Fractional Integrable and Related Discrete Nonlinear Schr odinger Equations

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Fractional Integrable and Related Discrete Nonlinear Schr¨odinger Equations

Mark J. Ablowitz,1Joel B. Been,2, 3, ∗and Lincoln D. Carr2, 3, 4

1Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, U.S.A.

2Department of Applied Mathematics and Statistics,

Colorado School of Mines, Golden, Colorado 80401, U.S.A.

3Department of Physics, Colorado School of Mines, Golden, Colorado 80401, U.S.A.

4Quantum Engineering Program, Colorado School of Mines, Golden, Colorado 80401, U.S.A.

Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schr¨odinger

equations are key to the intersection of nonlinear dynamics and fractional calculus. In this

manuscript, the ﬁrst discrete/diﬀerential diﬀerence equation of this type is found, the fractional inte-

grable discrete nonlinear Schr¨odinger equation. This equation is linearized; special soliton solutions

are found whose peak velocities exhibit more complicated behavior than other previously obtained

fractional integrable equations. This equation is compared with the closely related fractional aver-

aged discrete nonlinear Schr¨odinger equation which has simpler structure than the integrable case.

For positive fractional parameter and small amplitude waves, the soliton solutions of the integrable

and averaged equations have similar behavior.

Keywords: Integrable equations; Fractional calculus; Discrete nonlinear Schr¨odinger equation; Fourier split

step

INTRODUCTION

Integrable systems play a central role in nonlinear dy-

namics because they provide exactly solvable models for

important physical systems. Notable examples of inte-

grable equations are the Korteweg-deVries (KdV), ap-

plicable to shallow water waves, plasma physics, and

lattice dynamics among others [1–3], and the nonlinear

Schr¨odinger (NLS) equation, which ﬁnds applications in

nonlinear optics, Bose-Einstein condensates, spin waves

in ferromagnetic ﬁlms, plasma physics, water waves, etc.

[2–5]. These integrable nonlinear evolution equations

have an inﬁnite number of conservation laws and soli-

ton solutions [6]. Solitons, the fundamental solutions

of such equations, are stable, localized nonlinear waves

which propagate without dispersing and interact elas-

tically with other solitons. Nonlinear integrable evolu-

tion equations have these surprising properties because

of their deep mathematical structure described by the

inverse scattering transform (IST).

IST is a method of solving nonlinear equations which

generalizes Fourier transforms. It solves these equations

in three steps: mapping the initial condition into scat-

tering space, evolving the intial data in scattering space

in time, and mapping the evolved scattering data back

to physical space; i.e., inverse scattering. This pro-

cess gives the solution to nonlinear equations solvable

by IST in terms of linear integral equations; such non-

linear equations are called integrable. Recently, we used

the mathematical structure of IST associated with for

the Korteweg-deVries (KdV) and nonlinear Schr¨odinger

(NLS) equations to develop a method of ﬁnding and solv-

ing the fractional KdV (fKdV) and fractional NLS (fNLS)

∗Corresponding author: joelbeen@mines.edu

equations [7]. We also showed that this method could

be applied to ﬁnd fractional extensions of the modiﬁed

KdV, sine-Gordon, and sinh-Gordon equations [8]. These

equations represent the ﬁrst known fractional integrable

nonlinear evolution equations with smooth (physical) so-

lutions and deeply connect the ﬁelds of nonlinear dynam-

ics and fractional calculus.

Fractional calculus is a mathematical structure origi-

nally designed to deﬁne non-integer derivatives and inte-

grals. It has sense become an eﬀective way of modeling

many physical processes that exist in multi-scale media

[9, 10] or exhibit non-Gaussian statistics or power law

behavior [11–13]. A particularly important example is

anomalous diﬀusion, where the mean squared displace-

ment is proportional to tα,α > 0 [11, 14–16]. Trans-

port that follows this rule has been observed in biology

[17–20], amorphous materials [21–23], porous media [24–

27], climate science [28], and attenuation in materials [29]

amongst others. As we have shown, the merger of frac-

tional and nonlinear characteristics in integrable equa-

tions such as fKdV and fNLS predict anomalous disper-

sion, where the velocity and amplitude of solitonic solu-

tions are related by a power law [7].

In this article, we demonstrate how the method in-

troduced in Ref. [7] can be applied to discrete (or

diﬀerential-diﬀerence) systems to deﬁne integrable dis-

crete fractional nonlinear evolution equations by present-

ing a fractional generalization of the integrable discrete

nonlinear Schr¨odinger (IDNLS) equation. We do this by

demonstrating the three key mathematical ingredients of

our method — IST, power law dispersion relations, and

completeness relations — for the Ablowitz-Ladik (AL)

discrete scattering problem. In the linear limit, the frac-

tional IDNLS (fIDNLS) equation is a discretization of the

fractional Schr¨odinger equation which was derived with

Feynman path integrals over L´evy ﬂights [30, 31].

arXiv:2210.01229v1 [nlin.SI] 3 Oct 2022

The KdV equation was the ﬁrst equation shown to be

solvable by IST in Ref. [32]; it was soon followed by

the NLS equation in Ref. [33]. These two equations were

then found to be contained in a general class of equations

solvable by IST when associated to the Ablowitz-Kaup-

Newell-Segur (AKNS) system [3, 34]. Shortly thereafter

IST was used to solve families of discrete (or diﬀerential

diﬀerence) problems like the self-dual network [35]. In

particular, it was discovered that the AKNS system could

be discretized while maintaining integrability, leading to

the AL scattering problem which was used to solve a fam-

ily of discrete nonlinear evolution equations [36]. This

family contained important discrete evolution equations

— continuous in time but discretized in space — such as

integrable discretizations of the nonlinear Schr¨odinger,

KdV, modiﬁed KdV, and sine Gordon equations. Fur-

ther, this family of equations was shown to have soliton

solutions and an inﬁnite number of conservation laws [36].

We derive the fIDNLS equation from the AL scatter-

ing problem using three key components: linear disper-

sion relations, completeness relations, and IST. IST is

used to linearized the equation and obtain special soliton

solutions.

We also show how the characteristics of the fractional

IDNLS (fIDNLS) equation reach beyond integrability by

comparing the one-soliton solution of the fIDNLS equa-

tion to the solitary wave solution of the fractional aver-

aged discrete nonlinear Schr¨odinger (fADNLS) equation.

This equation is a diﬀerent fractional generalization of

the IDNLS equation in which the linear second order dif-

ference is replaced by the discrete fractional Laplacian

[37–40]. The fADNLS equation can be understood as a

discretization of a fractional NLS equation involving the

Riesz derivative which has been extensively studied in,

e.g., [41–45]; it is also is also closely related to the (likely)

non-integrable fractional DNLS equation, recently stud-

ied in [37, 46]. Though the fADNLS equation is likely not

integrable to our knowledge (apart from the limiting case

when fADNLS reduces to IDNLS), the similarity between

the two equations suggests that some of the physical pre-

dictions of fractional integrable equations are shared by

equations which are simpler to realize computationally.

THE DISCRETE FRACTIONAL LINEAR

SCHR ¨

ODINGER EQUATION

Consider the family of discrete linear evolution equa-

tions

∂tqn+γ(−∆n)qn= 0 (1)

for the function qn(t) which depends on the discrete vari-

able n∈Zand the continuous variable t∈R. Here, γ

is a suﬃciently regular function of the discrete laplacian,

−∆n, deﬁned by

(−∆n)qn(t) = 1

h2−qn+1(t)+2qn(t)−qn−1(t)(2)

where his the distance between lattice sites. Using the

Z-transform, which is equivalent to the discrete Fourier

transform, the solution to Eq. (1) can be explicitly writ-

ten as

qn(t) = 1

2πZπ/h

−π/h

dkˆq(k, 0)eiknh−γ(4 sin2(kh/2)/h2)t(3)

where ˆq(k, 0) = hP∞

n=−∞ qn(0)e−iknh is the Z-transform

of qn(t) at t= 0 and 4 sin2(kh/2)/h2is the Fourier sym-

bol of −∆n. Note that the Z-transform is often written

in terms of zwith the substitution z=e−ikh where in-

tegration in kbecomes integration with respect to zon

the unit circle. If we choose γto be power law, then

Eq. (1) becomes a fractional discrete equation in terms

of the discrete fractional laplacian. For example, if we

put γ(−∆n) = −i(−∆n)1+,||<1, then we obtain the

linear fractional discrete Schr¨odinger equation

i∂tqn+ (−∆n)1+qn= 0.(4)

Here, (−∆n)1+is the discrete fractional laplacian of or-

der 1 + which is deﬁned in terms of its Fourier sym-

bol [4 sin2(hk/2)/h2]1+and the Z-transform/discrete

Fourier transform as

(−∆n)1+qn=1

2πZπ/h

−π/h

dkˆq(k)eiknh[4 sin2(kh/2)/h2]1+.

(5)

Notice that the kintegral above can be evaluated to ex-

press the discrete fractional laplacian as a summation

over mof qmmultiplied by a weight vector. The solution

to Eq. (4) can still be written in the form Eq. (3) with

γ(4 sin2(kh/2)/h2) = −i[4 sin2(kh/2)/h2]1+

and, because 4 sin2(kh/2)/h2is real and positive, the

solution to equation (1) with this choice of γis well

posed. In deﬁning and solving the linear fractional dis-

crete Schr¨odinger equation, we used a power law disper-

sion relation, ingredient 1 of our method, and we deﬁned

the fractional operator using completeness of the discrete

Fourier transform/Z-transform, ingredient 2. Then we

solve the equation by the inverse discrete Fourier trans-

form, the analog of ingredient 3.

THE FRACTIONAL INTEGRABLE DISCRETE

SCHR ¨

ODINGER EQUATION

To develop the fIDNLS equation, the integrable non-

linear analog of Eq. (4), and solve it, we apply the three

key ingredients of our method, starting with writing the

equation in terms of a linear dispersion relation. Note

that h= 1 is taken in this section without loss of gener-

ality; to recover the scaling factor for h6= 1, replace qn

by hqnand rnby hrn.

As in the linear case, Eq. (1), we have a family of

nonlinear evolution equations for the solutions qn(t) and

rn(t) [47], see also [48],

σ3

dun

dt +γ(Λ+)un= 0,un= (qn,−rn)T(6)

where Trepresents transpose, σ3= diag(1,−1), and Λ+

Λ+xn=hnE+

0E−

n x(1)

x(2)

k!(7)

+qnP+

n−1rk−1qnP+

n−2qk+1

−rnP+

n−1rk−1−rnP+

n−2qk+1 x(1)

x(2)

k!(8)

+hn qn+1 P+

n+1

hkqn+1 P+

n+1

−rn−1P+

hk−rn−1P+

hk! x(1)

x(2)

k!(9)

where hn= 1 −rnqn,P+

n=P∞

k=n, and E±

nx(q)

k=x(q)

n±1

with q= 1,2. The inverse of this operator is

Λ−1

+xn=hnE−

0E+

n x(1)

x(2)

k!(10)

+−qnP+

nrk+1 −qnP+

n+1 qk−1

rnP+

n−1rk+1 rnP+

nqk−1 x(1)

x(2)

k!(11)

+hn −qn−1P+

hk−qn−1P+

rn+1 P+

n+1

hk−rn+1 P+

n+1

hk! x(1)

x(2)

k!.(12)

Here, γis a suﬃciently regular function of the operator

Λ+and is connected with the linearized dispersion re-

lation. Specifying this dispersion relation, or γdirectly,

picks out particular equations from this family. For ex-

ample, if we take

γ(Λ+) = −i(2 −Λ+−Λ−1

and let rn=∓q∗

n, then we obtain the IDNLS equation

i∂tqn+ ∆nqn± |qn|2(qn+1 +qn−1)=0.(13)

We can relate γto the dispersion relation of the lineariza-

tion of (6) by considering the linear limit qn→0. In this

limit, we have

Λ+→E+

0E−

n≡Dn,(14)

so the linearization of the nonlinear evolution equation is

σ3

dun

dt +γ(Dn)un= 0.(15)

Because Dnis a diagonal matrix, we have

γ(Dn) = γ(E+

n) 0

0γ(E−

n).(16)

Taking the ﬁrst component of (15) with

qn=z2ne−iω(z)t

gives

γ(z2) = iω(z).(17)

Therefore, by specifying the linear limit of the nonlinear

evolution equation, we obtain the nonlinear equation it-

self. To deﬁne the fIDNLS equation, we choose the linear

limit to be the discrete linear fractional Schr¨odinger equa-

tion in (4), which gives the dispersion relation ω(z) =

−(2 −z2−z−2)1+and, hence, γ(z2) = −i(2 −z2−

z−2)1+. So, the fIDNLS equation is

i∂tun+ (2 −Λ+−Λ−1

+)1+un(t) = 0.(18)

In fact, by choosing γ(z2) = −i(2 −z2−z−2)m+, for

integer m, we generate a hierarchy of fractional equations

i∂tun+ (2 −Λ+−Λ−1

+)m+un(t)=0.(19)

It can be shown that the limit of (18) as →0 is the

IDNLS equation (13). Notice that to deﬁne the fIDNLS

equation, we used a power law dispersion relation, ingre-

dient 1 of the method. However, this dispersion relation

leads to the operator (2 −Λ+−Λ−1

+)1+the meaning of

which is currently unclear. To deﬁne this operator, we

will need to use the 2nd ingredient: appropriate com-

pleteness relations. The third ingredient will be making

use of IST to ﬁnd solutions of the fIDNLS equation.

COMPLETENESS OF SQUARED

EIGENFUNCTIONS AND FRACTIONAL

OPERATORS

In this section we deﬁne the fIDNLS equation in (18)

and, in fact, any equation of the form (1) that is well-

posed in physical space. We do this using the observation

that γ(Λ+) is a multiplication operator when acting on

the eigenfunctions of Λ+and the fact that the eigen-

functions of Λ+are complete. This result is known as

completeness of squared eigenfunctions, and is the sec-

ond ingredient in our method. The resulting represen-

tation of γ(Λ+) will be similar to that of the discrete

fractional laplacian in (5). The eigenfunctions of Λ+are

Ψn(z) and Ψn(z) each with eigenvalue z2(note that time

tis suppressed throughout this section). Therefore, the

operation of γ(Λ+) on these eigenfunctions is given by

γ(Λ+)Ψn=γ(z2)Ψn, γ(Λ+)Ψn=γ(z2)Ψn.(20)

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摘要：

FractionalIntegrableandRelatedDiscreteNonlinearSchrodingerEquationsMarkJ.Ablowitz,1JoelB.Been,2,3,andLincolnD.Carr2,3,41DepartmentofAppliedMathematics,UniversityofColorado,Boulder,Colorado80309,U.S.A.2DepartmentofAppliedMathematicsandStatistics,ColoradoSchoolofMines,Golden,Colorado80401,U.S.A.3Dep...

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Fractional Integrable and Related Discrete Nonlinear Schr odinger Equations

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