Computer Physics Communications 179 (2008) 449вЂ“456 Contents lists available at ScienceDirect Computer Physics Communications www.elsevier.com/locate/cpc An exponential time differencing method for the nonlinear SchrГ¶dinger equation F. de la Hoz a , F. Vadillo b,в€— a b University of the Basque Country, Dpto. de MatemГЎtica Aplicada, Spain University of the Basque Country, Dpto. de MatemГЎtica Aplicada, Estad. e I.O., Spain a r t i c l e i n f o a b s t r a c t Article history: Received 15 February 2008 Received in revised form 4 April 2008 Accepted 11 April 2008 Available online 24 April 2008 MSC: primary 65M70 secondary 65M20 Keywords: SchrГ¶dinger ETD Stiff systems Integrating factor Nonlinear waves The spectral methods offer very high spatial resolution for a wide range of nonlinear wave equations, so, for the best computational eп¬ѓciency, it should be desirable to use also high order methods in time but without very strict restrictions on the step size by reason of numerical stability. In this paper we study the exponential time differencing fourth-order RungeвЂ“Kutta (ETDRK4) method; this scheme was derived by Cox and Matthews in [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys. 176 (2002) 430вЂ“455] and was modiп¬Ѓed by Kassam and Trefethen in [A. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comp. 26 (2005) 1214вЂ“1233]. We compute its ampliп¬Ѓcation factor and plot its stability region, which gives us an explanation of its good behavior for dissipative and dispersive problems. We apply this method to the SchrГ¶dinger equation, obtaining excellent results for the cubic equation and the critical exponent case and, later, as an experimental approach to describe the various possible asymptotic behaviors with two space variables. В© 2008 Elsevier B.V. All rights reserved. 1. Introduction The spectral methods have been shown to be remarkably successful when solving time-dependent partial differential equations (PDEs). The idea is to approximate a solution u (x, t ) by a п¬ЃN nite sum v (x, t ) = k=0 ak (t )П†k (x), where the function class П†k (x), k = 0, 1, . . . , N, will be trigonometric for x-periodic problems and, otherwise, an orthogonal polynomial of Jacobi type, with Chebyshev polynomial being the most important special case. To determine the expansion coeп¬ѓcients ak (t ), we will focus on the pseudospectral methods, where it is required that the coeп¬ѓcients make the residual equal zero at as many (suitably chosen) spatial points as possible. Three books [4,10] and [23] have been contributed to supplement the classic references [12] and [5]. When a time-dependent PDE is discretized in space with a spectral simulation, the result is a coupled system of ordinary differential equations (ODEs) in time: it is the notion of the method of lines (MOL) and the resulting set of ODEs is stiff; the stiffness problem may be even exacerbated sometimes, for example, using Chebyshev polynomials (see Chapter 10 of [23] and its references). The linear terms are primarily responsible for the stiffness with rapid exponential decay of some modes (as with a dissipative PDE) or a rapid oscillation of some modes (as with a dispersive PDE). * Corresponding author. E-mail addresses: [email protected] (F. de la Hoz), [email protected] (F. Vadillo). 0010-4655/$ вЂ“ see front matter doi:10.1016/j.cpc.2008.04.013 В© 2008 Elsevier B.V. All rights reserved. Therefore, for a time-dependent PDE which combines low-order nonlinear terms with higher-order linear terms it is desirable to use higher-order approximation in space and time. The structure of this paper is as follows. In Section 2 we describe the ETDRK4 (Exponential Time Differencing fourth-order RungeвЂ“Kutta) method by Cox and Matthews in [6] and the modiп¬Ѓcation proposed by Kassam and Trefethen in [14]. We discuss the stability of the ETDRK4 method in Section 3. In Sections 4 and 5 we test the method for the nonlinear SchrГ¶dinger equation in one and two space dimensions and, п¬Ѓnally, we summarize our conclusions. 2. Exponential time differencing fourth-order RungeвЂ“Kutta method The numerical method considered in this paper is an exponential time differencing (ETD) scheme. These methods arose originally in the п¬Ѓeld of computational electrodynamics [22]. Since then, they have recently received attention in [3] and [19], but the most comprehensive treatment, and in particular the ETD with RungeвЂ“Kutta time stepping, is in the paper by Cox and Matthews [6]. The idea of the ETD methods is similar to the method of the integrating factor (see, for example, [4] or [23]): we multiply both sides of a differential equation by some integrating factor, then we make a change of variable that allows us to solve the linear part exactly and, п¬Ѓnally, we use a numerical method of our choice to solve the transformed nonlinear part. 450 F. de la Hoz, F. Vadillo / Computer Physics Communications 179 (2008) 449вЂ“456 When a time-dependent PDE in the form ut = Lu + N (u , t ), (2.1) where L and N are the linear and nonlinear operators respectively, is discretized in space with a spectral method, the result is a coupled system of ordinary differential equations (ODEs), with F (u ) the nonlinear part, we suppose that there exists a п¬Ѓxed point u 0 ; this means that cu 0 + F (u 0 ) = 0. Linearizing about this п¬Ѓxed point, if u is the perturbation of u 0 and О» = F (u 0 ), then ut = cu + О»u (3.2) Multiplying (2.2) by the term e в€’Lt , known as the integrating factor, gives and the п¬Ѓxed point u 0 is stable if Re(c + О») < 0. The application of the ETDRK4 method to (3.2) leads to a recurrence relation involving un and un+1 . Introducing the previous notation x = О»h, y = ch and using the MathematicaВ© algebra package, we obtain the following ampliп¬Ѓcation factor e в€’Lt ut в€’ e в€’Lt Lu = e в€’Lt N(u , t ), un+1 ut = Lu + N(u , t ). (2.2) and with the new variable v = equation (2.3) e в€’Lt u, we п¬Ѓnd the transformed v t = e в€’Lt N e Lt v , t , un c0 = e y , c1 = c2 = un+1 = e un + e e в€’LП„ N u (tn + П„ ), tn + П„ dП„ . (2.5) c3 = an = e Lh/2 un + Lв€’1 e Lh/2 в€’ I N(un , tn ), cn = e Lh/2 an + Lв€’1 e Lh/2 в€’ I 2N(bn , tn + h/2) в€’ N(un , tn ) , в€’4I в€’ hL + e Lh 4I в€’ 3hL + (hL)2 N(un , tn ) + 2 2I + hL + e Lh (в€’2I + hL) Г— N(an , tn + h/2) + N(bn , tn + h/2) + в€’4I в€’ 3hL в€’ (hL)2 + e Lh (4I в€’ hL) N(cn , tn + h) . More detailed derivations of the ETD schemes can be found in [6]. Unfortunately, in this form ETDRK4 suffers from numerical instability when L has eigenvalues close to zero, because disastrous cancellation errors arise. Kassam and Trefethen have studied in [14] these instabilities and have found that they can be removed by evaluating a certain integral on a contour that is separated from zero. The procedure is basically to change the evaluation of the coeп¬ѓcients, which is mathematically equivalent to the original ETDRK4 scheme of [6], but in [9] it has been shown to have the effect of improving the stability of integration in time. Also, it can be easily implemented and the impact on the total computing time is small. In fact, we have always used this idea in our MATLABВ© codes. 3. On the stability of ETDRK4 method The stability analysis of the ETDRK4 method is as follows (see [3,11] or [6]). For the nonlinear ODE du dt = cu + F (u ), 8e в€’ y3 e 2y y3 16e 2 y4 1 в€’ y2 16e в€’ y5 в€’ 16e 2 4e 16e y y4 в€’ 16e y y5 3y 2 12e y6 в€’ 24e 2 20e y y5 y6 в€’ + y4 y 8 y 2 y2 + y5 3y 2 в€’ + y4 16e y + + y6 12e 3y 2 y5 + 3y 2 в€’ 2e в€’ 16e y5 y2 y + 20e 2 y y5 5y 2 в€’ y4 3y 2 y6 6e 2 y y2 12e 2 в€’ y3 4e + 3y 2 10e y + y3 e2 y в€’ y2 4e y2 , 3y 2 y3 , y5 6e 2 y y3 6e y в€’ y y2 8e 5 в€’ y4 4e 2 + y2 8e 2 y 3e y 1 в€’ y3 + y4 + 4e 2 y + y3 y 4 3y 2 в€’ в€’ 2e y5 2e y6 + 2 y4 5y 2 + y5 y 2 y3 24e 2 y 5y 2 8e + + + 4e y y3 8e в€’ 5y 2 y6 6e y 2 y4 y 2 y4 в€’ + + 10e 2 в€’ y4 3y 2 2e y3 6 y5 y в€’ 6e y y4 , 18e 2 в€’ y5 2e 3y 2 y4 . An important remark: computing c 1 , c 2 , c 3 and c 4 by the above expressions suffers from numerical instability for y close to zero. Because of that, for small y, instead of them, we will use their asymptotic expansions bn = e Lh/2 un + Lв€’1 e Lh/2 в€’ I N(an , tn + h/2), un+1 = e Lh un + hв€’2 Lв€’3 + y4 + 8e 2 y в€’8 + c4 = + y3 0 The various ETD methods come from how one approximates the integral in this expression. Cox and Matthews derived in [6] a set of ETD methods based on the RungeвЂ“Kutta time stepping, which they called ETDRK methods. In this paper we consider the ETDRK4 fourth-order scheme with the formulae y в€’4 в€’ h Lh (3.3) where (2.4) where the linear term is gone; now we can use a time stepping method of our choice to advance in time. However, the integrating factor methods can also be a trap, for example, to model the formation and dynamics of solitary waves of the KdV equation (see Chapter 14 of [4]). A second drawback is the large error constant. In the derivation of the ETD methods, following [3], instead of changing the variable, we integrate (2.3) over a single time step of length h, getting Lh = r (x, y ) = c 0 + c 1 x + c 2 x2 + c 3 x3 + c 4 x4 , (3.1) 7 5 y + O y6 , 960 1 1 1 247 3 131 4 479 5 c2 = + y + y2 + y + y + y + O y6 , 2 2 4 2880 5760 96768 1 1 61 2 1 3 1441 4 67 c3 = + y + y + y + y + y5 + O y6 , 6 6 720 36 241920 120960 1 1 7 2 19 25 311 c4 = + y+ y + y3 в€’ y4 в€’ y5 24 32 640 11520 64512 860160 c1 = 1 + y + 1 2 y2 + 1 6 y3 + 13 320 y4 + + O y6 . We make two observations: вЂў As y в†’ 0, our approximation becomes r (x) = 1 + x + 1 2 x2 + 1 6 x3 + 1 24 x4 , which is the stability function for all the 4-stage RungeвЂ“Kutta methods of order four. вЂў Because c and О» may be complex, the stability region of the ETDRK4 method is four-dimensional and therefore quite diп¬ѓcult to represent. Unfortunately, we do not know any expression for |r (x, y )| = 1; we will only be able to plot it. The most common idea is to study it for each particular case; for example, assuming c to be п¬Ѓxed and real in [3] or that both F. de la Hoz, F. Vadillo / Computer Physics Communications 179 (2008) 449вЂ“456 451 Fig. 1. Boundary of stability regions for several negative y. Fig. 2. Experimental boundaries and ellipse for y = в€’75. c and О» are pure imaginary numbers in [11]. We provide several examples of stability regions in situations that interest us. For dissipative PDEs with periodic boundary conditions, the scalars c that arise with a Fourier spectral method are negative. Let us take for example BurgerвЂ™s equation ut = u xx в€’ with 0 < 1 2 u2 , x в€€ [в€’ПЂ , ПЂ ], (3.4) x 1. Transforming it to the Fourier space gives ut = в€’ Оѕ 2 u в€’ iОѕ 2 u2 , в€ЂОѕ, (3.5) where Оѕ is the Fourier wave-number and the coeп¬ѓcients c = в€’ Оѕ 2 < 0 span over a wide range of values when all the Fourier modes are considered. For high values of |Оѕ |, the solutions are at1. tracted to the slow manifold quickly because c < 0 and |c | In Fig. 1 we draw the boundary stability regions in the complex plane x, for y = 0, в€’0.9, в€’5, в€’10, в€’18, which are similar to Figs. 3.2, 3.3 and 3.4 of [9]. When the linear part is zero ( y = 0), we recognized the stability region of the fourth-order RungeвЂ“Kutta methods and, as y в†’ в€’в€ћ, the region grows. Of course, these regions only give an indication of the stability of the method. In fact, for y < 0, | y | 1 the boundaries that are observed approach to ellipses whose parameters have been п¬Ѓtted numerically with the following result Re(x) 2 + Im(x) 0.7 2 = y2 . (3.6) In Fig. 2 we draw the experimental boundaries and the ellipses (3.6) with y = в€’75. The spectrum of the linear operator increases as Оѕ 2 , while the eigenvalues of the linearization of the nonlinear part lay on the 452 F. de la Hoz, F. Vadillo / Computer Physics Communications 179 (2008) 449вЂ“456 Fig. 3. Boundary of stability regions for y = в€’10i and y = 10i. imaginary axis and increase as Оѕ . On the other hand, according to (3.6), when Re(x) = 0, the intersection with the imaginary axis Im(x) increases as | y |, i.e., as Оѕ 2 . Since the boundary of stability grows faster than x, the ETDRK4 method should have a very good behavior to solve BurgerвЂ™s equation, which conп¬Ѓrms the results of paper [14]. A similar analysis is applicable to other dissipative equations like, for instance, the KuramotoвЂ“Sivashinsky equation or the AllenвЂ“Cahn equation of [6] or [14], where h = 1/4. r (ОІ i , в€’ОІ i ) = 1, 4. Nonlinear SchrГ¶dinger equation in one space variable The nonlinear SchrГ¶dinger equation arises in a number of situations in which the envelope solutions of weakly nonlinear dispersive systems are described. The one-dimensional case is iut + u xx + q|u | p u = 0, (x, t ) в€€ (в€’в€ћ, в€ћ) Г— (0, T ], (4.1) u (x, 0) = u 0 (x), x в€€ (в€’в€ћ, в€ћ), (4.2) where q and p are real constants with p > 0 and the sign of q gives two very different problems: the focusing case if q > 0 and the defocussing one for q < 0. The initial conditions include the case where |u 0 (x)| в†’ 0 as |x| в†’ в€ћ, which we call the inп¬Ѓnite line problem, and also the periodic case where u 0 (x + L ) = u 0 (x). The solution of (4.1) satisп¬Ѓes at least two conservation laws: the L 2 norm C = u (В·, t ) 2 в€Ђt в€€ R, , and H = u x (В·, t ) 2 2 в€’ 2q p+2 u (В·, t ) p +2 , p +2 в€Ђt в€€ R. Moreover, this equation has traveling wave solutions of the form x в€’ bt therefore, the eigenvalue of the linear operator is pure imaginary and the stability regions are quite different. In Figs. 3 and 4 we draw from top to bottom and left to right the boundary stability regions for y = в€’10i, y = 10i and y = в€’75i, y = 75i, respectively. It must be understood that these regions only give an indication of the stability of the method. We observe that the stability regions include an interval of the imaginary axis, till exactly the value в€’ y. Again, using MathematicaВ© , we have checked that в€ЂОІ. (4.6) This argument gives an indication of the stability of the ETDRK4 method applied to Eq. (4.1), with the excellent results that we expose next. The experiments were carried out in our 1.60 GHz Fujitsu Siemens laptop. The time step is 0.01 and 0.001 and the maximum computer time is about 30 or 40 seconds. We would like to point out that we do not try to preserve any invariants of the equation or any phase space properties, neither to compare this method with others as it has already been made in [1,14,15] and [2], all of them having coincided in the clear superiority of ETDRK4 in front of the other methods from the point of view of both eп¬ѓciency and speed. We will rather try to reproduce with the ETDRK4 method the theoretical results gathered from Ref. [17]. First, we consider p = 2, the so-called cubic nonlinear SchrГ¶dinger equation. In this case, the traveling waves (4.3) are solitons because they retain their shape even after interaction among themselves, i.e., they act somewhat like particles; see, for example, [16] or [8]. The initial condition used is g (x) = 2ОІ q sech( ОІ x). (4.7) When the spatial part is discretized using a Fourier spectral method, transforming it to the Fourier space gives This function is not mathematically periodic, but it can be regarded as being periodic in practice because it is very close to zero at the ends of the interval. To study the interaction of two solitons, we assume that the initial condition is the superposition of the solitons with different speeds. In Fig. 5 on the left, we can check how our method reproduces the elastic collision of both solitons with the very wellknown landslide in phase. A stringent test for numerical schemes is to consider the case of a bound state of multiple solitons in which the initial condition is ut = в€’i Оѕ 2 u + iq |u | p u , u 0 (x) = sech(x), u (x, t ) = f (x в€’ ct , ОІ) exp ic 2 (4.3) , which travel with speed c; b is an arbitrary parameter, ОІ = 2c ( 2c в€’ b) and f (x, ОІ) = p+2 2q ОІ sech2 p 2 в€ЂОѕ ; 1/ p ОІx . (4.4) (4.5) (4.8) F. de la Hoz, F. Vadillo / Computer Physics Communications 179 (2008) 449вЂ“456 453 Fig. 4. Boundary of stability regions for y = в€’75i and y = 75i. Fig. 5. Collision of two solitons and q = 8. Fig. 6. Cubic SchrГ¶dinger equation with q = 18 and q = 32. which, according to (4.7), gives rise to a stationary soliton provided that q = 2, but complex phenomena may be originated for other values of q. For q = 2Q 2 , Q = 2, 3, . . . , Miles has shown in [18] that (4.8) will evolve into a bound state of Q solitons, all moving at the same speed. In Fig. 5 on the right for q = 8 and in Fig. 6 for the cases q = 18, 32, we plot the modulus of the numerical solutions with initial condition sech(x). For q = 8 the graph reaches a spike, followed by a low area and a second spike and, then, it re- turns to its initial shape to begin another period. For q = 18 the graph reaches a spike, followed by a pair of symmetric ridges, followed by a second spike; п¬Ѓnally, for q = 32 we can observe three ridges between two spikes. As a п¬Ѓnal test, we have checked the behavior of our method for the problem of the critical exponent p = 4; this means that the asymptotic behavior of the solutions of (4.1) is very different. Theorem 6.2 at p. 124 of [17] proves the extension globally 454 F. de la Hoz, F. Vadillo / Computer Physics Communications 179 (2008) 449вЂ“456 This mathematical model occurs in many physical applications; it is especially relevant in Optics (more details in [21]). This equation has been intensively studied but its behaviors are not understood completely (see [13,20,21] and their references for details). A very important question concerns the conditions of stability or instability of the standing wave solutions. It is well known that for q > 0 (focusing case), the larger wave numbers are stabilized by dispersion, while the smaller ones are linearly unstable. This case is also the one studied in [24] and, when u 0 2 is above a threshold value, the solution of (5.1) can self-focus and become singular in п¬Ѓnite time; this phenomenon is called wave collapse or blow-up of the wave amplitude. In our п¬Ѓrst test with q = 1 and the initial condition u 0 (x, y ) = 2 + 0.01 sin x + ПЂ 4 sin y + (x, y ) в€€ [в€’ПЂ , ПЂ ] Г— [в€’ПЂ , ПЂ ], ПЂ 4 , (5.2) with u 0 2 в‰€ 12.56, using (256) collocations points, we have obtained Fig. 9, which reproduces Fig. 1.1 of [21]. The computer time takes about 64.30 seconds. In the second test we try to observe the differences between q = 1 and q = в€’1 and the blow-up case with the initial condition 2 Fig. 7. Defocussing case: p = 4 and q = в€’1. in time of the local solutions under any of the following hypotheses: вЂў Defocussing case: q < 0. In Fig. 7 we can appreciate the typical dispersion. вЂў Focusing case: q > 0 and u 0 2 c 0 . Moreover, Weinstein in [24] estimates the value of the constant c 0 = П€ 2 , where П€ is a positive solution of the equation П€ в€’П€ +П€ 5 = 0 of minimal L 2 norm, i.e., П€(x) = 4 3 4 sech(x) в‰€ 0.9306 sech(x). (4.9) In Fig. 8 we can check the numerical solutions with the iniП€(x) on the left and with the initial condition tial condition в€љ 0.94 sech(x) on the right with a blow-up in п¬Ѓnite time. 5. The two-dimensional nonlinear SchrГ¶dinger equation Bearing in mind the good behavior of our simulation for the one-dimensional problem and the fact that MATLABВ© also implements higher dimensional discrete Fourier transforms and their inverses (in two variables, we have fft2 and ifft2), we thought that small modiп¬Ѓcations of our program would allow us to simulate the two-dimensional cubic nonlinear SchrГ¶dinger equation iut + u xx + u y y + q|u |2 u = 0, (x, y ) в€€ R2 , t > 0. (5.1) u 0 (x, y ) = О± В· sech x2 + y 2 , (x, y ) в€€ [в€’20, 20] Г— [в€’20, 20], (5.3) в€љ where О± is a real parameter; now, u 0 2 = О± ПЂ . In Fig. 10, we have drawn the evolution in time for О± = 5, q = 1, with u 0 2 в‰€ 8.8623. We can observe that the dispersion process dominates. On the contrary, for the case q = в€’1 shown in Fig. 11, the dominating process is the dissipative one. Finally, in Fig. 12 we have represented the evolution with О± = 6; in that case, u 0 2 в‰€ 10.6347. The computed blow-up happens quite fast, at t = 0.1; the height of the solution is about 11.89 and for t = 0.11, it is 259.90. 6. Conclusions The proposers of the ETDRK schemes in [6] concluded that they are more accurate than other methods (standard integrating factor techniques or linearly implicit schemes); they have good stability properties and are widely applicable to nonlinear wave equations. However, Cox and Matthews were aware of the numerical instability for the ETDRK4 method when computing the coeп¬ѓcients. Later, Kassan and Trefethen in [14] modiп¬Ѓed the ETDRK4 method with very good results. In the opinion of these authors, the modiп¬Ѓed ETDRK4 is the best by a clear margin compared with others methods. Fig. 8. Focusing case: p = 4 and q = 1. F. de la Hoz, F. Vadillo / Computer Physics Communications 179 (2008) 449вЂ“456 Fig. 12. Initial condition (5.3) with Fig. 9. Initial condition (5.2) and q = 1. 455 О± = 6 q = 1. We have computed and studied the numerical stability function of the ETDRK4 methods and, besides their good stability properties, we have exposed the reasons of their good behavior for dissipative and dispersive problems. In addition, we have applied this method to the SchrГ¶dinger equations (4.1) and (5.1), achieving the excellent results that we have just mentioned. It would be interesting to implement this method for non-periodic boundary condition problems in two dimensions, for which another kind of orthogonal polynomials would be required (Chebyshev, Hermite. . .), and to simulate numerically the results obtained by [13]. We think that even the incompressible NavierвЂ“Stokes equations could be simulated with the proposed method (see, for example, [3]). Recently, in [7], we simulated the blow-up of semi-linear diffusion equations in one and two spatial dimensions with very good results. Acknowledgements Fig. 10. Initial condition (5.3) with О± = 5 and q = 1. The authors would like to thank Luis Vega for fruitful conversations concerning this paper and Carlos Gorria for his help with MathematicaВ© . This work was supported by the Basque Country University with the project 9/UPV 00127.310-15969/2004, and by a scholarship for the formation of researchers (Ref. BFI02.135), given to F. de la Hoz by the Basque Autonomous Government. References Fig. 11. Initial condition (5.3) with О± = 5 q = в€’1. [1] H. Berland, B. Skaftestad, Solving the nonlinear SchrГ¶dinger equations using exponential integrators, Tech. Report 5/2005, The Norwegian University of Science and Technology, 2005. [2] H. Bernard, A. Islas, C.M. Schober, Conservation of phase properties using exponential integrators on the cubic SchrГ¶dinger equation, J. Comp. Phys. 225 (2007) 284вЂ“299. [3] G. Beylkin, J.M. Keiser, L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comp. Phys. 147 (1998) 362вЂ“387. [4] J.P. Boyd, Chebyshev and Fourier Spectral Methods, second ed. (revised), Dover, 2001. [5] C. Canuto, M.Y. Hussain, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, 1988. [6] S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys. 176 (2002) 430вЂ“455. [7] F. de la Hoz, F. Vadillo, A numerical simulation for the blow-up of semi-linear diffusion equations, Int. J. of Comp. Math., in press. [8] P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge University Press, 1988. [9] Q. Du, W. Zhu, Analysis and applications of the exponential time differencing schemes and their contour integration modiп¬Ѓcations, BIT Numerical Mathematics 45 (2005) 307вЂ“328. 456 F. de la Hoz, F. Vadillo / Computer Physics Communications 179 (2008) 449вЂ“456 [10] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, 1998. [11] B. Fornberg, T.A. Driscoll, A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comp. Phys. 155 (1999) 456вЂ“467. [12] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, 1977. [13] Z. Hongian, Z. Hongqing, A new approach for solitons of the (2 + 1)-dimensional cubic nonlinear SchrГ¶dinger equation, Chaos, Solitons and Fractals (2007), in press. [14] A. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comp. 26 (2005) 1214вЂ“1233. [15] S. Krogstag, Generalized integrating factor methods for stiff PDEs, J. Comp. Phys. 203 (2005) 72вЂ“88. [16] G.L. Lamb, Elements of Soliton Theory, John Wiley, 1980. [17] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations, IMPA, Rio de Janeiro, 2004. [18] J.W. Miles, An envelope soliton problem, SIAM J. Appl. Math. 41 (1981) 227вЂ“ 230. [19] D.R. Mott, E.S. Oran, B. van Leer, A quasi-steady state solver for stiff ODE of reaction kinetics, J. Comp. Phys. 164 (2000) 407вЂ“428. [20] H. Sakaguchi, T. Higashiuchi, Two-dimensional dark in the nonlinear SchrГ¶dinger equation, Phys. Lett. A 359 (2006) 647вЂ“651. [21] C. Sulem, P.L. Sulem, The Nonlinear SchrГ¶dinger Equation, Springer, 1999. [22] A. Taп¬‚ove, Computational Electrodynamics: The Finite Difference Time-Domain Method, Artech House, Boston, 1995. [23] Lloyd N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. [24] M.I. Weinstein, Nonlinear SchrГ¶dinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983) 567вЂ“576.

© Copyright 2024 Paperzz