=1.
(a)
(b)


## 4. Conclusions

Our attention has been focused on (1) which describes the propagation of the subpicosecond or femtosecond optical pulse in a monomodal optical fiber. With symbolic computation, we have constructed N-fold Darboux transformation and derived two-soliton and breather solutions on the vanishing and two types of nonvanishing backgrounds, respectively. In addition, some figures have been plotted to display the dynamic characteristics of those solitons.



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