Recent work by Ablowitz et al. [1,2] has pointed outthe problem of numerically induced spatial and tem-poral chaos in numerical solutions of nonlinear waveequations such as the sine-Gordon equation [1], thenonlinear Schrödinger equation [3–5] and the modi-fied Korteweg–de Vries (KdV) equation [6]. A seri-ous implication is that at least some previous reportsof chaos, in fact, are numerical artifacts. This problemmight affect the mathematical modeling of many realproblems in physics, chemistry, biology and engineer-ing. Mathematically, for an integrable system, suchnumerical instability is associated with singularities inso-called phase space action–angle variables, whichproduce homoclinic orbits in the related phase spacegeometry [1–5,7–9]. The numerical solution near the
homoclinic orbits can be exponentially unstable [1]due to the frequent visiting of two solutions on “op-posite sides” of the homoclinic orbit. This visiting orhomoclinic orbit crossing can result from extremelysmall perturbations in the numerical parameters and/orfrom numerical errors in the calculated solution.
Conceptually, such “visiting” differs very muchfrom the Gibbs’ oscillations occurring in spectralmethod approximations of a step function or in thenumerical solution of Burgers’ equation with a highReynolds number. The Gibbs’ oscillations are usu-ally very regular but the homoclinic orbit crossing is,in general, very irregular. Moreover, the homoclinicorbit crossing is induced by the presence of phasespace singularities, whereas Gibbs’ oscillations arecaused by the sharp spatial changes inreal spacesolutions over a small region such as a boundarylayer. There, however, is a universal feature of theseinstabilities from the point of view of numerical inte-gration. Essentially, instabilities typically occur when
the numerical algorithm with the given space andtime meshes is not adequate for describing the ac-tual solution. In many cases, the rate of convergenceof a given method simply cannot match the rate ofdivergence of a solution near the singularity.
Recent analysis by Ablowitz et al. [2] found thatpseudo-spectral methods perform significantly bet-ter for the Sine-Gordon equation than lattice type,symplectic schemes. From this result they concludedthat numerical accuracy for approximating the spa-tial derivatives is more important than the symplecticschemes for the numerical integration of a generalclass of nonlinear wave equations. The importanceof this result is obvious. However, computation-ally, for nonlinear equations, the global spectral andpseudo-spectral methods are not nearly as simpleand robust to implement as various local methods.Moreover, global methods can only be directly usedin structured grids. For unstructured grids, whichare required for the case of complex geometry andboundary, only local methods can be directly imple-mented. Therefore, it would be extremely desirableto have accurate, efficient, and robust approaches forsolving the various nonlinear wave equations whichplay an important role in modern science and tech-nology. Wavelet theory has been expected to fulfillthis task and has been extensively studied recently forthis purpose [10–13], including the recently reportedadaptive wavelet algorithms [14,15]. However, theseefforts have been hindered either by the technical dif-ficulties of incorporating multiresolution analysis intothe treatment of boundary conditions or by the lackof accurate and efficient wavelets for solving linearand nonlinear partial differential equations (PDEs).For example, Beylkin and Keiser [14] reported thedifficulty of handling the KdV equation by using asophisticated wavelet algorithm.
Discrete singular convolution (DSC) [16] is a po-tential approach for numerically solving a few classesof problems, including Hilbert transform, processingof analytic signals, computational electromagnetics,computational tomography, and linear and nonlineardynamics. In fact, underlying mathematical structureof DSC is the theory of distributions. One of the dis-tributions used in the aforementioned applications is
the Dirac delta function which is a generalized func-tion following from the fact that it is an integrablefunction inside a particular interval but it need nothave a value. Heaviside introduced both the unit stepHeaviside function and the Dirac delta function as itsderivative, and referred to the latter as the unit im-pulse. Dirac, for the first time, explicitly discussed theproperties ofδ in his classic text on quantum mechan-ics; for this reasonδ is often called Dirac delta func-tion. However, delta distribution has a history whichantedates both Heaviside and Dirac. It appeared inexplicit form as early as 1822, in Fourier’sThéorieAnalytique de la Chaleur. The work of Heaviside, andsubsequently of Dirac, in the systematic but informalexploitation of the step function and delta function hasmade delta distribution familiar to physicists and engi-neers before Sobolev, Schwartz [17], Korevaar [18,19]and others put it into a rigorous mathematical form.In particular, the Hermite function expansion of Diracdelta function was proposed by Schwartz [17] andKorevaar [20] over 40 years ago and was used by Hoff-man et. al. [21] for numerical simulations. Generalorthogonal series analyses of the delta distributionhave been subsequentially studied by Walter [22] andothers [23–25]. The use of many delta sequences asprobability density estimators was discussed by Walterand Blum [25] and others [24,26,27].
The purpose of the present paper is to demonstratethat the discrete singular convolution (DSC) algorithmprovides a powerful tool for solving the sine-Gordonequation. This paper is organized as follows. Section2 is devoted to a brief review of the discrete singularconvolution algorithm. Numerical integration of thesine-Gordon equation is given in Section 3. Particularattention is paid to those initial values which are closeto the homoclinic orbits and have previously ledto numerically induced spatial and temporal chaos.Conclusional remarks are given in Section 4.
2. Discrete singular convolution
Singular convolutions appear in many science andengineering problems, such as Hilbert transform, Abeltransform and Radon transform. Discrete singular
G.W. Wei / Physica D 137 (2000) 247–259 249
convolution is a unified approach for numericallysolving singular convolution problems. The simplestway to introduce the theory ofsingular convolution(SC) is to work in the context of distributions. LetT be a distribution andη(t) be an element of thespace of test functions (e.g.,η(t) ∈ D). A singularconvolution is defined as
F(t) = (T ∗ η)(t) =∫ ∞
−∞T (t − x)η(x) dx. (1)
HereT (t − x) is a singular kernel. Depending on theform of the kernelT , the singular convolution is thecentral issue for a wide range of science and engi-neering problems. For example, singular kernels of theHilbert typehave a general form
T (x) = 1
xn(n > 0). (2)
Here, kernelsT (x) = 1/xa (0 < a < 1) definethe Abel transformwhich is closely connected witha generalization of the tautochrone problem. KernelT (x) = 1/x is commonly encountered in electrody-namics, theory of linear response, signal processing,theory of analytic functions, and the Hilbert transform.Kernel T (x) = 1/x2 is widely used in tomography.Other interesting examples are singular kernels of thedelta type
T (x) = δ(n)(x) (n = 0, 1, 2, . . . ). (3)
Here, kernelT (x) = δ(x) is important for the interpo-lation of surfaces and curves (including atomic, molec-ular and biological potential energy surfaces, aircraftand missile surfaces), andT (x) = δ(n)(x) (n =1, 2, . . . ) are essential for obtaining weak solutionsof partial differential equations. However, since thesekernels are singular, they cannot be directly digital-ized in computer. Hence, the singular convolution,(1), is of little numerical merit. To avoid the difficultyof using singular expressions directly in computer,sequences of approximations{Tα} of the distributionT can be constructed
limα→α0
Tα(x) → T (x), (4)
where α0 is a generalized limit. Obviously, in thecase ofT (x) = δ(x), the sequence,Tα(x), is a delta
sequence. Furthermore, with a sufficiently smoothapproximation, it makes sense to consider adiscretesingular convolution(DSC)
Fα(t) =∑
k
Tα(t − xk)f (xk), (5)
whereFα(t) is an approximation toF(t) and{xk} isan appropriate set of discrete points on which the DSC(5) is well defined. Note that the original test functionη(x) has been replaced byf (x). The mathematicalproperty or requirement off (x) is determined by theapproximate kernelTα. In general, the convolution isrequired being Lebesgue integrable.
A sequence of approximation can be improved bya regularizer [28,29]:
limσ→∞Rσ (x) = 1. (6)
The regularizer is designed to increase the regularity ofconvolution kernels. For the delta sequence, it followsfrom Eq. (4) that∫
limα→α0
Tα(x)Rσ (x) dx = Rσ (0) = 1, (7)
whereRσ (0) = 1 is the special requirement for adeltaregularizer.
As an interesting example, Shannon’s kernelsinαx/πx is a delta sequence kernel
limα→∞
⟨sinαx
πx, η(x)
⟩= η(0). (8)
Other important examples include the Dirichlet kernel
sin [(l + 12)(x − x)]
2π sin [12(x − x′)]
,
the modified Dirichlet kernel
sin [(l + 12)(x − x′)]
2π tan [12(x − x′)],
and the de la Vallée Poussin kernel
1
πα
cos [α(x − x′)] − cos [2α(x − x′)](x − x′)2
.
For sequences of both the delta type and theHilbert type, an interpolating (or quasi interpolating)algorithm sampling atNyquist frequency, α = π/∆,
250 G.W. Wei / Physica D 137 (2000) 247–259
has a great advantage over a non-interpolating dis-cretization
sin [α(x − x′)]π(x − x′)
→ sin(π/∆)(x − xk)
(π/∆)(x − xk). (9)
The interpolating nature not only guarantees the high-est accuracy on the set of grid points but also pro-vides the highest possible computational efficiency ofa grid. This is because theNyquist intervalgiven by[−π/∆, π/∆] is the largest possible sampling inter-val that is free of alias whenever theL2 functionf (x)
under study satisfies theNyquist condition:
suppf̂ (k) ⊂{−π
∆,π
∆
}. (10)
This fact can be formally given by Shannon’s samplingtheorem
f (x) =∞∑
k=−∞f (xk)
sin(π/∆)(x − xk)
(π/∆)(x − xk). (11)
The significance of Shannon’s sampling theorem isthat by a discrete but infinite set of sampling data{f (xk)}, one can actually recover a bandlimitedL2
function on the real line. Shannon’s sampling theo-rem has great impact on information theory, signaland image processing because the Fourier transformof Shannon’s kernel is an ideal low-pass filter for sig-nals bandlimited to [−π/∆, π/∆].
The uniform, Nyquist rate, interpolating discretiza-tion is also used for the Dirichlet kernel:
sin [(l + 12)(x − x′)]
2π sin [12(x − x′)]
→ sin((π/∆)(x − xk))
(2M + 1) sin((π/∆)((x − xk)/(2M + 1))).
(12)
In a comparison to Shannon’s kernel, the Dirichlet ker-nel has one more parameterM which can be optimizedto achieve better results in computations. Usually, weset a sufficiently largeM for various numerical ap-plications. Obviously, the Dirichlet kernel converts toShannon’s kernel at the limitM → ∞. This uni-form interpolating discretization will also be used fordiscretizing the modified Dirichlet kernels
sin [(l + 12)(x − x′)]
2π tan [12(x − x′)]
→ sin((π/∆)(x − xk))
(2M + 1) tan((π/∆)((x − xk)/(2M + 1))),
(13)
and for the de la Vallée Poussin kernels
1
πα
cos [α(x − x′)] − cos [2α(x − x′)](x − x′)2
→ 2
3
cos(π/∆̄)(x − xk) − cos(2π/∆̄)(x − xk)
[(π/∆̄)(x − xk)]2,
(14)
where ∆̄ = 32∆. Sinceπ/∆ is proportional to the
highest frequency which can be reached in the Fourierrepresentation, the∆ should be very small for a givenproblem involving very oscillatory functions or veryhigh frequency components.
We use a symmetrically (or antisymmetrically) trun-cated, translationally invariant singular kernel
f (n)(x) ≈W∑
k=−W
δ(n)α (x − xk)f (xk)
(n = 0, 1, 2, . . . ), (15)
where{xk} are centered aroundx and 2W + 1 is thecomputational bandwidth, or effective kernel support,which is usually smaller than the whole computationaldomain [a, b]. Hereδ
(n)α (x−xk) is a collective symbol
for thenth derivative of any of the right-hand side ofEqs. (9) and (12)–(14).
In the DSC approach, it is convenient to discretizean operator on a grid of the coordinate representation.This is illustrated by using a Hamiltonian. As such, thepotential part,V (x), of the Hamiltonian is diagonal.Hence, its discretization is simply given by a directinterpolation on the grid
V (x) → V (xk)δm,k. (16)
The differentiation matrix of an operator or the Hamil-tonian on the coordinate grid is then given in terms offunctional derivatives
− ~2
2µ
d2
dx2→ − ~
2
2µδ(2)α (xm − xk), (17)
G.W. Wei / Physica D 137 (2000) 247–259 251
whereµ is the mass of the Hamiltonian system andδ(2)α (xm − xk) areanalyticallygiven by
δ(2)α (xm − xk) =
[(d
dx
)2
δα(x − xk)
]x=xm
. (18)
Thus, the full DSC grid representation for the Hamil-tonian operator,H , is given by
H(xm, xk) = − ~2
2µδ(2)α (xm − xk) + V (xm)δm,k. (19)
All other operators which consist of a non-diagonaldifferentiation part and/or a diagonal part can betreated similarly.
In the present study we limit our attention to thesingular kernels of Shannon (Shannon), the de laVallée Poussin (DLVP), Dirichlet (Dirichlet) and themodified Dirichlet (MD). Nevertheless, various otherdelta sequence kernels can be similarly employed. Itis noted that the singular kernels of Shannon (Shan-non) and de la Vallée Poussin (DLVP) are parameterfree, which is convenient for applications. The 2M +1parameter used for the other two kernels is chosen as71 for all calculations. We note that as long as the2M + 1 value is chosen sufficiently large (2M + 1 >
W , where 2W + 1 is the matrix bandwidth), the nu-merical results are not sensitive to the specific valuesused. The time discretization is obtained by using thefourth order Runge–Kutta scheme.
3. Application and results
The sine-Gordon equation is one of the most im-portant nonlinear wave equations that can be used tomodel the soliton waves in nature. Although Stokes[30] described in detail the Stokes waves in 1847,the formal development of soliton theory was in1960s [31–33]. Benjamin studied the instability ofStokes waves in deep water. Such an instability isalso described by the nonlinear Schrödinger equation.Zabusky and Kruskal [32] pioneeringly studied theinstability in the Korteweg–de Vries (KdV) equationdescribing nonlinear waves in shallow water [34,35].A variety of other natural phenomena can be modeled
by the sine-Gordon equation. These include the ro-tator phase dynamics of then-heneicosane (C22H44)polymer [36], H–D exchange in DNA [37,38] andr.f. radiation from a Josephson junction [39]. Thenumerical instability of nonlinear wave equations,including numerically induced spatial and temporalchaos of these equations, has been carefully investi-gated by Ablowitz et al. [1–4] recently. The purposeof this section is both to test the accuracy and relia-bility of the DSC algorithm for temporal integrationof nonlinear wave equations and to numerically studythe waveforms of the sine-Gordon equation whenthe initial values are either exponentially close to or“exactly” on the most unstable homoclinic orbit.
The sine-Gordon equation is given by
∂2u(x, t)
∂t2− ∂2u(x, t)
∂x2+ sin [u(x, t)] = 0 (20)
with periodic boundary conditions
u(x, t) = u(x + L, t). (21)
It is a completely integrable Hamiltonian system [1,7]and the associated Hamiltonian can be defined as
H =∫ L
0
[1
2v2 + 1
2
(∂u
∂x
)2
+ 1 − cos(u)
]dx. (22)
The sine-Gordon Hamiltonian operator maps from theinfinite dimensional phase space
F (L) ={
u(x) =(
u(x)
v(x)
), eiu(x+L) = eiu(x),
v(x + L) = v(x)
}(23)
to the real lineR. Herev = ut andu are related to thephase space variable(q, p) according toq = u andp = v. It is noted that Hamilton’s equations hold as
∂q
∂t= δH
δp,
∂p
∂t= −δH
δq. (24)
Ercolani et al. [7] analyzed the sine-Gordon phasespace geometry. In particular, homoclinic orbits wereshown to be associated with numerical instabilities[1,7] and even chaos. The simplest homoclinic orbit
252 G.W. Wei / Physica D 137 (2000) 247–259
can be obtained by examining the spatially homoge-neous sine-Gordon equation, i.e., the periodic pendu-lum equation
∂2u
∂t2+ sin(u) = 0. (25)
The pendulum equation is integrable and has awell-known homoclinic orbit
u(x, t) = π − 4 tan−1[e(t+t0)] (26)
corresponding to the phase space separatrix at (π, 0).A family of homoclinic orbits can be constructed [1,6]by using the sine-Gordon symmetry
(x, t, u) → (t, x, u + π). (27)
By doing this one starts with a breather solution
u(x, t) = 4 tan−1[
tan(ν) cos [ cos(ν)t ]
cosh [ sin(ν)x]
], (28)
where the parameterν satisfies
|ν| � 1. (29)
This breather can be viewed as a kink–antikinkbounded pair in space with a 2π/ cos(ν) periodicoscillation in time. The sine-Gordon space–time sym-metry (27) leads to a family of real valued homoclinicorbits
u(x, t) = π + 4 tan−1[
tan(ν) cos [ cos(ν)x]
cosh [ sin(ν)t ]
]. (30)
In contrast to the spatially homogeneous homoclinicorbit, Eq. (26), this family of homoclinic orbits alsohas a 2π/ cos(ν) periodic spatial structure, i.e., a tan-gent cone associated with the phase space limit point(π, 0). We refer the reader to Ref. [7] for the construc-tion of more general homoclinic states and a detailedspectral analysis.
For numerical purpose, the sine-Gordon equation(20) is rewritten as a pair of coupled equations
∂u(x, t)
∂t= v(x, t),
∂v(x, t)
∂t= ∂2u(x, t)
∂x2− sinu(x, t). (31)
Two types of initial conditions are used in the presentcomputations: one is chosen to be extremely close to or
even “exactly” on the homoclinic orbits and the othercorresponds to a case where there is a breather-kinkand antikink transition. Both cases were used previ-ously by Ablowitz et al. [1] to demonstrate the possi-ble appearance of numerical chaos. However, they didnot provide stable numerical waveforms of the system.
3.1. Near homoclinic orbit states
We first consider the numerical solution of Eq. (20)with the initial values [1]
u(x, 0) = π + ε cos(µx), ut (x, 0) = 0, (32)
with µ = 2π/L andL = 2√
2π . It is noted that thevalue of π is numerically generated as cos−1(−1)
to the normal double precision. For this reason weuse the quotation marks for the wordexactly whenwe say exactly on the homoclinic orbit. The periodicboundary condition, Eq. (21), is numerically imple-mented. Ablowitz et al. [1] showed that for smallε’s,these initial values are exponentially close to the ho-moclinic manifold and produce numerical instabilitywhen one uses the Hirota algorithm [40], which is adoubly discrete, integrable discretization scheme stan-dardly used in numerical integration of nonlinear waveequations. Whenε was chosen to be small, both spa-tial and temporal chaos were easily excited by verysmall perturbations, even on the order of round-offerror. These authors found a more troubling aspectthat these numerical instabilities persisted as the meshwere refined and cannot be detected by monitoring theconserved quantities of the equation. In other words,the temporal evolution of the numerical solution re-mained unstable even if all the conserved quantitieswere well preserved (by employing a very fine grid).They showed that the chaos persists even if a spatiallycompletely integrable scheme is employed.
To demonstrate the numerical stability of the DSCalgorithm, we first consider those twoε values (ε =0.05, 0.1) that have previously been used by Ablowitzet al. [1]. In the case ofε = 0.05, Ablowitz et al. [1]found numerically induced chaos in a very early timeof the integration by using 64 grid points. As plotted inFig. 1, the DSC algorithm produces stable wave formseven if a small number of grid points is chosen as
G.W. Wei / Physica D 137 (2000) 247–259 253
Fig. 1. The DSC solution of the sine-Gordon equation (ε = 0.05, N = 32, τ = 0.02): (a) betweent = 100 andt = 200; (b) betweent = 10 000 andt = 10 020.
N = 32. Obviously no irregular oscillation occurredin our results. Actually, no trace of any numerical in-stability has been detected even when the waveformsare integrated up to 10 020 time units.
The case ofε = 0.1 was also studied by Ablowitzet al. [1]. In particular, they found that this initial valueled to numerically induced spatial and temporal chaosafter a relatively longer time integration (t ≥ 300 timeunits). In the present study, we still choose a relativelylarge grid mesh (N = 32) for the delta sequence ker-nels. As plotted in Fig. 2, there is no trace of numericalinstability in our results. In fact, our DSC solution is
regular and stable even if the wave form is propagatedto 10 020 time units (see Fig. 2b).
To verify our results, we further consider a largeε
value (ε = 1000). This value is far from the low orderhomoclinic manifold. As expected, our results are stillvery stable even up to 10 020 time units. These aredepicted in Fig. 3. It is noted that when largeε valuesare used, the corresponding sine-Gordon waveformsare highly oscillative in the spatial domain.
For the last test, we chooseε = 0 so that the initialvalues for the sine-Gordon equation is “exactly” on themost unstable homoclinic manifold. Surprisingly, we
254 G.W. Wei / Physica D 137 (2000) 247–259
Fig. 2. The DSC solution of the sine-Gordon equation (ε = 0.1, N = 32, τ = 0.02): (a) betweent = 100 andt = 200; (b) betweent = 10 000 andt = 10 020.
obtain another set of stable waveforms for this initialvalue as given in Fig. 4, which is free from any traceof numerical instability.
By a comparison of waveforms for variousε val-ues from 0 to 1000, we conclude that all of the resultsare consistent with each other and sound. All resultsin these figures are obtained by using 32 grid pointsand a time mesh ofτ = 0.02. Note that we have fixedthe number of plots as 100 in all figures. To confirmour results further, we have also tested a number ofdifferent initial values (ε = 0.001, 1, and 100), a num-ber of different (even and odd) grid points and some
very small or large time meshes, such asτ = 0.00001,0.25. We have also changed the number of plots andintegration time units. All of these results are stable,and consistent with regard to these variations.
Our results are completely regular and free of nu-merical instability for all four delta sequence kernelsnumerically tested in this work. We noted that graph-ically, there was no difference between the results ob-tained by employing any of the delta sequence kernelsof Shannon, Dirichlet and modified Dirichlet. We havealso obtained the same type of results by means ofthe de la Vallée Poussin delta sequence kernel using
G.W. Wei / Physica D 137 (2000) 247–259 255
Fig. 3. The DSC solution of the sine-Gordon equation (ε = 1000.0, N = 32, τ = 0.02): (a) betweent = 100 andt = 200; (b) betweent = 10 000 andt = 10 020.
slightly more grid points. These results provide verystrong evidence that the DSC algorithm is very reliableand robust for integrating nonlinear wave equationswith low-lying unstable homoclinic orbits. As is wellknown, the low-lying homoclinic orbits are most eas-ily excited by extremely small perturbations, even thecomputer round-off errors for the sine-Gordon equa-tion. Ablowitz et al. [1] pointed out that occurrence ofnumerical chaos is primarily the result of inaccurateapproximation of the spatial derivatives in discretizingthe partial differential equation [2]. Hence, the presentDSC algorithm must provide very high accuracy for
approximating the spatial derivatives. This is consis-tent with our results on eigenvalue problems of theSchrödinger equation and the Fokker–Planck equationwhere the second order derivative is required.
3.2. Breather-kink and antikink transition
To explore the accuracy and reliability of the DSCalgorithm for the integration of the sine-Gordon equa-tion further, we consider different initial values whereprevious work indicates the occurrence of a differ-ent type of numerical instability [1]. The analytical
256 G.W. Wei / Physica D 137 (2000) 247–259
Fig. 4. The DSC solution of the sine-Gordon equation (ε = 0.0, N = 32, τ = 0.02): (a) betweent = 100 andt = 200; (b) betweent = 10 000 andt = 10 020.
solution represents a breather-kink and antikink tran-sition associated with a double point in the nonlinearspectrum of the sine-Gordon equation [1]:
u(x, t) = 4 tan−1 [sech(x)t ] , −∞ < x < ∞, (33)
with initial values
u(x, 0) = 0, ut (x, 0) = 4 sech(x), (34)
and periodic boundary conditions. In this case thesolution is far away from the unstable homoclinicmanifolds. Using the doubly discrete, integrable dis-cretization scheme [40], Ablowitz et al. [1] found that
numerically induced chaos occurred for 64 grid pointsafter a number of integrations. However, unlike theprevious examples of their calculations, the chaos dis-appeared when a refined grid mesh(N = 128) wasemployed. A careful examination of their results in-dicates that the breather-kink and antikink transitionoccurred much earlier than it should in their refinedcalculation. This phenomena can be characterized asnumerical catalyzed breather-kink and antikink tran-sition. In our calculation, a total of 64 grid points(N = 64) is used in the interval [−20, 20] for the deltasequence kernels of Shannon, Dirichlet and modified
G.W. Wei / Physica D 137 (2000) 247–259 257
Fig. 5. Numerical and exact solution of the sine-Gordon equation with initial values (34) describing a breather-kink and antikink transition(betweent = 0.05 andt = 100, N = 64, τ = 0.05).
Dirichlet. To achieve the same level of accuracy, asmany as 1.5 times of grid points (N = 96) are used forthe de la Vallée Poussin delta sequence kernel. Boththe exact results and the Dirichlet delta sequence ker-nel calculations obtained by using 64 grid points areplotted in Fig. 5. There is no visible difference in thetwo solutions. The computational results obtained byusing other delta sequence kernels are graphically thesame as that in Fig. 5. Actually, the present methodprovides smooth, stable numerical solutions by using40 or even fewer grid points. The accuracy of theDSC algorithm for the sine-Gordon equation is con-trollable and theL∞ andL2 errors are very small withthe present choice of spatial and temporal mesh sizes.These results are listed in Table 1. It is seen that theDSC algorithm achieves an accuracy of four or five
Table 1L∞ andL2 errors of the numerical solutions for the sine-Gordon equation
significant figures for the sine-Gordon equation usinga reasonably small number of grid points and rela-tively large time increment.
4. Conclusion
By focusing on the delta sequence kernels of Shan-non, Dirichlet, modified Dirichlet and the de la ValléePoussin, the utility of the DSC algorithm is exploredfor the nonlinear dynamics of the sine-Gordon equa-tion, which is a challenging case when the initialvalues are close to the homoclinic orbits. The DSCalgorithm is used for spatial discretization which isin association with the fourth order Runge–Kuttascheme for time integration. We have chosen the
258 G.W. Wei / Physica D 137 (2000) 247–259
sine-Gordon equation because Ablowitz et al. [2]have recently called attention to the numerical dif-ficulties of this problem. They showed that wheninitial values are chosen to be close to the homoclinicmanifolds, previous integrable discretization schemesencountered difficulties, including numerically in-duced spatiotemporal chaos [1]. There is no reporton the numerical method which is totally free fromhomoclinic-orbit-crossing in the literature.
A number of new initial values are considered, in-cluding a case where the initial value is “exactly”on the homoclinic orbit. Important numerical issuesexamined in this paper are the accuracy of approx-imation, the speed of convergence, the simplicity ofimplementation, the stability of integration and the re-liability of application. The DSC algorithm performsextremely well for all issues.
For two troubling initial values which are closeto the sine-Gordon homoclinic orbit and have beenused previously by Ablowitz et al. [2], the DSC al-gorithm provides stable, smooth, chaos free and evenhomoclinic-orbit-crossing free results by using a smallnumber of grid points(N = 32). In both cases weactually integrate the waveforms up to 10 020 timeunits without encountering any trace of numerical in-stability. To verify our results, we considerε valuesas large as 1000. There is still no trace of numericalinstability in our solution for this case. In a dramat-ical case, the initial value is chosen to be “exactly”on the homoclinic orbit (ε = 0). Our DSC resultsremain smooth, stable, and consistent with those ob-tained with non-zeroε values. Our results are verifiedby using a variety of different time meshes, number ofgrid points, number of plots and integrating time units.The complete dynamical behavior of the sine-Gordonequation with this set of initial values can be under-stood from our numerical simulations.
The other type of initial values considered in thiswork is the situation where a solitary transition from abreather to a kink and antikink occurs during the timeevolution of the sine-Gordon soliton. Numerically in-duced spatial chaos was also reported in the literaturefor this problem [1] when the number of grid pointsis chosen asN = 64 in the interval of [−20, 20]. TheDSC algorithm provides stable results by using only
40 grid points in the same interval. Our DSC resultsare accurate up to five significant figures for this ini-tial value using a slightly larger number of grid point(N = 64) in the interval of [−20, 20] and a relativelylarge time increment of 0.05.
Since the symplectic numerical schemes are de-signed to preserve the phase space structures ofHamiltonian systems, they have been the main fo-cus of enormous research in the past and have beenregarded as superior to explicit methods such asthe Runge–Kutta scheme. Only very recently after asystematical comparison of a number of numericalschemes have Ablowitz et al. [2] pointed out that theaccuracy of the spatial discretization is more impor-tant than the symplectic property. The superior resultsobtained in this work indicate that the DSC algorithmprovides highly accurate spatial discretizations forintegrating nonlinear partial differential equations.However, it is our experience that not only the ac-curacy but also the implementation of a numericalalgorithm that determine the numerical stability forintegration of nonlinear wave equations. This pointwill be formally explored in our future work.
Acknowledgements
This work is supported in part by the National Uni-versity of Singapore. The correspondence with Prof.M.J. Ablowitz and discussion with Prof. C. Schoberabout the sine-Gordon equation are gratefully ac-knowledged.
References
[1] M.J. Ablowitz, B.M. Herbst, C. Schober, J. Comput. Phys.126 (1996) 299.
[2] M.J. Ablowitz, B.M. Herbst, C. Schober, J. Comput. Phys.131 (1997) 354.
[9] M.G. Forest, C.G. Goedde, A. Sinha, Physica D 67 (1993)347.
[10] R.L. Schult, H.W. Wyld, Phys. Rev. A 46 (1992) 12.[11] J. Liandrat, V. Perrier, P. Tchamitchian, in: M.B. Ruskai, G.
Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer,L. Raphael (Eds.), Wavelets and their Applications, Jones &Bartlett, Boston, 1992.
[12] B. Jawerth, W. Sweldens, SIAM Rev. 36 (1994) 377.[13] W.K. Liu, Int. J. Numer. Meth. Fluids 21 (1995) 901.[14] G. Beylkin, J. Keiser, J. Comput. Phys. 132 (1997) 233.[15] J. Fröhlich, K. Schneider, J. Comput. Phys. 130 (1997) 174.[16] G.W. Wei, J. Chem. Phys. 110 (1999) 8930.[17] L. Schwartz, Théorie des Distributions, Hermann, Paris, 1951.[18] J. Korevaar, Nederl. Akad. Westensh. Proc. Ser. A 8 (1955)
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