On a Discrete Version of the Korteweg-De Vries Equation

DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 22–29

ON A DISCRETE VERSION OF THE KORTEWEG-DE VRIES EQUATION

M. Agrotis University of Cyprus Department of Mathematics and Statistics P.O.Box 20537 Nicosia 1678 Cyprus S. Lafortune Department of Mathematics College of Charleston 66 George Street Charleston, SC 29424-0001, USA P.G. Kevrekidis University of Massachusetts Lederle Graduate Research Tower Department of Mathematics and Statistics Amherst, MA 01003, USA

Abstract. In this short communication, we consider a discrete example of how to perform multiple scale expansions and by starting from the discrete nonlinear Schr¨odingerequation (DNLS) as well as the Ablowitz-Ladik nonlinear Schr¨odinger equation (AL-NLS), we obtain the corresponding discrete versions of a Korteweg-de Vries (KdV) equation. We analyze in particular the equation obtained from the AL- NLS and discuss its integrability, as well as its connections with previously studied discrete versions of the KdV equation.

1. Introduction. In a seminal paper [1], Zakharov and Kuznetsov demonstrated how, starting from an integrable system and performing multiple scale expansions, one obtains other integrable systems. For example, they showed how from the KdV equation, one can get the NLS equation (scalar, as well as vector) and vice-versa (for small amplitude waves on top of a finite amplitude background), or how from the Kadomtsev-Petviashvili (KP) equation one can get the Davey-Stewardson (DS) equation or the equation for the interaction of N waves. All the above are celebrated examples of integrable systems that at one time made famous the inverse scattering transform [2, 3, 4]. It is well-known that many of the systems of interest to applications in nonlinear optics [5] or DNA [6], in Josephson junctions [7] or in the study of dislocations [8], are inherently discrete. However, discreteness in breaking one of the important symmetries of the continuum model, namely translational invariance [9], typically breaks integrability as well. As a result, there are relatively few known examples of integrable differential-difference equations. The most famous among them are the Toda lattice [10], the Lotka-Volterra model [11], and the so-called Ablowitz-Ladik

2000 Mathematics Subject Classification. Primary: 34K99, 35Q53, 35Q55. Key words and phrases. Nonlinear SchrödingerEquation, Korteweg-de Vries equation, Multiple Scale Expansions, Singularity Confinement. The third author is supported in part by NSF-DMS 0204585, an NSF CAREER award and the Eppley Foundation for Research.

22 ON A DISCRETE VERSION OF THE KORTEWEG-DE VRIES EQUATION 23 discretization of the NLS equation (AL-NLS) [12]. For a recent review exposition of especially discrete integrable models and the notion of integrability in discrete systems, see e.g. [13] and references therein. Naturally, then, the question arises as to, if one performs multiple scale expansions for one of these models, whether or not one will get a new discrete integrable system. In this brief report, we address this question in a case study. We start from the DNLS and from the AL-NLS, and obtain two discrete forms of the KdV equation1. We focus, in particular, on the latter equation and study analytical as well as numerical diagnostics to examine its integrability. We conclude that the derived equation does not appear to fulfill relevant integrability criteria (such as the singularity confinement property [15]), even though its behavior is numerically verified to be near integrable. We should stress here that the purpose of this exercise is not one of mere mathematical curiosity. It is, firstly, important to establish (even at the level of a case example) whether or not multi-scale expansions with a discrete integrable system as a starting point still result in integrable systems. It is also of interest to have a discrete integrable (or near integrable) version of the KdV equation, since as has been shown by Ablowitz and co-workers for NLS type equations [16], a numerical discretization scheme that preserves the integrability feature (and/or a number of invariants) will be much closer to the actual continuum system for the purposes of numerical computations, than a typical discretization thereof. Therefore such a scheme will be useful for the numerical study of water wave problems, where the continuum KdV equation is the relevant model of interest [3, 17]. Finally, as is well-known, the behavior of small amplitude waves for problems modeled by the defocusing NLS is described by a form of the KdV equation. It would then be of physical interest to obtain the equation corresponding to the discrete case (which is relevant not only to arrays of optical waveguides but also to the study of Bose- Einstein condensates, in the presence of the so-called optical lattice [18]) both for the ubiquitous DNLS model, as well as for the integrable AL-NLS model.

2. Discretizations of the KdV Equation. Our starting point will be the DNLS equation 2 iu˙ n = −∆2un + |un| un, (1) where un is the complex ﬁeld at site n (and time t). The overdot denotes temporal 2 derivative while the discrete Laplacian is ∆2un = (un+1 + un−1 − 2un)/h . From here on we will rescale the lattice spacing h to 1, but it will be evident how to restore it in the various cases (expressions whose continuum limits correspond to derivatives should be rescaled by 1/ha, where a is the order of the corresponding derivative). 1/2 Using the polar coordinate expansion un = ρn exp(iφn), as well as the approximations exp(iφn±1) ≈ exp(iφn)(1 + i(φn±1 − φn)), we obtain ρ−1/2 n ρ˙ = −ρ1/2 (φ − φ ) − ρ1/2 (φ − φ ) (2) 2 n n+1 n+1 n n−1 n−1 n 1/2 ˙ 1/2 1/2 1/2 3/2 −ρn φn = −(ρn+1 + ρn−1 − 2ρn ) + ρn . (3)

1Naturally, if different discretizations of NLS were used, involving different numbers of neighbors and/or differencing schemes, the results would be different; here we only focus on the typical discretization schemes commonly studied in the literature [14]. 24 M. AGROTIS, S. LAFORTUNE AND P.G. KEVREKIDIS

Equations (2)-(3) are the starting point for our multiscale perturbation theory analysis. In particular, we use the expansion near the uniform steady state of the defocusing problem un = exp(−it), of the form

2 ρn = 1 + ²ρ1,n + ² ρ2,n + ... (4) 1/2 3/2 φn = −t + ² φ1,n + ² φ2,n + ... (5) t0 = ²3/2t; h0 = ²1/2h ≡ ²1/2 (6)

Notice that this expansion is a discrete analog of a multiple-scale expansion used in the corresponding continuum problem. However, there are some interesting diﬀer- ences: 1. In the discrete case there are no spatial derivatives. However, to preserve the scaling, for terms corresponding in the continuum case to spatial derivatives, a mul- tiplicative scaling factor of ²1/2 has been used for each derivative (hence the term corresponding to the second derivative in each of Eqs. (2)-(3) has been rescaled by ²). 2. Continuum temporal derivatives should be transformed (through the multiscale 3/2 1/2 expansion applied in the travelling wave frame) as ∂t = ² ∂t0 − c² ∂x0 . However, for the discrete model, the derivative with respect to x0 is replaced by a centered diﬀerence approximation (i.e., (un+1 − un−1)/2).

3/2 5/2 2 Comparing the O(² ) and O(² ) equations with the ones of O(²) and O(² ),√ we obtain the solvability condition c2 = 2, as well as the equations (choosing c = 2)

1 ρ1,n = √ (φ1,n+1 − φ1,n−1) (7) 2 1 h i φ˙ = (φ − φ )2 + (φ − φ )2 1,n 2 n+1 n n−1 n 1 = √ (φ1,n+2 − 2φ1,n+1 + 2φ1,n−1 − φ1,n−2) . (8) 2 2 √ Eq. (7) is a discrete analog of the equation ρ1 = 2φ1,x0 that is obtained in [1], while Eq. (8) has as its√ continuum limit the partial differential equation (PDE) 2 0 φ1,t + φ1,x0 = φ1,x0x0x0 / 2, whose differentiation with respect to x (alongside a time and field trivial rescaling) shows that φ1,x0 satisfies the KdV equation. This discrete version of KdV, as we will argue below is not expected to be integrable. If, on the other hand, we start from the integrable AL-NLS, of the form u + u iu˙ = −∆ u + |u |2 n+1 n−1 , (9) n 2 n n 2 and perform the analogous multiple scale calculations, we obtain the discrete analog of ρ1x = φ1xx , as well as the dynamical equation for ρ1,n. The resulting evolution equation (up to O(h2) corrections and upon suitable straightforward time and field rescalings) reads:

1 ρ˙ = (ρ − 2ρ + 2ρ − ρ ) 1,n 2 1,n+2 1,n+1 1,n−1 1,n−2 3 + (ρ (ρ + ρ ) − ρ (ρ + ρ )) . (10) 4 1,n+1 1,n 1,n+2 1,n−1 1,n 1,n−2 ON A DISCRETE VERSION OF THE KORTEWEG-DE VRIES EQUATION 25

We are thus led to consider the discrete version of the KdV equation 1 u˙ = (u − 2u + 2u − u ) n 2h3 n+2 n+1 n−1 n−2 3 + [u (u + u ) − u (u + u )] . (11) 4h n+1 n n+2 n−1 n n−2 3. Conservation Laws. Notice that the choice of the rescalings in Eq. (10) has been such that the continuum limit of Eq. (11) is the regular form of the continuum KdV equation (ut = uxxx + 6uux). Let us ﬁrst consider the static problem of Eq. (11). Notice that the corresponding static problem can be rewritten as

H(un, un+1, un+2) = H(un−2, un−1, un), (12) where H is deﬁned as 1 3β H(α, β, γ) = (α + γ − 2β) + (α + γ) (13) 2h3 4h The “conservation” of H between any three (consecutive) sites of the lattice leads us to the conclusion that (for the static problem) H = H∞ = 0. Moreover, the time dependent version of the model can be re-written as

T˙n = Hn+1 − Hn−1, (14) where Tn = un. So this model equation can be written in the form of conservation laws [19], where the density is equal to the value of the field, and the flux is H. It is worth noting here, in support of our non-integrability claim for the dKdV arising from the multiscale expansion calculation starting from DNLS, that in the former case such a conservation laws’ formulation is not possible. It should also be remarked that a dynamically conserved quantityP straighforwardly stemming from the formulation of Eq. (14) is the mass M = n un. Perhaps more interestingly, in the exception of a (somewhat unusual as it in- volves only nearest neighbors) discretization of the modified KdV equation that appeared in [12] and of the Lotka-Volterra model, the only other well-known integrable discrete KdV that we were able to trace was due to Ohta and Hirota [20]; the latter closely resembles Eq. (11). However, the left hand side was of the form 2 2 2 u˙ n/(1 + h un), while a contribution of un+1 − un−1 to the uux term appears on the right hand side of the equation of [20]. Hence in the latter case also the conservation laws’ formulation of Eq. (14) is present (as is the conservation of the total mass), 2 2 but Tn has the logarithmic form Tn = (1/h ) log(1 + h un). 4. Numerical Results. To support the analytical findings above about the near integrability of Eq. (11) and its close proximity to the behavior of the equation of [20], we performed a series of numerical experiments. The computations focused on identifying discrete travelling solutions for the equation (11). In analogy with the continuum equation, solutions travelling with speed c will be in the latter case of the form µ√ ¶ c c u(X) = sech2 (X − X ) (15) 2 2 0 in the travelling wave frame variable X. To evaluate the discrete solution and show that its form is also the one of hyperbolic secant squared, we calculated the solution as a steady state of the discretization of the continuum equation (already written) in the travelling wave frame. We also performed linear stability analysis which demonstrated the stability of the resulting solutions. A typical solution example 26 M. AGROTIS, S. LAFORTUNE AND P.G. KEVREKIDIS for h = 0.5 is shown in the top left panel of Fig. 1 for c = 1.2. The solution was obtained for various values of c and assuming a functional form of the solution (in 2 −1 the travelling frame) un = A sech (ρ (n − n0)h) we fitted the dependence of the amplitude A and the width ρ as a function of c in the top right and the bottom left panels of Fig. 1. We find that A = P1c + Q1, with P1 = 0.53 and Q1 = −0.02 −2 (compared to the continuum predictions of P1 = 0.5 and Q1 = 0), while ρ = P2c+ Q2, with numerical fits of P1 = 0.234, Q1 = 0.013 compared with the continuum ones of P2 = 0.25 and Q2 = 0. These results and the very good fit of the solution to a square hyperbolic secant function indicate that the solution maintains a close form to the continuum problem solution. The minor discrepancies can be well-justified by the approximate nature of the numerical computations. Finally, we have performed numerical experiments of collisions of two solitary waves. An example of this type is shown in the bottom right panel of Fig. 1. It is clear from the graph that the two solitary waves pass through each other and continue their trajectory unaltered in shape or speed by the collision process. Furthermore no visible signs of extended wave radiation are generated in the process. In view of the “working definition” for solitons [2, 21] (and its implications on integrability) according to which, the solitary waves are really solitons if their collisions are elastic, our numerical findings provide additional support to the claim for the near integrable nature of the model of Eq. (11).

5. Singularity Confinement Method. To determine whether the model is completely integrable, we use a recently developed method for examining the complete integrability of differential-difference equations, namely the singularity confinement criterion of [15]. In order to illustrate the method of singularity confinement, we first consider the equation of Ohta and Hirota [20] described earlier. The confinement deals with the spontaneous appearance of a singularity at some step n. First, we re-interpret the equation of [20] as an iteration for un

3 2 2h u˙ n un+1(2 − h (un + un+1)) un+2 = 2 2 + 2 (h un+1 + 1)(h un + 1) h un+1 + 1 2 2 un−1(2 − h (un + un−1)) + un−2(1 + h un−1) − 2 . (16) h un+1 + 1

2 Then we study what happens if, at some iteration n, un+1(t) takes the value −1/h for some time t = t0 and the other previous iterates (un−2, un−1 and un) are non- divergent and free. We only consider the simplest possible behaviour around the singularity when the first nonzero term in the Taylor expansion of un+1 around the 2 singularity is of first degree un+1 = −1/h + α(t)τ where α(t0) 6= 0 and τ = t − t0. Then, the two following iterates un+2 and un+3 as calculated with (16) are diverging 1 in the limit t → t0 and un+4 = − h2 + O(τ). It is then possible to verify that the three following iterates have finite values when t → t0 and thus the singularity is confined. Just as in the previous example, equation (11) can be written as an iteration for un whose right-hand-side has also a singularity for un+1(t) equal to 2 2 −1/h . By considering the same behaviour un+1 = −1/h +α(t)τ one finds that the −1 two following iterates un+2 and un+4 are diverging in the limit t → t0 (as τ and τ −2 respectively). Examination of further iterations demonstrates that subsequent “even iterates” (un+2k, k = 3, 4, 5,... ) invariably diverge. These results imply that the model of Eq. (11) is not completely integrable. ON A DISCRETE VERSION OF THE KORTEWEG-DE VRIES EQUATION 27

0.6 1.1 0.5 1

| 0.4 n

A 0.9 |u 0.3 0.8 0.2

0.1 0.7

0 0.6 180 190 200 210 220 1.2 1.4 1.6 1.8 2 2.2 n c

0.55 0 200

0.5 −0.2 250 0.45 −0.4 300 −2

ρ 0.4 −0.6

350 0.35 −0.8

0.3 400 −1

1.2 1.4 1.6 1.8 2 2.2 50 100 150 200 c

Figure 1. The top left panel shows the (numerically obtained) solution of Eq. (11) for h = 0.5 and c = 1.2, as a function of the lattice site index n. The top right panel shows the amplitude of the numerical solution as a function of the speed (for the best ﬁt and relevant discussion see text); similarly the bottom left panel shows the square inverse width (ρ−2) of the solution as a function of c, showing once again a theoretically expected linear dependence. Fi- nally, the bottom right panel shows the space-time evolution of the ﬁeld contour plot for the collision of two solitons with speeds c = 1.2 and c = 2.2. As time evolves, the taller and faster wave overtakes the slower and smaller amplitude one, through an apparently quasi- elastic collision and the solitons continue their trajectories with the same speed, without emitting any (visible) “radiation”.

6. Summary and Conclusions. In conclusion, in this work we have considered a case example of performing multiple scale expansion calculations in discrete models. We have demonstrated the rescalings and approximations that are necessary for such considerations. Starting from the (defocusing) DNLS equation, we have obtained the corresponding non-integrable dKdV equation, while starting from the AL-NLS, we have obtained a near-integrable dKdV equation whose connections to the integrable discretization of [20], and properties/conservation laws we have examined on the basis of analytical as well as numerical evidence. It is interesting to note that in this case example starting from an integrable discrete model, through multiscale expansions we do not obtain a new integrable discrete model. While this result is not too surprising, it would be interesting to examine whether this method can yield any novel discrete integrable systems. In that vein, a natural extension of the present work would be to consider the possibility of obtaining previously unexplored new discrete models, using as a starting point the Toda lattice, the Lotka-Volterra model, or integrable discretizations of the sine-Gordon [22] equation. It would also be very interesting to understand where the presented discrete version of the multiscale expansion “loses” the integrability of the original equation and whether alternative such expansions can be developed to maintain the integrable 28 M. AGROTIS, S. LAFORTUNE AND P.G. KEVREKIDIS structure. Finally, another interesting direction would be to also examine “doubly discrete” schemes, i.e., schemes discrete both in space and in time (or coupled map lattices, as they are otherwise known). Integrable such schemes also exist (see e.g. [16] for an example in the sine-Gordon equation case) and it would be particularly interesting to examine what types of modiﬁed schemes can be derived in this con- text, upon application of the discrete multiple scale expansions developed herein. Such studies are currently in progress in will be reported in future publications.

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