# The Soliton Solutions and Long-Time Asymptotic Analysis for an Integrable Variable Coefficient Nonlocal Nonlinear Schrödinger Equation

Authors: Guiying Chen; Xiangpeng Xin; Feng Zhang
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

An integrable variable coefficient nonlocal nonlinear Schrödinger equation (NNLS) is studied; by employing the Hirota’s bilinear method, the bilinear form is obtained, and theN-soliton solutions are constructed. In addition, some singular solutions and period solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions, the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic.


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## Body

## 1. Introduction

In 1998, Bender and coworker first proposed thePT- (parity-time-) symmetry for non-Hermitian quantum mechanics [1]. Now, PT-symmetry has been extensively studied in diverse areas such as lasers [2], acoustics [3], nonlinear optics [4], Bose-Einstein condensation [5], and quantum mechanics [6, 7]. The nonlinear Schrödinger equation has been regarded as the basic model to describe the propagation of solitons in optical fiber, and its spatial solitons have become the research frontier of nonlinear science [8, 9]. In 2013, Ablowitz and Musslimani incorporated the PT-symmetry with nonlinear integrable systems and proposed the nonlocal or PT-symmetry nonlinear Schrödinger equation (NLS) [10],
(1)iqtx,t+qxxx,t+2q2x,tq∗−x,t=0,where ∗ represents complex conjugation. Obviously, Equation (1) is invariant under the parity-time (PT) transformation, and its solution is evaluated at (x,t) and (−x,t). Since Equation (1) was proposed, many researchers have carried out a lot of work on it. The integrability [10, 11], the Cauchy problem [12], the inverse scattering transform [13], and exact solutions, such as breathers, periodic, and rational solutions [14], general rogue waves [15], multiple bright soliton [16], higher order rational solutions [17], and N-soliton solutions [18] of (1) have been derived. Moreover, other nonlocal integrable systems have also been investigated like nonlocal modified Korteweg-de Vries equation [19, 20], nonlocal KP equation [21], nonlocal vector nonlinear nonlinear Schrödinger equation [22, 23], nonlocal discrete nonlinear Schrödinger equation [24–26], nonlocal Davey-Stewartson I equation [27], etc.Although much advance has been made in nonlocal systems, there are very few studies on nonlocal equations with variable coefficients. From the realistic point of view, it is more accurate to describe the physical phenomena by using the variable coefficient equations in many physics situations [28]. So it is a meaningful work to study the exact solutions for nonlocal equations with variable coefficients. In [29], authors constructed the soliton solutions for the variable coefficient nonlocal NLS equation by using Darboux transformation. In [30], analytical matter wave solutions of a (2+1)-dimensional nonlocal Gross-Pitaevskii equation are investigated. In this paper, we consider the variable coefficient nonlocal NLS equation,
(2)iqtx,t−δtqxxx,t−2δtqx,t2q∗−x,t+αtqx,t=0,where the dispersion coefficient δt and the gain/loss coefficient αt are arbitrary real continuous even functions of variable t. Obviously, Equation (2) keeps the parity-time transformation x⟶−x, t⟶−t, qx,t⟶q∗−x,−t invariant, so it is PT-symmetric. When δt=−1 and αt=0, Equation (2) reduces to the constant coefficient self-focusing nonlocal NLS equation (1). When αt=0, Equation (2) becomes variable coefficient nonlocal NLS equation which has been solved by Darboux transformations in [29]. The novelty of this paper is that the variable coefficient NLS equation is firstly solved by Hirota’s bilinear method, the more general two-soliton solution and N-soliton solution are reported, and the collision of the two solitons is firstly discussed.The paper is organized as follows: In Section2, the bilinear form and the one-soliton, two-soliton, and N-soliton solutions of Equation (2) are obtained based on the Hirota’s direct method. In Section 3, the asymptotic behavior is studied to prove that the two-soliton collision is elastic. Finally, conclusions are given in Section 4.


## 2. The Bilinear Form and Soliton Solutions

We implement the following dependent variable transformation to Equation (2)
(3)q=eiβtgf,where g and f are complex functions and βt is a real function; then, the following bilinear equations of Equation (2) are obtained as follows:
(4)iDt−δtDx2g·f=0,f∗−x,tDx2f·f=2fgg∗−x,t,where βt=∫αtdt and D is the bilinear operator [24]:
(5)DtlDxmfx,t·gx,t=∂∂t−∂∂t′l∂∂x−∂∂x′mfx,tgx′,t′x′=x,t′=t.


### 2.1. One-Soliton Solution

In order to construct the soliton solutions for Equation (2), we expend f and g as follows:
(6)f=1+ε2f2+ε4f4+ε6f6+⋯,g=εg1+ε3g3+ε5g5+⋯,where ε is an arbitrary small parameter. Then, substituting Equation (6) into the bilinear equations (4) and collecting the same power coefficients in ε, we get the following equations:
(7)ε1:iDt−δtDx2g1·1=0,(8)ε2:Dx2f2·1+1·f2=2g1g1∗−x,t,(9)ε3:iDt−δtDx2g3·1+g1·f2=0,(10)ε4:Dx21·f4+f4·1+f2·f2+f2∗−x,tDx21·f2+f2·1=2g1g3∗−x,t+g1∗−x,tg3+2f2g1g1∗−x,t,(11)ε5:iDt−δtDx2g5·1+g3·f2+g1·f4=0,(12)ε6:Dx21·f6+f6·1+f2·f4+f4·f2+f2∗−x,tDx21·f4+f4·1+f2f2+f4∗−x,t×Dx21·f2+f2·1=2g5g1∗−x,t+g3g3∗−x,t+g1g5∗−x,t+2f2g1g3∗−x,t+g3g1∗−x,t+2f4g1g1∗−x,t.Now, we construct the one-soliton solution for Equation (2). Assuming g1=eη with η=kx+wt, η∗−x,t=−k∗x+w∗t, Equation (7) yields the dispersive relation with wt=−ik2∫δtdt. Then, substituting the obtained g1 into Equation (8), we get f2=Aeη+η∗−x,t with A=1/k−k∗2. Hence, g1 and f2 can be expressed as
(13)g1=ekx−ik2∫δtdt,f2=1k−k∗2ek−k∗x−ik2−k∗2∫δtdt.Other left equations are satisfied if we setg3=g5=⋯=0 and f4=f6=⋯=0. Hence, we get the one-soliton solution for Equation (2) as
(14)q=ei∫αtdtεekx−ik2∫δtdt/1+ε21/k−k∗2ek−k∗x−ik2−k∗2∫δtdt.Now, settingε=1 and αt=0, we get several special solutions for Equation (2):
(i)
Ifk=−2λ2i and −4λ2i=γ2, where λ2 is a real number, Equation (14) turns into the following period solution which has been reported in [29],(15)q=−4iλ2γ2e4iλ22∫δtdtγ22e2iλ2x+e−2iλ2x.(ii)
Ifk=a+ib (a,b∈R, and ab≠0), Equation (14) becomes(16)q=eax+2ab∫δtdteibx+∫αt−a2−b2δtdt1−1/4b2e4ab∫δtdte2ibx.Obviously, Equation (16) is the one-soliton solution with the singular point x0,t0=lπ/b,t0, where t0 satisfies ∫δtdt=−ln4b2/4ab, and l∈Z.
(iii)
Ifk=ib, b∈R, and b≠0, we get the spatial period solution(17)q=4b216b4−8b2cos2bx+1,where the period M=π/b.To show the characteristics of the one-soliton solution, we illustrate the singular solution (16) and the period solution (17) in Figure 1 (when δt=−1) and Figure 2 (when δt=t2).Figure 1
(a) Soliton solution with singularity with parametersa=0.12 and b=−0.35. (b) Spatial period soliton solution with parameters a=0 and b=0.2, period M=5π.
(a)
Singular solution (δt=−1)(b)
Period solution (δt=−1)Figure 2
(a) Soliton solution with singularity whena=0.1 and b=−0.3. (b) Spatial period soliton solution with parameter a=0 and b=0.2, period M=5π.
(a)
Singular solution (δt=t2)(b)
Period solution (δt=t2)


### 2.2. Two-Soliton Solution

To get the two-soliton solution, we letg1=eη1+eη2 with ηj=kjx+wjt, ηj∗−x,t=−kj∗x+wj∗t, j=1,2. From Equation (7), we have wjt=−ik2∫δtdt, j=1,2. Plugging the obtained g1 into Equation (8) leads to
(18)f2=a1,1∗eη1+η1∗−x,t+a1,2∗eη1+η2∗−x,t+a2,1∗eη2+η1∗−x,t+a2,2∗eη2+η2∗−x,t,where al,m∗=1/kl−km∗2, l,m=1,2.Then, plugging the knowng1 and f2 into Equation (9) and Equation (10), we derive g3 and f4 as
(19)g3=a1,2,1∗eη1+η2+η1∗−x,t+a1,2,2∗eη1+η2+η2∗−x,t,f4=a1,2,1∗,2∗eη1+η2+η1∗−x,t+η2∗−x,t.where
(20)al,m=1kl−km2,al,m∗=1kl−km∗2,l,m=1,2,a1,2,1∗=a1,2a1,1∗a2,1∗,a1,2,2∗=a1,2a1,2∗a2,2∗,a1,2,1∗,2∗=a1,2a1,1∗a1,2∗a2,1∗a2,2∗a1∗,2∗.Other equations are satisfied if we letf6=f8=⋯=0 and g5=g7=⋯=0. Therefore, for ε=1, we get the two-soliton solution as
(21)q=ei∫αtdteη1+eη2+γ1eη1+η2+η1∗−x,t+γ2eη1+η2+η2∗−x,t1+ρ1eη1+η1∗+ρ2eη1+η2∗+ρ3eη2+η1∗+ρ4eη2+η2∗+θeη1+η2+η1∗+η2∗.where γ1=a1,2,1∗, γ2=a1,2,2∗, ρ1=a1,1∗, ρ2=a1,2∗, ρ3=a2,1∗, ρ4=a2,2∗, ρ5=a1,2, and θ=a1,2,1∗,2∗. Specially, if k1=b1i and k2=b2i, the solution Equation (21) becomes a double spatial-period solution which is illustrated in Figure 3 (when δt=t2).Figure 3
(a) Double spatial-period soliton solution with parameters:k1=0.4i, k2=0.6i. (b) Cross-sectional shots of solution (a) at t=0 (red), t=5 (blue), and t=10 (green).
(a)
Two-soliton solution (δt=t2)(b)
Cross-sectional display diagrams (δt=t2)


### 2.3.N-Soliton Solution

TheN-soliton solution for Equation (2) can be shown as follows:
(22)q=ei∫αtdtgf,where
(23)f=∑μ=0,1eexp∑l=12Nμlηl+∑l<m2NμlμmAlm,g=∑μ=0,1oexp∑l=12Nμlηl+∑l<m2NμlμmAlm,g∗−x,t=∑μ=0,1cexp∑l=12Nμlηl+∑l<m2NμlμmAlm,where
(24)ηl=klx+ωlt,ωlt=−ik2∫δtdt,ηl+N=ηl∗−x,t,kl+N=kl∗l=1,2,⋯,N,Alm=ln1kl−km2l=1,2,⋯,N,m=N+1,⋯,2N,Alm=lnkl−km2l,m=1,2,⋯,N,orl,m=N+1,⋯,2N,and for μl=0 or 1 (l=1,2,⋯N), ∑μ=0,1e, ∑μ=0,1o, and ∑μ=0,1c satisfy the following conditions, respectively,
(25)∑l=1Nμl=∑l=1Nμl+N,∑l=1Nμl=1+∑l=1Nμl+N,1+∑l=1Nμl=∑l=1Nμl+N.


## 2.1. One-Soliton Solution

In order to construct the soliton solutions for Equation (2), we expend f and g as follows:
(6)f=1+ε2f2+ε4f4+ε6f6+⋯,g=εg1+ε3g3+ε5g5+⋯,where ε is an arbitrary small parameter. Then, substituting Equation (6) into the bilinear equations (4) and collecting the same power coefficients in ε, we get the following equations:
(7)ε1:iDt−δtDx2g1·1=0,(8)ε2:Dx2f2·1+1·f2=2g1g1∗−x,t,(9)ε3:iDt−δtDx2g3·1+g1·f2=0,(10)ε4:Dx21·f4+f4·1+f2·f2+f2∗−x,tDx21·f2+f2·1=2g1g3∗−x,t+g1∗−x,tg3+2f2g1g1∗−x,t,(11)ε5:iDt−δtDx2g5·1+g3·f2+g1·f4=0,(12)ε6:Dx21·f6+f6·1+f2·f4+f4·f2+f2∗−x,tDx21·f4+f4·1+f2f2+f4∗−x,t×Dx21·f2+f2·1=2g5g1∗−x,t+g3g3∗−x,t+g1g5∗−x,t+2f2g1g3∗−x,t+g3g1∗−x,t+2f4g1g1∗−x,t.Now, we construct the one-soliton solution for Equation (2). Assuming g1=eη with η=kx+wt, η∗−x,t=−k∗x+w∗t, Equation (7) yields the dispersive relation with wt=−ik2∫δtdt. Then, substituting the obtained g1 into Equation (8), we get f2=Aeη+η∗−x,t with A=1/k−k∗2. Hence, g1 and f2 can be expressed as
(13)g1=ekx−ik2∫δtdt,f2=1k−k∗2ek−k∗x−ik2−k∗2∫δtdt.Other left equations are satisfied if we setg3=g5=⋯=0 and f4=f6=⋯=0. Hence, we get the one-soliton solution for Equation (2) as
(14)q=ei∫αtdtεekx−ik2∫δtdt/1+ε21/k−k∗2ek−k∗x−ik2−k∗2∫δtdt.Now, settingε=1 and αt=0, we get several special solutions for Equation (2):
(i)
Ifk=−2λ2i and −4λ2i=γ2, where λ2 is a real number, Equation (14) turns into the following period solution which has been reported in [29],(15)q=−4iλ2γ2e4iλ22∫δtdtγ22e2iλ2x+e−2iλ2x.(ii)
Ifk=a+ib (a,b∈R, and ab≠0), Equation (14) becomes(16)q=eax+2ab∫δtdteibx+∫αt−a2−b2δtdt1−1/4b2e4ab∫δtdte2ibx.Obviously, Equation (16) is the one-soliton solution with the singular point x0,t0=lπ/b,t0, where t0 satisfies ∫δtdt=−ln4b2/4ab, and l∈Z.
(iii)
Ifk=ib, b∈R, and b≠0, we get the spatial period solution(17)q=4b216b4−8b2cos2bx+1,where the period M=π/b.To show the characteristics of the one-soliton solution, we illustrate the singular solution (16) and the period solution (17) in Figure 1 (when δt=−1) and Figure 2 (when δt=t2).Figure 1
(a) Soliton solution with singularity with parametersa=0.12 and b=−0.35. (b) Spatial period soliton solution with parameters a=0 and b=0.2, period M=5π.
(a)
Singular solution (δt=−1)(b)
Period solution (δt=−1)Figure 2
(a) Soliton solution with singularity whena=0.1 and b=−0.3. (b) Spatial period soliton solution with parameter a=0 and b=0.2, period M=5π.
(a)
Singular solution (δt=t2)(b)
Period solution (δt=t2)


## 2.2. Two-Soliton Solution

To get the two-soliton solution, we letg1=eη1+eη2 with ηj=kjx+wjt, ηj∗−x,t=−kj∗x+wj∗t, j=1,2. From Equation (7), we have wjt=−ik2∫δtdt, j=1,2. Plugging the obtained g1 into Equation (8) leads to
(18)f2=a1,1∗eη1+η1∗−x,t+a1,2∗eη1+η2∗−x,t+a2,1∗eη2+η1∗−x,t+a2,2∗eη2+η2∗−x,t,where al,m∗=1/kl−km∗2, l,m=1,2.Then, plugging the knowng1 and f2 into Equation (9) and Equation (10), we derive g3 and f4 as
(19)g3=a1,2,1∗eη1+η2+η1∗−x,t+a1,2,2∗eη1+η2+η2∗−x,t,f4=a1,2,1∗,2∗eη1+η2+η1∗−x,t+η2∗−x,t.where
(20)al,m=1kl−km2,al,m∗=1kl−km∗2,l,m=1,2,a1,2,1∗=a1,2a1,1∗a2,1∗,a1,2,2∗=a1,2a1,2∗a2,2∗,a1,2,1∗,2∗=a1,2a1,1∗a1,2∗a2,1∗a2,2∗a1∗,2∗.Other equations are satisfied if we letf6=f8=⋯=0 and g5=g7=⋯=0. Therefore, for ε=1, we get the two-soliton solution as
(21)q=ei∫αtdteη1+eη2+γ1eη1+η2+η1∗−x,t+γ2eη1+η2+η2∗−x,t1+ρ1eη1+η1∗+ρ2eη1+η2∗+ρ3eη2+η1∗+ρ4eη2+η2∗+θeη1+η2+η1∗+η2∗.where γ1=a1,2,1∗, γ2=a1,2,2∗, ρ1=a1,1∗, ρ2=a1,2∗, ρ3=a2,1∗, ρ4=a2,2∗, ρ5=a1,2, and θ=a1,2,1∗,2∗. Specially, if k1=b1i and k2=b2i, the solution Equation (21) becomes a double spatial-period solution which is illustrated in Figure 3 (when δt=t2).Figure 3
(a) Double spatial-period soliton solution with parameters:k1=0.4i, k2=0.6i. (b) Cross-sectional shots of solution (a) at t=0 (red), t=5 (blue), and t=10 (green).
(a)
Two-soliton solution (δt=t2)(b)
Cross-sectional display diagrams (δt=t2)


## 2.3.N-Soliton Solution

TheN-soliton solution for Equation (2) can be shown as follows:
(22)q=ei∫αtdtgf,where
(23)f=∑μ=0,1eexp∑l=12Nμlηl+∑l<m2NμlμmAlm,g=∑μ=0,1oexp∑l=12Nμlηl+∑l<m2NμlμmAlm,g∗−x,t=∑μ=0,1cexp∑l=12Nμlηl+∑l<m2NμlμmAlm,where
(24)ηl=klx+ωlt,ωlt=−ik2∫δtdt,ηl+N=ηl∗−x,t,kl+N=kl∗l=1,2,⋯,N,Alm=ln1kl−km2l=1,2,⋯,N,m=N+1,⋯,2N,Alm=lnkl−km2l,m=1,2,⋯,N,orl,m=N+1,⋯,2N,and for μl=0 or 1 (l=1,2,⋯N), ∑μ=0,1e, ∑μ=0,1o, and ∑μ=0,1c satisfy the following conditions, respectively,
(25)∑l=1Nμl=∑l=1Nμl+N,∑l=1Nμl=1+∑l=1Nμl+N,1+∑l=1Nμl=∑l=1Nμl+N.


## 3. Asymptotic Analysis on Two-Soliton Solution

The asymptotic behavior of the two-soliton solution is dependent onδt. In this section, under certain assumption that limt⟶+∞∫δtdt=+∞, we investigate the asymptotic behavior of the two-soliton solution. Since δt is an even real function, we have limt⟶−∞∫δtdt=−∞. For simplicity, we denote −ikj2 by ωj, j=1,2, then ηj=kjx+ωj∫δtdt,j=1,2.For fixedη1, we get
(26)η2=k2k1η1+w2−k2k1w1∫δtdt,η2∗−x,t=k2∗k1∗η1∗−x,t+w2∗−k2∗k1∗w1∗∫δtdt,η2+η2∗−x,t=2Rek2k1ξ1+2Rew2−k2k1w1∫δtdt.where w2−k2/k1w1=i−k22+k2k1.Suppose thatRew2−k2/k1w1>0, that is, Imk22−k2k1<0. The two-soliton solution asymptotically tends to be one-soliton solution as follows:
(27)q~12eη1−η1∗−x,t−lnρ1/2+i∫δtdtsechη1+η1∗−x,t+lnρ12,t⟶−∞,(28)q~ρ2ρ52eη1−η1∗−x,t−lnθ/ρ4/2+i∫δtdtsechξ1+ξ1∗−x,t+lnθ/ρ42,t⟶+∞.For fixedη2, suppose that Rew2−k2/k1w1>0, in a similar way, we get the asymptotic expressions of Equation (21):
(29)q~12eη2+η2∗−x,t−lnρ4/2+i∫δtdtsechη2+η2∗−x,t+lnρ42,t⟶−∞,(30)q~ρ3ρ52eη2+η2∗−x,t−lnθ/ρ1/2+i∫δtdtsechη2+η2∗−x,t+lnθ/ρ12,t⟶+∞.We can see that the asymptotic solutions Equation (27) and Equation (28), Equation (29) and Equation (30) have the same form, which implies that the two-soliton collision is elastic. But the two-soliton solution is not a travelling wave. If we suppose that limt⟶+∞∫δtdt=−∞, the same conclusion can be drawn.


## 4. Conclusion and Remarks

In the current paper, we studied an integrable variable coefficient nonlocal nonlinear Schrödinger equation via the Hirota’s bilinear method. We first constructed the bilinear form and then theN-soliton solution. Furthermore, under certain conditions, we analyzed the asymptotic behavior of the two-soliton solution and proved that the collision of the two soliton is elastic. Also, we demonstrated that by choosing different special parameters, the obtained soliton solutions can reduce to spatial period solution or singular solution. We know that sometimes the higher-dimensional nonlinear systems are more suitable to model the physical phenomena such as ultrafast nonlinear optics, so we hope to investigate the 2+1-dimensional variable coefficient nonlocal partial equations in the future.



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