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Breakdown of modulational approximations in nonlinear wave interaction
Computer Physics Communications 121–122 (1999) 420–422 www.elsevier.nl/locate/cpc
Breakdown of modulational approximations in nonlinear wave interaction F.B. Rizzato 1 Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Pôrto Alegre, Rio Grande do Sul, Brazil
Abstract In this work we investigate the validity limits of modulational approximations as a method to describe the nonlinear interaction of conservative wave fields. We focus on a nonlinear Klein–Gordon equation and suggest that the breakdown of the approximation is accompanied by a transition to regimes of spatiotemporal chaos. Integration is performed through a sympletic algorithm which is faster than explicit FFT methods. 1999 Elsevier Science B.V. All rights reserved.
Modulational instability of high-frequency nonlinear waves is present in a variety of circumstances involving wave propagation in continuous systems. One can often obtain a simplified integrable dynamics if the modulations are much slower than the carrier frequency [1,2]. Slowness can be examined from the following dimensionless nonlinear equation: ∂t2 A(x, t) − ∂x2 A(x, t) + ω2 A(x, t) − A(x, t)3 + A(x, t)5 = 0,
(1)
where ω is a linear frequency. Eq. (1) describes the interaction of electronic and ionic waves in plasmas, but different settings are governed by a similar rule. Write A(x, t) as ˜ t)eiωt + c.c. A(x, t) = A(x, Then if one assumes slowness to discard terms like ˜ one obtains ∂t2 A, ˜ t) − ∂x2 A(x, ˜ t) − 3 A(x, ˜ t) 2 A(x, ˜ t) = 0, 2iω∂t A(x, which apart from rescalings is the Nonlinear Schrödinger Equation, NLS. One can perform a stability 1 E-mail: [email protected].
analysis on the NLS by perturbing an homogeneous self-sustained state, Ah with small fluctuations of a given wavevector k. One concludes that the system √ 2ωρ, where Ah = becomes unstable when k < √ 2ρ cos(2ρt). When unstable, the homogeneous state typically evolves towards a state populated by solitons. The integrable approximation is obtained when there is a great disparity between the time scales of the highfrequency ω and the modulational frequency, Ω, such that Ω ω. Ω can be estimated from an order of magnitude analysis of NLS; ω∂t A˜ ∼ k 2 A˜ − A˜ 3 . For 2 . Therefore, the modulational ˜ Ω/ω ∼ (A/ω) ˜ k ∼ A, approximation is valid when A˜ ω. The natural question is on the breakdown of the validity. To address the issue, we start from the basic equation, Eq. (1), and examine what happens as the ratio ˜ A/ω grows from small values compared to the unity. Our first task is to examine how the modulational instability arises. To perform the analysis, we write the solution as, A = Ah (t) + A1 (t) eikx + e−ikx , Ah,1 real. Ah is spatially homogeneous and A1 depends on space through the wave vector k. After some
0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 3 7 2 - 0
F.B. Rizzato / Computer Physics Communications 121–122 (1999) 420–422
421
Fig. 1. Low-dimensional Poincaré plots on the projected phase-space (p1 , q1 ). ω = 0.1; Ah,max = ω/10 (a), and Ah,max = ω (b). k = Ah,max /2. Ah,max : maximum initial Ah out of 20 plotted orbits.
Fig. 2. Full spatio-temporal simulations using the same parameters as in Fig. 1, and Ah,0 /ω = 0.1 (a) and Ah,0 /ω = 1.0 (b). A(x, t) in the vertical axes.
lengthy algebra, one finds out that the coupled nonlinear dynamics of the fields Ah and A1 is governed by the Hamiltonian p02 ω2 q02 q04 2q06 p12 + − + + 2 2 2 3 2 χ 2 q12 3q14 5q16 − + + 2 4 3 (2) − 3q02 q12 + 10q04q12 + 15q02q14 , √ χ 2 ≡ ω2 + k 2 , q0 = Ah / 2, q1 = A1 and where the p are conjugate to the q. One can now determine the stability properties of the homogeneous pump, as we had mentioned before. To see this, assume that in average, q0 q1 and solve the dynamics H=
perturbatively with help of action-angle variables. One obtains the resonant form h1,r = (k 2 /ω − ρ)I /2 + ρI /2 cos(2φ), with φ = θ − (ω − 3/2ρ)t, (p1 , χq12 ) = √ 2I (cos θ, sin θ ), for χ ≈ ω + k 2 /(2ω) and k ω. From the resonant form, instability requires k < √ 2ωρ, which is nothing else but the modulational result anticipated earlier. If the reduced dynamics is unstable but of the regular type, it is likely that in a full spatio-temporal simulation we see only a series of solitons, or soliton-like structures, periodically formed. We consider then ω = 0.1 and make, in Fig. 1, Poincaré plots based on Hamiltonian (2) [3]. When Ah,0 ω (Ah,0 ≡ Ah (t = 0)) where the modulational approximation is good, one has a plot like that of
422
F.B. Rizzato / Computer Physics Communications 121–122 (1999) 420–422
√ Fig. 1(a) where Ah,0 /ω = 0.1; k = ωρ/2 here and in all other cases, which sets up the maximum growth rate situation. It is seen that the point q1 = p1 = 0 is unstable hyperbolic, but the orbital motion is regular. Now, in the case of Fig. 1(b) where Ah,0 /ω = 1.0, the dynamics is mostly chaotic. Now to the full simulation of Eq. (1). The simulations are made through a discretization of the spatial domain via a finite difference method. The dynamics is resolved in time by means of a symplectic integrator. The results are quite robust and energy is conserved to 10−5 parts in one. Our results are displayed in Fig. 2. In the case of Fig. 2(a) where Ah,0 /ω = 0.1, it is seen that the spatio-temporal dynamics is unstable but regular as predicted by the NLS and by Fig. 1(a). As we go to other extreme, Ah,0 /ω = 1, Fig. 1(b) indicates full, widespread chaos. Accordingly, the full simula-
tions shown in Fig. 2(b) display a highly disorganized space-temporal chaotic dynamics.
Acknowledgments This work was partially supported by FINEP and CNPq, and CESUP-UFRGS, Brazil. References [1] F.B. Rizzato, G.I. de Oliveira, R. Erichsen, Phys. Rev. E 57 (1998) 2776; R. Erichsen, G.I. de Oliveira, F.B. Rizzato, Phys. Rev. E (1998) accepted. [2] S.R. Lopes, F.B. Rizzato, Physica D 117 (1998) 13. [3] A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Motion (Springer, 1991).
Breakdown of modulational approximations in nonlinear wave interaction F.B. Rizzato 1 Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Pôrto Alegre, Rio Grande do Sul, Brazil
Abstract In this work we investigate the validity limits of modulational approximations as a method to describe the nonlinear interaction of conservative wave fields. We focus on a nonlinear Klein–Gordon equation and suggest that the breakdown of the approximation is accompanied by a transition to regimes of spatiotemporal chaos. Integration is performed through a sympletic algorithm which is faster than explicit FFT methods. 1999 Elsevier Science B.V. All rights reserved.
Modulational instability of high-frequency nonlinear waves is present in a variety of circumstances involving wave propagation in continuous systems. One can often obtain a simplified integrable dynamics if the modulations are much slower than the carrier frequency [1,2]. Slowness can be examined from the following dimensionless nonlinear equation: ∂t2 A(x, t) − ∂x2 A(x, t) + ω2 A(x, t) − A(x, t)3 + A(x, t)5 = 0,
(1)
where ω is a linear frequency. Eq. (1) describes the interaction of electronic and ionic waves in plasmas, but different settings are governed by a similar rule. Write A(x, t) as ˜ t)eiωt + c.c. A(x, t) = A(x, Then if one assumes slowness to discard terms like ˜ one obtains ∂t2 A, ˜ t) − ∂x2 A(x, ˜ t) − 3 A(x, ˜ t) 2 A(x, ˜ t) = 0, 2iω∂t A(x, which apart from rescalings is the Nonlinear Schrödinger Equation, NLS. One can perform a stability 1 E-mail: [email protected].
analysis on the NLS by perturbing an homogeneous self-sustained state, Ah with small fluctuations of a given wavevector k. One concludes that the system √ 2ωρ, where Ah = becomes unstable when k < √ 2ρ cos(2ρt). When unstable, the homogeneous state typically evolves towards a state populated by solitons. The integrable approximation is obtained when there is a great disparity between the time scales of the highfrequency ω and the modulational frequency, Ω, such that Ω ω. Ω can be estimated from an order of magnitude analysis of NLS; ω∂t A˜ ∼ k 2 A˜ − A˜ 3 . For 2 . Therefore, the modulational ˜ Ω/ω ∼ (A/ω) ˜ k ∼ A, approximation is valid when A˜ ω. The natural question is on the breakdown of the validity. To address the issue, we start from the basic equation, Eq. (1), and examine what happens as the ratio ˜ A/ω grows from small values compared to the unity. Our first task is to examine how the modulational instability arises. To perform the analysis, we write the solution as, A = Ah (t) + A1 (t) eikx + e−ikx , Ah,1 real. Ah is spatially homogeneous and A1 depends on space through the wave vector k. After some
0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 3 7 2 - 0
F.B. Rizzato / Computer Physics Communications 121–122 (1999) 420–422
421
Fig. 1. Low-dimensional Poincaré plots on the projected phase-space (p1 , q1 ). ω = 0.1; Ah,max = ω/10 (a), and Ah,max = ω (b). k = Ah,max /2. Ah,max : maximum initial Ah out of 20 plotted orbits.
Fig. 2. Full spatio-temporal simulations using the same parameters as in Fig. 1, and Ah,0 /ω = 0.1 (a) and Ah,0 /ω = 1.0 (b). A(x, t) in the vertical axes.
lengthy algebra, one finds out that the coupled nonlinear dynamics of the fields Ah and A1 is governed by the Hamiltonian p02 ω2 q02 q04 2q06 p12 + − + + 2 2 2 3 2 χ 2 q12 3q14 5q16 − + + 2 4 3 (2) − 3q02 q12 + 10q04q12 + 15q02q14 , √ χ 2 ≡ ω2 + k 2 , q0 = Ah / 2, q1 = A1 and where the p are conjugate to the q. One can now determine the stability properties of the homogeneous pump, as we had mentioned before. To see this, assume that in average, q0 q1 and solve the dynamics H=
perturbatively with help of action-angle variables. One obtains the resonant form h1,r = (k 2 /ω − ρ)I /2 + ρI /2 cos(2φ), with φ = θ − (ω − 3/2ρ)t, (p1 , χq12 ) = √ 2I (cos θ, sin θ ), for χ ≈ ω + k 2 /(2ω) and k ω. From the resonant form, instability requires k < √ 2ωρ, which is nothing else but the modulational result anticipated earlier. If the reduced dynamics is unstable but of the regular type, it is likely that in a full spatio-temporal simulation we see only a series of solitons, or soliton-like structures, periodically formed. We consider then ω = 0.1 and make, in Fig. 1, Poincaré plots based on Hamiltonian (2) [3]. When Ah,0 ω (Ah,0 ≡ Ah (t = 0)) where the modulational approximation is good, one has a plot like that of
422
F.B. Rizzato / Computer Physics Communications 121–122 (1999) 420–422
√ Fig. 1(a) where Ah,0 /ω = 0.1; k = ωρ/2 here and in all other cases, which sets up the maximum growth rate situation. It is seen that the point q1 = p1 = 0 is unstable hyperbolic, but the orbital motion is regular. Now, in the case of Fig. 1(b) where Ah,0 /ω = 1.0, the dynamics is mostly chaotic. Now to the full simulation of Eq. (1). The simulations are made through a discretization of the spatial domain via a finite difference method. The dynamics is resolved in time by means of a symplectic integrator. The results are quite robust and energy is conserved to 10−5 parts in one. Our results are displayed in Fig. 2. In the case of Fig. 2(a) where Ah,0 /ω = 0.1, it is seen that the spatio-temporal dynamics is unstable but regular as predicted by the NLS and by Fig. 1(a). As we go to other extreme, Ah,0 /ω = 1, Fig. 1(b) indicates full, widespread chaos. Accordingly, the full simula-
tions shown in Fig. 2(b) display a highly disorganized space-temporal chaotic dynamics.
Acknowledgments This work was partially supported by FINEP and CNPq, and CESUP-UFRGS, Brazil. References [1] F.B. Rizzato, G.I. de Oliveira, R. Erichsen, Phys. Rev. E 57 (1998) 2776; R. Erichsen, G.I. de Oliveira, F.B. Rizzato, Phys. Rev. E (1998) accepted. [2] S.R. Lopes, F.B. Rizzato, Physica D 117 (1998) 13. [3] A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Motion (Springer, 1991).