Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential

# Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential

## Abstract

The integrable nonlocal nonlinear Schrödinger (NNLS) equation with the self-induced parity-time-symmetric potential [Phys. Rev. Lett. 110 (2013) 064105] is investigated, which is an integrable extension of the standard NLS equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation -fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical behaviors of these higher-order rational solitons with some small noises on the basis of numerical simulations.

The study of nonlinear waves and soliton theory has become a more and more significant subject in many branches of nonlinear science. The fundamental solitons (e.g., bright and dark solitons) are usually expressed in terms of fractional formals of exponential functions. They more quickly tend to some constants than the localized rational solutions as the variables approach to infinity. The localized rational solutions of nonlinear wave equations admit the special properties, one of which is that they may have the finite critical points (e.g., rational rogue waves) or infinite many critical points (e.g., rational solitons). Recently, a new integrable nonlocal nonlinear Schrödinger (NNLS) equation with the self-induced parity-time-(-) symmetric potential was presented [Phys. Rev. Lett. 110 (2013) 064105]. In this paper, we present the nonlocal version of the generalized perturbation Darboux transformation to study the novel higher-order rational solitons of the NNLS equation, which exhibit the abundant wave structures for the different choices of parameters (e.g., interactions of bright and dark rational solitons). Moreover, we also investigate their dynamical behaviors with some small noises by using numerical simulations. These results would be useful for understanding the corresponding rational soliton phenomena in the many fields of nonlocal nonlinear dynamical systems such as nonlinear optics, Bose-Einstein condensates, ocean, and other relevant fields.

## I Introduction

Recently, rogue waves (RWs, a special type of rational solitons), originally occurring in the deep ocean (1); (2); (3); (4), attracted more and more theoretical and experimental attention in many other fields such as nonlinear optics (5); (6); (7), hydrodynamics (8), Bose-Einstein condensates (9); (10), plasma (11), and even finance (12); (13). RWs are also called freak waves (14), giant waves, great waves, huge waves, ginormous waves, ghost waves, killer waves, etc. The danger of oceanic RWs is due to their sudden appearance and disappearance as ‘waves appear from nowhere and disappear without a trace’ (15); (16). Moreover, the word ‘rogon’ was coined for the RWs if they reappear virtually unaffected in size or shape shortly after their interactions (17), which is similar to ‘soliton’.

The integrable nonlinear Schrödinger (NLS) equation (18); (19); (22); (20); (21)

 iqt−12qxx−σ|q|2q=0,q≡q(x,t) (1)

appears in many fields of nonlinear science such as nonlinear optics, the deep ocean, DNA, and Bose-Einstein condensates, where the subscript denotes the partial derivative with respect to the variables and . It is an importantly nonlinear integrable model admitting explicit first-order RW solution (also called Peregrine’s RW solution) for the focusing case  (23) , which can be regarded as the parameter limit of its breathers (24); (25); (26); (27), and higher-order RW solutions (28); (29); (30); (31); (32); (33); (34). The intensity, , is localized in both space and time and approaches to one not zero as , which differs from its bright soliton (), , in which as . It has been shown that the Peregrine’s RW solution has a good agreement with the numerical and experimental results of Eq. (1(7). But the NLS equation with the defocusing case was shown to possess the singularly rational solutions.

Recently, a new nonlocal nonlinear Schrödinger (NNLS) equation with the self-induced -symmetric potential was presented in the form (35)

 iqt(x,t)−12qxx(x,t)−σq2(x,t)q∗(−x,t)=0, (2)

where the subscript denotes the partial derivative with respect to the variables, the star stands for the complex conjugation, and corresponds to the self-focusing case and defocusing case , respectively. Eq. (2) can be regarded as an integrable extension of Eq. (1) with . Eq. (2) was still verified to be completely integrable, that is, it admits the Lax pair, infinite conversation laws, etc., but Eq. (2) and Eq. (1) are different. The solitons and breather solutions of Eq. (2) have been studied (36). The semi-linear operator related to Eq. (2) is -symmetric for any solutions of Eq. (2), where the complex -symmetric potential is regarded to be self-induced and is a parameter for the linear spectral problem, and the operators and are defined by : and :  (37).

We now simply compare the NLS equation (1) with NNLS equation (2). (i) If , then Eq. (2) becomes the NLS equation (1). That is, if the solutions of NLS equation (1) are the even functions for , then its solutions must be ones of the NNLS equation Eq. (2); (ii) If , then Eq. (2) becomes the NLS equation (1) with . That is, if the solutions of NLS equation (1) are the odd functions for , then its solutions must be ones of the NNLS equation (2) with . For example, the first-order RW solution and higher-order RW solutions of NLS equation (1) also solve Eq. (2) with . The above-mentioned bright solution is not an even or odd function for for the non-zero wave speed parameter , but it is an even function for for the zero speed such that with is also a bright soliton of NNLS equation (2) with (see Ref. (48) for other special solutions). To the best of our knowledge, the higher-order rational solitons (which are the neither even nor odd functions for ) and dynamical behaviors of Eq. (2) were not considered before.

Recently, some power methods have been developed to investigate the higher-order RW solutions of nonlinear wave equations such as the modified and generalized Darboux transformation (DT)  (28); (29); (30); (31); (32); (33); (34); (38); (39); (40), the Hirota’s bilinear method with the -function (41); (42), the similarity (symmetry) transformation (17); (43); (44), and so on. Recently, we presented a generalized perturbation -fold Darboux transformation to find higher-order RW solutions of modified NLS equation (45) and generalized integrable coupled AB system (46), which are both local models.

In this paper, we will extend our previous method used in the local equations (45); (46) to present the nonlocal version of the local -fold Darboux transformation in terms of the Taylor series expansion for the parameter and a limit procedure to directly obtain higher-order rational solitons of the NNLS equation (2). The biggest advantage of our method is to obtain the higher-order rational solitons in terms of determinants without complicated iterations, and the relationships between the multi-rational solitons and the ‘seed’ solutions are established clearly.

## Ii The nonlocal nonlinear Schrödinger equation

### ii.1 Lax pair and gauge transformation

To study the novel localized solutions (e.g., regular rational solitons) of Eq. (2), we need to present its generalized perturbation -fold DT of Eq. (2) in terms of its Lax pair. We firstly consider the Lax pair (or the linear iso-spectral problems) of Eq.  (2) in the form (35)

 φx=Uφ,U=(λq(x,t)−σq∗(−x,t)−λ), (3)
 φt=Vφ,V=(V11V12V21V22), (4)

with

 V11=−iλ2−i2σq(x,t)q∗(−x,t),V12=−iλq(x,t)−i2qx(x,t),V21=iσλq∗(−x,t)+i2σq∗x(−x,t),V22=iλ2+i2σq(x,t)q∗(−x,t)

where the star represents the complex conjugation, (the superscript denotes the vector transpose) is the vector eigenfunction, is the spectral parameter, and . It is easy to show that the compatibility condition , that is, zero curvature equation , of Lax pair (3)-(4) just leads to Eq. (2).

We now consider the gauge (symmetry) transformation (47)

 ˜φ=T(λ)φ,˜φ=(˜ϕ,˜ψ)T (5)

of the Lax pair (3) and (4), where is the unknown Darboux matrix, and the new eigenfunction satisfies the same Lax pair (3) and (4) with the old potential function being replaced by the new one , i.e.,

 ˜φx=˜U˜φ,˜U=U|q(x,t)→˜q(x,t),q(−x,t)→˜q(−x,t), (6)
 ˜φt=˜V˜φ,˜V=V|q(x,t)→˜q(x,t),q(−x,t)→˜q(−x,t). (7)

Therefore, on the basis of Eqs. (6)-(7) we have

 Tx+[T,{U,˜U}]=0,Tt+[T,{V,˜V}]=0, (8)

where we have introduced the generalized bracket . In particular, the generalized bracket reduces to Lie bracket for the case . Therefore we have which yields the same equation (2) with , that is, in the spectral problem (6) and (7) is also a solution of Eq. (2).

### ii.2 Nonlocal Darboux matrix and generalized perturbation (1,n−1)-fold Darboux transformation

To construct the nonlocal -fold DT of Eq. (2), we consider the nonlocal Darboux matrix in Eq. (5) in the form

 T=T(λ)= ⎛⎜ ⎜ ⎜ ⎜ ⎜⎝λN−N−1∑j=0A(j)(x,t)λj−N−1∑j=0B(j)(x,t)(−λ)j−N−1∑j=0B(j)∗(−x,t)(−λ)jσ[λN−N−1∑j=0A(j)∗(−x,t)λj]⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ (9)

with the complex functions and solving the linear algebraic system and the eigenfunctions are the solutions of the linear spectral problem (3) and (4) for the distinct spectral parameter and the same initial solution . This case is not considered here. Notice that the entries in the second rows in matrix are all nonlocal functions, which differ from the local cases (45); (46).

To find the generalized (new) Darboux transformation, we here consider the nonlocal Darboux matrix (9) with only one spectral parameter . Thus the condition leads to the linear algebraic system

 [λN1−N−1∑j=0A(j)(x,t)λj1]ϕ(λ1,x,t) −N−1∑j=0B(j)(x,t)(−λ1)jψ(λ1,x,t)=0, (10) σ[λN∗1−N−1∑j=0A(j)(x,t)λj∗1]ψ∗(λ1,−x,t) −N−1∑j=0B(j)(x,t)(−λ1)j∗ϕ∗(λ1,−x,t)=0, (11)

where is a solution of the Lax pair (3) and (4) with the spectral parameter and an initial solution .

We now know that the two linear algebraic equations (10) and (11) contain the unknown functions and . (i) if , then we can determine only two complex functions and from Eqs. (10) and (11) such that we can not deduce the different functions and compared with ones from the usual DT; (ii) if , then we have free functions for and in system (10) and (11), which seems to be useful to determine the nonlocal Darboux matrix , but it may be difficult to show the invariant conditions (8).

To uniquely determine the functions and , we need to find more (e.g., ) proper constraints for complex functions and except for the given constraints (10) and (11). We now expand the expression

 T(λ1)φ(λ1)∣∣λ1=λ1+ε=+∞∑k=0k∑j=0T(j)(λ1)φ(k−j)(λ1)εk

at , where and with

 T(k)11=CkNλN−k1−N−1∑j=kCijA(j)(x,t)λj−k1, (12) T(k)12=−N−1∑j=kCkjB(j)(x,t)(−λ1)j−k, (13) T(k)21=−N−1∑j=kCkjB(j)∗(−x,t)(−λ1)j−k, (14) T(k)22=σ⎡⎣CkNλN−k1−N−1∑j=kCkjA(j)∗(−x,t)λj−k1⎤⎦ (15)

with   ().

To determine the unknown functions in Eq. (9) with , let which lead to the linear algebraic system with equations

 T(0)(λ1)φ(0)(λ1)=0,T(0)(λ1)φ(1)(λ1)+T(1)(λ1)φ(0)(λ1)=0,⋯⋯,N−1∑j=0T(j)(λ1)φ(N−1−j)(λ1)=0, (16)

in which the first matrix system is just system  (10)-(11). Up to now, we have introduced system (16) containing algebraic equations with unknowns functions and . When the eigenvalue is suitably chosen so that the determinant of coefficients for system (16) is nonzero, hence the Darboux matrix is uniquely determined by system (16).

Theorem 1. Let be column vector solutions of the spectral problem (3)-(4) for the spectral parameters and initial solution of Eq. (2), respectively, then the generalized perturbation -fold Darboux transformation of Eq. (2) is defined by

 ˜qN(x,t)=q0(x,t)+2(−1)N−1B(N), (17)

where with

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣λ1(N−1)ϕ(0)λ1(N−2)ϕ(0)…ϕ(0)λ1(N−1)ψ(0)λ1(N−2)ψ(0)…ψ(0)Δ2,1Δ2,2…ϕ(1)Δ2,N+1Δ2,N+2…ψ(1)……………………ΔN,1ΔN,2…ϕ(N−1)ΔN,N+1ΔN,N+2…ψ(N−1)σλ1∗(N−1)ψ(0)∗(−x,t)σλ1∗(N−2)ψ(0)∗(−x,t)…σψ(0)∗(−x,t)(−λ∗1)(N−1)ϕ(0)∗(−x,t)(−λ∗1)(N−2)ϕ(0)∗(−x,t)…ϕ(0)∗(−x,t)σΔN+2,1σΔN+2,2…σψ(1)∗(−x,t)(−1)(N−1)ΔN+2,N+1(−1)(N−2)ΔN+2,N+2…ϕ(1)(−x,t)……………………σΔ2N,1σΔ2N,2…σψ(N−1)∗(−x,t)(−1)(N−1)Δ(i)2N,N+1(−1)(N−2)Δ2N,N+2…ϕ(N−1)(−x,t)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

where ) are given by the following formulae:

 Δj,s=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩j−1∑k=0CkN−sλ1(N−s−k)ϕ(j−1−k)for1≤j,s≤N,j−1∑k=0Ck2N−sλ1(2N−s−k)ψ(j−1−k)for1≤j≤N,N+1≤s≤2N,σj−(N+1)∑k=0CkN−sλ1∗(N−s−k)ψ(j−N−1−k)∗(−x,t)forN+1≤j≤2N,1≤s≤N,j−(N+1)∑k=0Ck2N−sλ1∗(2N−s−k)ϕ(j−N−1−k)∗(−x,t)forN+1≤j,s≤2N

and is determined using the determinant by replacing its -th column with the column vector , where

 bj=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩j−1∑k=0CkNλ1(N−k)ϕ(j−1−k)for1≤j≤N,σj−(N+1)∑k=0CkNλ1∗(N−k)ψ(j−N−1−k)∗(−x,t)forN+1≤j≤2N.

### ii.3 Generalized perturbation (n,M=N−n)-fold Darboux transformation

We further extend the above-obtained nonlocal -fold DT of Eq. (2), in which we use only one spectral parameter and the th-order derivatives of and with . Nowadays we use () distinct spectral parameters and their corresponding highest order ( derivatives, where these non-negative integers are required to satisfy with , where is the same as one in the Darboux matrix (9).

Similarly, we still consider the Darboux matrix (9) and the eigenfunctions are the solutions of the linear spectral problem (3) and (4) for the distinct spectral parameter and the same initial solution . Thus we have

 T(λi+ε)φi(λi+ε)=+∞∑k=0k∑j=0T(j)(λi)φ(k−j)i(λi)εk,

where . Let with and that we obtain the linear algebraic system with the equations ():

 T(0)(λi)φ(0)i(λi)=0,T(0)(λi)φ(1)i(λi)+T(1)(λi)φ(0)i(λi)=0,⋯⋯,mi∑j=0T(j)(λi)φ(mi−j)i(λi)=0, (18)

, in which we know that some first systems for every index , i.e., with are just some ones in Eqs. (10) and (11), but they are different if there exist at least one index . For the chosen spectral parameters , the derterminant of coefficients of system (18) for the variables are non-zero such that we can determine them by using the Cramer’s rule.

Theorem 2. Let be column vector solutions of Lax pair (3) and (4) for the spectral parameters and initial solution of Eq. (2), respectively, then the generalized perturbation -fold Darboux transformation of Eq. (2) is given by

 ˜qN(x,t)=q0(x,t)+2(−1)N−1B(N), (19)

where ( with and

 =⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣λi(N−1)ϕ(0)iλi(N−2)ϕ(0)i…ϕ(0)iλi(N−1)ψ(0)iλi(N−2)ψ(0)i…ψ(0)iΔ(i)2,1Δ(i)2,2…ϕ(1)iΔ(i)2,N+1Δ(i)2,N+2…ψ(1)i……………………Δ(i)mi+1,1Δ(i)mi+1,2…ϕ(mi)iΔ(i)mi+1,N+1Δ(i)mi+1,N+2…ψ(mi)iσλi∗(N−1)ψ(0)∗i(−x,t)σλi∗(N−2)ψ(0)∗i(−x,t)…σψ(0)∗i(−x,t)(−λ∗i)(N−1)ϕ(0)∗i(−x,t)(−λ∗i)(N−2)ϕ(0)∗i(−x,t)…ϕ(0)∗i(−x,t)σΔ(i)mi+3,1σΔ(i)mi+3,2…σψ(1)∗i(−x,t)(−1)(N−1)Δ(i)mi+3,N+1(−1)(N−2)Δ(i)mi+3,N+2…ϕ(1)i(−x,t)……………………σΔ(i)2(mi+1),1σΔ(i)2(mi+1),2…ψ(mi)∗i(−x,t)(−1)(N−1)Δ(i)2(mi+1),N+1(−1)(N−2)Δ(i)2(mi+1),N+2…ϕ(mi)i(−x,t)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

where ) are given by the following formulae:

 Δ(i)j,s=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩j−1∑k=0CkN−sλi(N−s−k)ϕ(j−1−k)ifor1≤j≤mi+1,1≤s≤N,j−1∑k=0Ck2N−sλi(2N−s−k)ψ(j−1−k)ifor1≤j≤mi+1,N+1≤s≤2N,σj−(N+1)∑k=0CkN−sλi∗(N−s−k)ψ(j−N−1−k)∗i(−x,t)formi+2≤j≤2(mi+1),1≤s≤N,j−(N+1)∑k=0Ck2N−sλi∗(2N−s−k)ϕ(j−N−1−k)∗i(−x,t)formi+2≤j≤2(mi+1),N+1≤s≤2N (20)

and is given by the determinant by replacing its -th column with the column vector , where in which

 b(i)j=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩j−1∑k=0CkNλi(N−k)ϕ(j−1−k)ifor1≤j≤mi+1,σj−(N+1)∑k=0CkNλi∗(N−k)ψ(j−N−1−k)∗i(−x,t)formi+2≤j≤2(mi+1). (21)

Remark.  (i) if the number of the spectral parameters is one, that is, , in which , then Theorem 2 reduces to Theorem 1; (ii) if and , then Theorem 2 reduces to the usual -fold DT (36), that is, the generalized perturbation -fold DT is a new extension of the usual -fold DT. Thus the generalized perturbation -fold DT can be used to obtain not only solitons (which are similar to ones obtained by using the usual -fold DT) but also new solutions including higher-order rational solutions (see Sec.IID).

### ii.4 Higher-order rational soliton structures and dynamical behaviors

We here study the rational solitons of Eq. (2) by using the obtained nonlocal -fold DT in Theorem 1. We start from the general ‘seed’ plane-wave solution of Eq. (2) in the form , where are both real-valued constant. Without loss of generality, we set and substitute the initial plane-wave solution into the Lax pair (3) and (4), we can give its the solution of Eqs. (3) and (4) with the spectral parameter as follows:

 φ=⎡⎢ ⎢ ⎢⎣(C1eA+C2e−A)e−iσρ2t/2(C1√λ21−σρ2−λ1ρeA−C2√λ21−σρ2+λ1ρe−A)eiσρ2t/2⎤⎥ ⎥ ⎥⎦,

where with , ,