- Email: [email protected]
Coherent solutions for the fundamental resonance of the Boussinesq equation
Chaos, Solitons & Fractals 54 (2013) 57–64
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Coherent solutions for the fundamental resonance of the Boussinesq equation Attilio Maccari ⇑ Technical Institute Cardano, Department of Physics, Monterotondo 00015, Rome, Italy
a r t i c l e
i n f o
Article history: Received 5 July 2011 Accepted 21 May 2013 Available online 15 June 2013
a b s t r a c t We consider the Boussinesq equation in an infinite wall under an external resonant weak excitation. Two slow flow equations are obtained and amplitude and phase modulation equations as well as external force-response and frequency–response curves are determined. Energy considerations are used in order to study the global behavior of the two slow flow equations and to demonstrate the existence of closed orbits around the equilibrium points with a second frequency in addition to the forcing frequency. Two period quasi-periodic motions are present with amplitudes depending on the initial conditions. Moreover, in certain cases librations can occur with a frequency depending on the amplitude of the external excitation. If the external excitation increases, the modulation period for the slow flow equations becomes infinite and an infinite-period bifurcation occurs, while the modulation amplitude remains finite and the libration transforms into a closed orbit. Finally, we demonstrate that the separatrix of the two slow flow equations corresponds to an asymptotically steady-state and phase-locked periodic solution. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Nonlinearities and resonances are responsible for very interesting behaviors and in the last years an increasing number of studies has been performed on these subjects. The Boussinesq equation (1872) describes the propagation of small amplitude, long nonlinear waves on the surface of shallow water
utt uxx þ auxxxx bðu2 Þxx ¼ 0;
ð1Þ
where u = u(x, t) is an elevation of the free surface of fluid and the constant coefficients a and b depend on the depth of fluid and the characteristic speed of long waves. If a < 0 the Eq. (1) is called the ‘‘bad’’ Boussinesq equation and if a > 0 the ‘‘good’’ Boussinesq equation. In the following both cases (a > 0 and a < 0) are studied. Boussinesq derived the Eq. (1) in the case a < 0, while the case a > 0 describes the small nonlinear oscillations of elastic beams and is linearly stable. It is well known that the Eq. (1) was deduced earlier ⇑ Tel.: +39 69091369. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.016
than the Korteweg–de Vries equation and represents the first mathematical explanation of solitary waves discovered in 1834 by Scott Russell. Cauchy and spatially periodic problems have been studied for the Eq. (2) and the longtime asymptotics of their global in time solutions have been deduced [1–4]. In this paper we seek approximate solutions for the Boussinesq equation in an infinite wall under the action of a fundamental excitation in resonance with the frequency of a generic mode. The relevant partial differential equation is
utt uxx þ auxxxx bðu2 Þxx 2F cosðXtÞ ¼ 0;
ð2Þ
where u = u(x, t), F = F(x) and the parameters a, b and X are constant and the boundary conditions are
uð0; tÞ ¼ 0; uxx ð0; tÞ ¼ 0;
uðL; tÞ ¼ 0; uxx ðL; tÞ ¼ 0:
ð3aÞ ð3bÞ
The Boussinesq equation has been previously considered [5].
58
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
In the unforced version (F(x) = 0) of Eq. (2), the equation is integrable by the inverse scattering transform [2], while in the linear version (F(x) = b = 0) the normalized eigenmodes and natural frequencies are
Un ðxÞ ¼
rffiffiffi np 2 sin x ; L L
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np np2 xn ¼ : 1þa L L
ð4Þ
ð5Þ
The general solution for arbitrary initial conditions is obtained in the linear case through a mode superposition as
uðx; tÞ ¼
N X An cosðxn t þ #n ÞUn ðxÞ;
ð6Þ
n¼1
where An and #n are determined by the initial conditions. The most important property of linear modes is the selfsimilarity, i.e., at any instant the displacement pattern is a multiple of some basic spatial pattern, the mode shape Un(x), and the multiplying factor is a harmonic function of time. However, it is well known that the nonlinear vibration modes can deviate from those of the associated linear problem and the classical term ‘mode shape’ may not be very meaningful for nonlinear systems. We use a perturbation method illustrated in [6] and also in [7–11] in order to obtain analytic approximate solutions and, in particular, we consider the externally forced Boussinesq equation
utt uxx þ auxxxx ebðu2 Þxx 2e2 F cosðXtÞ ¼ 0;
ð7Þ
with the resonance condition xn X and the boundary conditions (3), where e is a small nondimensional parameter that is artificially introduced to serve as bookkeeping device and will be set equal to unity in the final analysis. Note that the perturbed Boussinesq equation was studied in various papers (see [12] and references therein). In Section 2 we calculate an approximate analytic solution and to avoid computational difficulties we shall limit our treatment to a first order approximation. We derive a model system of two coupled differential equations in the phase and amplitude of solutions. Fixed points of the model system correspond to periodic solutions of the fundamental resonance of the Boussinesq equation (2). We find frequency–response and external force-response curves, which include the results of stability analysis. Three steps are used by the AP method: (i) obtaining the form of solution in terms of harmonic components, (ii) introducing a slow time scale, and (iii) solving directly for the various harmonic components via harmonic balance. A systematic means is then provided to derive increasingly accurate solutions by increasing the order of approximation in terms of the small parameter e. In the framework of the AP method, no assumptions are made a priori on the form of the spatial part of the solution. The principal advantage is the treatment of the boundary conditions at higher orders and moreover the predicted behavior is not self-similar. We demonstrate the existence of stable periodic solutions and jump phenomenona both for the ‘good’ and the ‘bad’ Boussinesq equation.
In Section 3, a global analysis of the model system by energy considerations is performed and we demonstrate the existence of closed orbits, corresponding to modulated motions for the Boussinesq equation (2). Another type of modulated motion (libration) can arise also for low values of the parametric excitation. In this case the two-period quasiperiodic motion is characterized by a modulation of the fundamental oscillation, with a modulation amplitude proportional to the magnitude of the external excitation. In certain cases an infinite-period bifurcation may occur, because if we increase the external excitation, while the modulation amplitude remains finite, the modulation period lengthens and becomes infinite and then librations disappear and are substituted by closed orbits. Finally we demonstrate that the separatrix of the model system corresponds to an asymptotically steady-state phase-locked periodic solution for the Boussinesq equation. The last section is devoted to recapitulate the most important results and indicate some possible generalizations. 2. The approximate solution for the fundamental resonance of the nth mode We consider the externally excited Boussinesq equation in an infinite wall and in order to express the nearness of the excitation frequency to the natural frequency, we define a detuning parameter r through the relation
xn ¼ X þ e2 r:
ð8Þ
We introduce the slow time
s ¼ e2 t;
ð9Þ
because we need to look on larger time scales, in order to obtain a non negligible contribution by nonlinear and excitation terms. The solution u(x, t) of Eq. (7) with the boundary conditions (3) can be expressed by means of a power series in the expansion parameter e,
uðx; tÞ ¼
þ1 X
ecm wm ðx; s; eÞ expðimXtÞ;
ð10Þ
m¼1
where cm = jm 1j for m – 0, c0 = r and wm ðx; s; eÞ ¼ w m ðx; s; eÞ. Note that r is the order of the zero-mode term and can be set equal to 1 for the self-consistency of the perturbation method. Eq. (10) can be written more explicitly (r = 1 and c1 = 0)
uðx; tÞ ¼ ew0 ðx; s; eÞ þ ðw1 ðx; s; eÞ expðiXtÞ þ ew2 ðx; s; eÞ expð2iXtÞ þ c:c:Þ þ h:o:t:;
ð11Þ
where h.o.t. = higher order terms and c.c. stands for complex conjugate of the preceding terms. The functions wm(x, s, e) depend on the parameter e and we suppose that the limit of the functions wm for e ? 0 exists and is finite and moreover they can be expanded in power series of e, i.e.
wm ðx; s; eÞ ¼
1 X
ei wmðiÞ ðx; sÞ:
i¼0
ð12Þ
59
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
In the following for simplicity we use the abbreviations ð0Þ wð0Þ m ¼ wm m – 1 for and w1 ¼ w for m = 1. In the lowest order calculations, only the functions corresponding to i = 0 appear. We have assumed only one mode in the expansion, because we suppose that the system is excited near the natural frequency of a specific linear mode and that mode is not involved in an internal resonance with any other mode. After inserting the assumed solution (10) in the starting Eq. (7), we obtain some equations involving the coefficients of every harmonic exp(imXt). For m = 1 we obtain the linear equation
wxx awxxxx þ x2n w ¼ 0; ðorder wð0; sÞ ¼ 0;
e0 Þ;
wðL; sÞ ¼ 0;
2
a8 ¼ a11
2
! 2 a11 ðeð2a6 LÞ 1Þða26 þ a29 þ 4k Þ : þ 2 2 sinhða6 LÞ ða6 þ a29 Þ sinða9 LÞ ð25Þ 2
ð13bÞ
ð14Þ
2
eðw2;xx aw2;xxxx þ 4X w2 Þ ¼ ebW U2n þ Oðe2 Þ; 2
2 n
ð15Þ
2
eðw0;xx aw0;xxxx Þ ¼ 2ebjWj U þ Oðe Þ
ð16Þ
where w0 and w2 satisfy the same boundary conditions as w (Eq. (3)). The solutions of the linear Eqs. (15) and (16) are easily derived
w2 ðx; sÞ ¼ bg 2 ðxÞW2 ðsÞ;
dW 2bðw0 WUn ðxÞ þ w2 W Un ðxÞÞxx ds þ2rXWUn ðxÞ F ¼ 0:
Making use of the normalization of the eigenmodes and with the definition
fn ¼
Z
g 0 ðxÞ ¼ a1 þ a2 sinhða4 xÞ þ a2 sinhða4 ðL xÞÞ þ a3 cosð2kxÞ;
i
dW fn rW NjWj2 W þ ¼ 0; ds 2X
2
N ¼ b ð2G0 þ G2 Þ; G0 ¼
ð19Þ
and 2
a3 ¼
2
Lð1 þ 4ak Þ
;
a2 ¼
4ak a3 ; sinhða4 LÞ
ð20Þ
2
ð21Þ 2
4k a29 1 eða6 LÞ 4k a5 ¼ a11 þ 1 2 sinhða6 LÞ a26 þ a29 a26 þ a29 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t1 1 4X2 ; þ þ a6 ¼ 2 2a 4a a ! 2 eða6 LÞ 1 4k a7 ¼ a11 1 ; 2 sinhða6 LÞ a26 þ a29 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t 1 1 4X2 ; a9 ¼ þ þ 2 2a 4a a
ð28Þ
Z
ð30Þ
Un ðxÞðg 2 ðxÞUn ðxÞÞxx dx:
ð31Þ
L
The effective nonlinearity coefficient depends on the overall spatial dependence of the motion. Expressing the complex-valued function into real and imaginary parts, we obtain
ð32Þ
and we arrive at the model equations
dq fn ¼ sin #; ds 2X
!!
ð29Þ
Un ðxÞðg 0 ðxÞUn ðxÞÞxx dx;
WðsÞ ¼ qðsÞ expði#ðsÞÞ
rffiffiffi 1 ; a4 ¼ a
ð27Þ
L
0
þ a10 sinða9 xÞ þ a11 cosð2kxÞ
2
Z 0
G2 ¼
þ a8 cosða9 xÞ
a1 ¼ a3 ð1 þ 4ak Þ;
FðxÞUn ðxÞdx;
we arrive at the ordinary differential equation
ð18Þ þ a7 e
L
0
where (for simplicity we show them in the case a > 0)
ða6 xÞ
ð26Þ
where
w0 ðx; sÞ ¼ 2bg 0 ðxÞjWðsÞj2 ; ð17Þ
1
2
þ ða29 4k Þeða6 LÞ 4k Þ
2iXUn ðxÞ
2
np ; L
ð24Þ
2
If we consider Eq. (7) for m = 1 and at the order e , then we find that the function W(s) satisfies the nonlinear differential equation
where Un(x) is given by Eq. (4) and W(s) is an arbitrary function of time. Considering Eq. (7) for m = 2 and m = 0 yields
k¼
1 3ak L
ð13aÞ
wðx; sÞ ¼ Un ðxÞWðsÞ;
g 2 ðxÞ ¼ a5 e
a11 ¼
a11 2 ðða2 þ 4k Þ cosða6 LÞ ða26 þ a29 Þ sinða9 LÞ 6
a10 ¼
whose solution is
ða6 xÞ
a26 þ 4k ; a26 þ a29
ð33Þ
;
q ð22Þ
d# N fn ¼ q3 rq þ cos #: ds X 2X
ð34Þ
The Eqs. (33) and (34) can be found in standard books about perturbation methods [6–21], where the primary resonance is carefully studied. Note that from Eqs. (11) and (32) we can express the field to the second approximation as
uðx; tÞ ¼ 2Un ðxÞqðtÞ cosðXt #Þ 2bq2 ðtÞðg 0 ðxÞ ð23Þ
þ g 2 ðxÞ cosð2Xt 2#ÞÞ; where q and # are given by Eqs. (33) and (34).
ð35Þ
60
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
The validity of the approximate solution should be expected to be restricted on bounded intervals of the s-variable and on time-scale t = O(1/e2). If one wishes to construct solutions on intervals such that s = O(1/e), then the higher terms must be included, because they will in general affect the solution. Periodic solutions of the complete system described by Eq. (7) correspond to the fixed points of Eqs. (33) and (34), which are obtained by the conditions dq/ds = d#/ds = 0. We find the equilibrium points (steady state solutions) q0, #0,
q30 þ
#0 ¼ 0 or #0 ¼ p;
rX N
q20
fn ¼ 0; 2N
ð36Þ
where the minus sign corresponds to #0 = 0 and the plus sign to #0 = p. In order to establish the stability of steady state solutions, we superpose small perturbations in the amplitudes and the phases on the steady state solutions and the resulting equations are then linearized. Subsequently we consider the eigenvalues of the corresponding system of first order differential equations with constant coefficients (the Jacobian matrix). The eigenvalue equation is 2
k þ Q ¼ 0:
ð37Þ
Fig. 1. Amplitude of the response (q) as function of the external excitation (f) of the first mode. Solid lines stand for stable solutions.
if F n < 3p4 ffiffi3, then we obtain three equilibrium points, given by (Fig. 2)
pffiffiffi p u 3 ðq1E ; #1E Þ ¼ 2 R cos ;p 3
where
Q¼
rþ
3Nq20
X
rþ
N q20
and then only saddle (Q < 0) or elliptic (Q > 0) points occur for the model system (33) and (34). We observe that the modulation Eqs. (33) and (34) have only two independent parameters, because trough the rescaling
q ! Lq; s ! T s;
Lfn fn ! ; T
pffiffiffi u 3 ;0 ; ðq2E ; #2E Þ ¼ 2 R cos 3 pffiffiffi u 4 3 ðq3E ; #3E Þ ¼ 2 R cos þ p ;0 ; 3 3
ð38Þ
X
sffiffiffiffiffiffiffiffiffiffi jrjX L¼ ; jNj
T¼
1 jrj
;
ð42Þ ð43Þ
ð44Þ
where
pffiffiffi 3 3F n cos u ¼ ; 4
1 R ¼ pffiffiffi : 3 3
ð45Þ
ð39aÞ we can always set
N
X
¼ 1;
f
r ¼ 1; F n ¼ n :
ð39bÞ
X
We distinguish four cases: (i) N/X = 1, r = 1: there is only an elliptic equilibrium point, given by (Fig. 1)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u u u u 2 3 3 F n 1 tF n F 2n 1 C B tF n ðqE ; #E Þ ¼ @ þ þ þ þ ; pA 4 16 27 4 16 27 ð40Þ 4 ffiffi p 3 3
(ii) N/X = 1, r = 1: if F n > there is only an elliptic equilibrium point, given by (Fig. 2)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u u u u 2 3 3 F n 1 tF n F 2n 1 C B tF n ðqE ; #E Þ ¼ @ þ þ ; pA 4 16 27 4 16 27
ð41Þ
Fig. 2. Amplitude of the response (q) as function of the external excitation (f) of the first mode. Solid lines stand for stable solutions, dashed lines for unstable solutions.
61
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
The equilibrium point given by Eq. (43) is a saddle point, while the other two equilibrium points given by Eqs. (42) and (44) are elliptic points. In Fig. 2, we observe an interesting behavior, because if we consider the smaller elliptic point and increase the amplitude of the external force, suddenly in correspondence with the critical value F n ¼ 3p4 ffiffi3, the solution jumps into the second elliptic point. (iii) N/X = 1, r = 1: there is only an elliptic equilibrium point, given by Eq. (40) but with #E = 0. (iv) N/X = 1, r = 1: if F n > 3p4 ffiffi3 there is only an elliptic point, given by Eq. (41) but with #E = 0; if F n < 3p4 ffiffi3 there are three equilibrium points, the first (saddle) and the second (elliptic) given by Eqs. (43) and (44) but with #E = p, and the third, which is always elliptic, given by Eq. (42) but with #E = 0. The last two situations are connected by a simple temporal inversion (and the transformation # ? p + #) to the first two cases. A jump phenomenon is clearly observable in Fig. 2 because for increasing values of the external excitation we observe a discontinuous transition from the low amplitude solution to the high amplitude steady-state periodic response. When f < 0.78, the response depends on the initial conditions and the system tends asymptotically to oscillate around the smaller steady-state periodic response or to the larger one. If we want to observe the jump phenomenon we must vary f from zero and the response will be the low amplitude solution if the initial conditions correspond to an amplitude q near the low amplitude and until f reaches 0.78. A small increase in f beyond 0.78 causes an upward jump to the high amplitude steady-state response. As f increases further, the solution amplitude continues to increase slowly (hysteresis effect). The frequency–response equation follows from Eq. (34), 2
r¼
Nq
X
fn : 2Xq
ð46Þ
In Figs. 3 and 4 we show the frequency–response curves (as usual, solid lines stand for stable solutions and dashed lines for unstable solutions). We see in Fig. 3 that only a steady-state periodic response is possible, on the contrary one, two or three solutions are possible in Fig. 4, depending on the value of r. If r < 0.010, only a stable solution is possible. If 0.010 < r < 0.003, there are three possible solutions: an intermediate unstable solution and tow stable, low and high amplitude, solutions. For r > 0.003 only the stable high amplitude solution is possible. 3. Modulated motion Using the global analysis [11–18] we investigate the model system (33) and (34) and demonstrate the existence of closed orbits, which correspond to two-period quasiperiodic solutions of the Boussinesq equation (7). We find the following integral of motion
3 Nq fn Hðq; #Þ ¼ q þ rq cos # : 2X X
ð47Þ
Fig. 3. The frequency–response curve for the fundamental resonance of the first mode, i.e., the response (q) as function of the detuning (r). Solid line stands for stable solutions.
Fig. 4. The frequency–response curve for the fundamental resonance of the first mode, i.e., the response (q) as function of the detuning (r). Solid line stands for stable solutions, dashed lines for unstable solutions.
As a consequence the energy-like function H(q, #) is constant along the solution curves. In Figs. 5 and 6 we show different solution curves in the plane (q, #) and along each solution curve the energy-like function H(q, #) of the Eqs. (33) and (34) is constant. The case F n < 3p4 ffiffi3 is represented in Fig. 5 and the case F n > 3p4 ffiffi3 in Fig. 6. The frequency of the small oscillations around the elliptic point in the plane (q, #) is
X1 ¼
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3Nq20 N q2 rþ rþ 0 :
X
X
ð48Þ
The periodic motion around the elliptic point for the Eqs. (33) and (34) corresponds to a two period quasi-periodic motion for the Boussinesq equation (7). If we perform a linearization of Eqs. (33) and (34) near the elliptic point, we can write explicitly the solution for the small oscillations
qðtÞ ¼ qE þ ðqI qE Þ cosðX1 tÞ þ sinðX1 tÞ;
fn ð#I #E Þ cos #E 2XX1 ð49Þ
62
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
ut ðx; 0Þ ¼ u_ 0 ðxÞ ¼ 2XUn ðxÞqI sinð#I Þ 4Xbq2I g 2 ðxÞ sinð2#I Þ;
ð52Þ
The solution is then characterized by two frequencies and only if qI = qE, #I = #E, the solution is simply periodic and corresponds to the elliptic equilibrium point. Another interesting possibility which can be easily treated is the case of the libration
N 3 q þ rq fn ; I X I 2X
ð53Þ
because we find that the Eqs. (33) and (34) yield
e t; #ðtÞ ¼ #I X
qðtÞ ¼ qI
ð54Þ
e t þ #I ÞÞ fn ðcos #I cosð X ; e 2X X
ð55Þ
where
N
e ¼ q2 þ r : X X I Fig. 5. Phase space representation of solutions for the system of Eqs. (33) and (34) in the case (iii). Only an elliptic points is present.
ð56Þ
e The solution is quasi-periodic with the frequencies 1 and X and is always given by the Eq. (35). The transition from oscillation around the elliptic point and the libration occurs through an infinite-period bifurcation. For very low value of the external excitation we have demonstrated the existence of a modulated motion with frequency (56). A better approximation for the modulation b can be obtained directly from Eq. (34), after frequency X substituting q with qI (being the modulation of q(t) very small, see Eq. (55)),
2p Te ¼ ¼ e X
Z 2p 0
2XqI d# fn2 cos # 2XqI r 2Nq3i
4pXqI ffi: ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 4qi Xr þ Nq2i fn2
ð57Þ
The period increases with fn and diverges as fn approaches
fC ¼ 2qI Xr þ N q2I . We can estimate the order of divergence by noting that
pffiffiffi p 2 2XqI e T pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; fC fC fn
Fig. 6. Phase space representation pffiffiffiof solutions for the system of Eqs. (33) and (34) in the case (iv), F < 4=3 3. Two elliptic points and a saddle point are present.
#ðtÞ ¼ #E þ ð#I #E Þ cosðX1 tÞ
ð3Nq2E þ XrÞðqI qE Þ
qE XX1
sinðX1 tÞ;
ð50Þ
where qI, #I are connected to the initial conditions trough the following relations
uðx; 0Þ ¼ u0 ðxÞ ¼ 2Un ðxÞqI cosð#I Þ 2bq2I ðg 0 ðxÞ þ g 2 ðxÞ cosð2#I ÞÞ;
ð51Þ
ð58Þ
e blows up following a square-root scalwhich shows that T ing law while the amplitude of motion remains finite (Fig. 7). For the Boussinesq equation (7) we observe a modulated motion with the carrier frequency and the modulae ¼ 2p= T e . When fn is slightly less than fC, tion frequency X the fixed points of Eqs. (33) and (34) make themselves felt because they slow the passage through the ‘bottleneck’ near them. If f continues to increase then an infinite-period bifurcation occurs, because the modulation period lengthens and becomes infinite. After this point, the asymptotic motion is governed by the steady state response given by an oscillation around the elliptic point of the system (33) and (34) while the asymptotic motion corresponds to a two-period quasiperiodic behavior for the Boussinesq equation (7).
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
63
The solution of the Boussinesq equation (7) is then for large times
uðx; tÞ ¼ 2Un ðxÞq2E cosðXt #2E Þ 2bq22E ðg 0 ðxÞ þ g 2 ðxÞ cosð2Xt 2#2E ÞÞ;
ð64Þ
i.e. a phase-locked periodic solution corresponding to the saddle point. Another particular case is the perfect resonance with the external excitation (r = 0). The rescaling (39) cannot be introduced and we simply consider the Eqs. (33) and (34) with r = 0. We find only an elliptic equilibrium point, given by
rffiffiffiffiffiffiffi 3 fn ; qE ¼ 2N
#E ¼ 0;
ð65Þ
while the frequency of the small oscillations is
Fig. 7. The effective potential corresponding to a solutions along the separatrix. Note that the maximum value is zero. A material point with energy equal to zero reaches the maximum in an infinite time.
Global analysis of Eqs. (33) and (34) can be also performed through the study of the motion of a material point in an appropriately chosen potential. In fact, Eqs. (33), (34) and (47) yield after a lengthy calculation
ðRs Þ2 H2 þ VðRÞ ¼ ; 2 2
R ¼ q2 ;
ð59Þ
i.e. the temporal evolution of R is equivalent to that of a material point (mass = 1) with energy given by H2/2 in the potential
f2 HN R2 NrR3 VðRÞ ¼ rH þ n 2 R þ r2 þ X 2 2X 2X þ
N 2 R4
ð60Þ
8X2
and taking into account Eqs. (39) we find
VðRÞ ¼ s2 H þ
! F 2n R2 s1 s2 R3 R4 R þ ð1 s1 HÞ þ þ ; 2 2 2 8
ð61Þ
where
Fn ¼
fn
X
;
s1 ¼ sgnðN=XÞ;
s2 ¼ sgnðrÞ:
ð62Þ
We consider now the important case of the motion along the separatrix of Eqs. (33) and (34), in this case Eq. (61) yields
5 1 Rs 3 2 VðRÞ ¼ Rs 1 Rs þ 2R2s R þ þ Rs 2 2 2 4 R2
R3 R4 þ ; 2 8
Rs ¼ q22E ;
ð63Þ
and q2E is given by Eq. (43). The equivalent energy H2/2 is equal to the potential of the saddle point and if the initial conditions are chosen along the separatrix then in the limit of large times the motion arrive at the saddle point.
X1 ¼
pffiffiffi 3Nq20
X
:
ð66Þ
Note that it coincides with the case r = 0 of the Eq. (48). The approximate analytic solution (49) and (50) with the relations (51) and (52) for the initial conditions is also valid, after taking r = 0. 4. Conclusion We have investigated the properties of the approximate analytic solutions to the externally excited Boussinesq equation in an infinite wall. No assumptions are made a priori on the spatial dependence of the motion. We have employed the asymptotic perturbation method that is able to calculate with excellent precision the behavior of a weakly nonlinear partial differential equation, in resonance or in quasi-resonance with an external force. A system of nonlinear model equations describing the modulation of the amplitude and of the phase of the oscillation has been derived and the position of the equilibrium points and their dependence on the external force can be easily deduced. It is also worthwhile to notice that the AP method furnishes a way to go beyond the first approximation and to calculate the solution with an arbitrarily large increase of precision. We have shown external force-response curves and observed the presence of jump phenomena. We have demonstrated that in certain cases (oscillations around the elliptic point or librations) a second low frequency appears in addition to the forcing frequency and then stable two period quasi-periodic motions are present with amplitudes depending on the initial conditions. The value of the low frequency depends on the initial conditions. Librations are also possible with a frequency depending on the external periodic excitation. Moreover, an infinite-period bifurcation is possible when the external excitation is increased and the initial conditions are appropriately chosen, because the modulation period of the libration lengthens and becomes infinite, while the attractor settles down into a periodic motion (oscillation around the elliptic point) for the system model Eqs. (33) and (34).
64
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
Possible extensions and generalizations can be accomplished in many different ways. We can study the damping effects that has been not considered in the Boussinesq equation (6) and is the main cause of the quasi-periodic observed behavior. Finally, other nonlinear equations can be analyzed, for example the Kadomtsev–Petviashvili or Davey–Stewartson equation. References [1] Boussinesq J. Theorie des ondes et de remous qui se propagent le long d’un canal rectangulaire horizontal . . .. J Math Pures Appl 1872;17(Series 2):55–108. [2] Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press; 1991. [3] Guenther RB, Lee JW. Partial Differential Equations of Mathematical Physics and Integral Equations. NJ: Prentice Hall; 1988. [4] Maccari A. The Boussinesq equation as a source of equations integrable by the spectral transform. Nonlinearity 1997;10(4):849–56. [5] Maccari A. Periodic behavior for the parametrically excited Boussinesq equation. Chaos Solitons Fractals 2004;22(1):25–34. [6] Nayfeh A. Perturbation Methods. New York: Wiley-Interscience; 1973. [7] Maccari A. Non-resonant interacting ion acoustic waves in a magnetized plasma. Journal of Physics 1999;32A(4):693–700.
[8] Maccari A. Approximate particle-like solutions for a nonlinear relativistic scalar complex field model in 3 + 1 dimensions. Physics Letters 2000;276A:79–90. [9] Maccari A. Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Liénard systems. Int J NonLinear Mech 2000;35:239–62. [10] Maccari A. Non-resonant interacting waves for the nonlinear Klein– Gordon equation in three-dimensional space. Physica D 2000;135(3–4):331–44. [11] Maccari A. Non-resonant interacting water waves in 2 + 1 dimensions. Chaos Solitons Fractals 2002;14(1):105–16. [12] Gorbatova NK, Kiselev OM, Glebov SG. Resonant excitations of nonlinear waves. AIP Conf Proc 2006;834:196–205. [13] Hurewicz W. Lectures on Ordinary Differential Equations. Cambridge, MA: MIT Press; 1958. [14] Cesari L. Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. New York: Academic; 1963. [15] Chowand S-N, Hale JR. Methods of Bifurcation Theory. New York: Springer-Verlag; 1982. [16] Jordan DW, Smith P. Nonlinear Ordinary Differential Equations. second ed. Oxford: Oxford University Press; 1987. [17] Grimshaw R. Nonlinear Ordinary Differential Equations. Oxford: Blackwell; 1990. [18] Perko L. Differential Equations and Dynamical Systems. New York: Springer; 1991. [19] Strogatz SH. Nonlinear Dynamics and Chaos. Reading, MA: AddisonWesley; 1994. [20] Nayfeh AH, Balachandran B. Applied Nonlinear Dynamics. New York: John Wiley; 1995. [21] Bogoliubov NN, Mitropolski YA. Asymptotic Methods in the Theory of Non-Linear Oscillations. New York: Gordon and Breach; 1961.
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Coherent solutions for the fundamental resonance of the Boussinesq equation Attilio Maccari ⇑ Technical Institute Cardano, Department of Physics, Monterotondo 00015, Rome, Italy
a r t i c l e
i n f o
Article history: Received 5 July 2011 Accepted 21 May 2013 Available online 15 June 2013
a b s t r a c t We consider the Boussinesq equation in an infinite wall under an external resonant weak excitation. Two slow flow equations are obtained and amplitude and phase modulation equations as well as external force-response and frequency–response curves are determined. Energy considerations are used in order to study the global behavior of the two slow flow equations and to demonstrate the existence of closed orbits around the equilibrium points with a second frequency in addition to the forcing frequency. Two period quasi-periodic motions are present with amplitudes depending on the initial conditions. Moreover, in certain cases librations can occur with a frequency depending on the amplitude of the external excitation. If the external excitation increases, the modulation period for the slow flow equations becomes infinite and an infinite-period bifurcation occurs, while the modulation amplitude remains finite and the libration transforms into a closed orbit. Finally, we demonstrate that the separatrix of the two slow flow equations corresponds to an asymptotically steady-state and phase-locked periodic solution. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Nonlinearities and resonances are responsible for very interesting behaviors and in the last years an increasing number of studies has been performed on these subjects. The Boussinesq equation (1872) describes the propagation of small amplitude, long nonlinear waves on the surface of shallow water
utt uxx þ auxxxx bðu2 Þxx ¼ 0;
ð1Þ
where u = u(x, t) is an elevation of the free surface of fluid and the constant coefficients a and b depend on the depth of fluid and the characteristic speed of long waves. If a < 0 the Eq. (1) is called the ‘‘bad’’ Boussinesq equation and if a > 0 the ‘‘good’’ Boussinesq equation. In the following both cases (a > 0 and a < 0) are studied. Boussinesq derived the Eq. (1) in the case a < 0, while the case a > 0 describes the small nonlinear oscillations of elastic beams and is linearly stable. It is well known that the Eq. (1) was deduced earlier ⇑ Tel.: +39 69091369. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.016
than the Korteweg–de Vries equation and represents the first mathematical explanation of solitary waves discovered in 1834 by Scott Russell. Cauchy and spatially periodic problems have been studied for the Eq. (2) and the longtime asymptotics of their global in time solutions have been deduced [1–4]. In this paper we seek approximate solutions for the Boussinesq equation in an infinite wall under the action of a fundamental excitation in resonance with the frequency of a generic mode. The relevant partial differential equation is
utt uxx þ auxxxx bðu2 Þxx 2F cosðXtÞ ¼ 0;
ð2Þ
where u = u(x, t), F = F(x) and the parameters a, b and X are constant and the boundary conditions are
uð0; tÞ ¼ 0; uxx ð0; tÞ ¼ 0;
uðL; tÞ ¼ 0; uxx ðL; tÞ ¼ 0:
ð3aÞ ð3bÞ
The Boussinesq equation has been previously considered [5].
58
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
In the unforced version (F(x) = 0) of Eq. (2), the equation is integrable by the inverse scattering transform [2], while in the linear version (F(x) = b = 0) the normalized eigenmodes and natural frequencies are
Un ðxÞ ¼
rffiffiffi np 2 sin x ; L L
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np np2 xn ¼ : 1þa L L
ð4Þ
ð5Þ
The general solution for arbitrary initial conditions is obtained in the linear case through a mode superposition as
uðx; tÞ ¼
N X An cosðxn t þ #n ÞUn ðxÞ;
ð6Þ
n¼1
where An and #n are determined by the initial conditions. The most important property of linear modes is the selfsimilarity, i.e., at any instant the displacement pattern is a multiple of some basic spatial pattern, the mode shape Un(x), and the multiplying factor is a harmonic function of time. However, it is well known that the nonlinear vibration modes can deviate from those of the associated linear problem and the classical term ‘mode shape’ may not be very meaningful for nonlinear systems. We use a perturbation method illustrated in [6] and also in [7–11] in order to obtain analytic approximate solutions and, in particular, we consider the externally forced Boussinesq equation
utt uxx þ auxxxx ebðu2 Þxx 2e2 F cosðXtÞ ¼ 0;
ð7Þ
with the resonance condition xn X and the boundary conditions (3), where e is a small nondimensional parameter that is artificially introduced to serve as bookkeeping device and will be set equal to unity in the final analysis. Note that the perturbed Boussinesq equation was studied in various papers (see [12] and references therein). In Section 2 we calculate an approximate analytic solution and to avoid computational difficulties we shall limit our treatment to a first order approximation. We derive a model system of two coupled differential equations in the phase and amplitude of solutions. Fixed points of the model system correspond to periodic solutions of the fundamental resonance of the Boussinesq equation (2). We find frequency–response and external force-response curves, which include the results of stability analysis. Three steps are used by the AP method: (i) obtaining the form of solution in terms of harmonic components, (ii) introducing a slow time scale, and (iii) solving directly for the various harmonic components via harmonic balance. A systematic means is then provided to derive increasingly accurate solutions by increasing the order of approximation in terms of the small parameter e. In the framework of the AP method, no assumptions are made a priori on the form of the spatial part of the solution. The principal advantage is the treatment of the boundary conditions at higher orders and moreover the predicted behavior is not self-similar. We demonstrate the existence of stable periodic solutions and jump phenomenona both for the ‘good’ and the ‘bad’ Boussinesq equation.
In Section 3, a global analysis of the model system by energy considerations is performed and we demonstrate the existence of closed orbits, corresponding to modulated motions for the Boussinesq equation (2). Another type of modulated motion (libration) can arise also for low values of the parametric excitation. In this case the two-period quasiperiodic motion is characterized by a modulation of the fundamental oscillation, with a modulation amplitude proportional to the magnitude of the external excitation. In certain cases an infinite-period bifurcation may occur, because if we increase the external excitation, while the modulation amplitude remains finite, the modulation period lengthens and becomes infinite and then librations disappear and are substituted by closed orbits. Finally we demonstrate that the separatrix of the model system corresponds to an asymptotically steady-state phase-locked periodic solution for the Boussinesq equation. The last section is devoted to recapitulate the most important results and indicate some possible generalizations. 2. The approximate solution for the fundamental resonance of the nth mode We consider the externally excited Boussinesq equation in an infinite wall and in order to express the nearness of the excitation frequency to the natural frequency, we define a detuning parameter r through the relation
xn ¼ X þ e2 r:
ð8Þ
We introduce the slow time
s ¼ e2 t;
ð9Þ
because we need to look on larger time scales, in order to obtain a non negligible contribution by nonlinear and excitation terms. The solution u(x, t) of Eq. (7) with the boundary conditions (3) can be expressed by means of a power series in the expansion parameter e,
uðx; tÞ ¼
þ1 X
ecm wm ðx; s; eÞ expðimXtÞ;
ð10Þ
m¼1
where cm = jm 1j for m – 0, c0 = r and wm ðx; s; eÞ ¼ w m ðx; s; eÞ. Note that r is the order of the zero-mode term and can be set equal to 1 for the self-consistency of the perturbation method. Eq. (10) can be written more explicitly (r = 1 and c1 = 0)
uðx; tÞ ¼ ew0 ðx; s; eÞ þ ðw1 ðx; s; eÞ expðiXtÞ þ ew2 ðx; s; eÞ expð2iXtÞ þ c:c:Þ þ h:o:t:;
ð11Þ
where h.o.t. = higher order terms and c.c. stands for complex conjugate of the preceding terms. The functions wm(x, s, e) depend on the parameter e and we suppose that the limit of the functions wm for e ? 0 exists and is finite and moreover they can be expanded in power series of e, i.e.
wm ðx; s; eÞ ¼
1 X
ei wmðiÞ ðx; sÞ:
i¼0
ð12Þ
59
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
In the following for simplicity we use the abbreviations ð0Þ wð0Þ m ¼ wm m – 1 for and w1 ¼ w for m = 1. In the lowest order calculations, only the functions corresponding to i = 0 appear. We have assumed only one mode in the expansion, because we suppose that the system is excited near the natural frequency of a specific linear mode and that mode is not involved in an internal resonance with any other mode. After inserting the assumed solution (10) in the starting Eq. (7), we obtain some equations involving the coefficients of every harmonic exp(imXt). For m = 1 we obtain the linear equation
wxx awxxxx þ x2n w ¼ 0; ðorder wð0; sÞ ¼ 0;
e0 Þ;
wðL; sÞ ¼ 0;
2
a8 ¼ a11
2
! 2 a11 ðeð2a6 LÞ 1Þða26 þ a29 þ 4k Þ : þ 2 2 sinhða6 LÞ ða6 þ a29 Þ sinða9 LÞ ð25Þ 2
ð13bÞ
ð14Þ
2
eðw2;xx aw2;xxxx þ 4X w2 Þ ¼ ebW U2n þ Oðe2 Þ; 2
2 n
ð15Þ
2
eðw0;xx aw0;xxxx Þ ¼ 2ebjWj U þ Oðe Þ
ð16Þ
where w0 and w2 satisfy the same boundary conditions as w (Eq. (3)). The solutions of the linear Eqs. (15) and (16) are easily derived
w2 ðx; sÞ ¼ bg 2 ðxÞW2 ðsÞ;
dW 2bðw0 WUn ðxÞ þ w2 W Un ðxÞÞxx ds þ2rXWUn ðxÞ F ¼ 0:
Making use of the normalization of the eigenmodes and with the definition
fn ¼
Z
g 0 ðxÞ ¼ a1 þ a2 sinhða4 xÞ þ a2 sinhða4 ðL xÞÞ þ a3 cosð2kxÞ;
i
dW fn rW NjWj2 W þ ¼ 0; ds 2X
2
N ¼ b ð2G0 þ G2 Þ; G0 ¼
ð19Þ
and 2
a3 ¼
2
Lð1 þ 4ak Þ
;
a2 ¼
4ak a3 ; sinhða4 LÞ
ð20Þ
2
ð21Þ 2
4k a29 1 eða6 LÞ 4k a5 ¼ a11 þ 1 2 sinhða6 LÞ a26 þ a29 a26 þ a29 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t1 1 4X2 ; þ þ a6 ¼ 2 2a 4a a ! 2 eða6 LÞ 1 4k a7 ¼ a11 1 ; 2 sinhða6 LÞ a26 þ a29 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t 1 1 4X2 ; a9 ¼ þ þ 2 2a 4a a
ð28Þ
Z
ð30Þ
Un ðxÞðg 2 ðxÞUn ðxÞÞxx dx:
ð31Þ
L
The effective nonlinearity coefficient depends on the overall spatial dependence of the motion. Expressing the complex-valued function into real and imaginary parts, we obtain
ð32Þ
and we arrive at the model equations
dq fn ¼ sin #; ds 2X
!!
ð29Þ
Un ðxÞðg 0 ðxÞUn ðxÞÞxx dx;
WðsÞ ¼ qðsÞ expði#ðsÞÞ
rffiffiffi 1 ; a4 ¼ a
ð27Þ
L
0
þ a10 sinða9 xÞ þ a11 cosð2kxÞ
2
Z 0
G2 ¼
þ a8 cosða9 xÞ
a1 ¼ a3 ð1 þ 4ak Þ;
FðxÞUn ðxÞdx;
we arrive at the ordinary differential equation
ð18Þ þ a7 e
L
0
where (for simplicity we show them in the case a > 0)
ða6 xÞ
ð26Þ
where
w0 ðx; sÞ ¼ 2bg 0 ðxÞjWðsÞj2 ; ð17Þ
1
2
þ ða29 4k Þeða6 LÞ 4k Þ
2iXUn ðxÞ
2
np ; L
ð24Þ
2
If we consider Eq. (7) for m = 1 and at the order e , then we find that the function W(s) satisfies the nonlinear differential equation
where Un(x) is given by Eq. (4) and W(s) is an arbitrary function of time. Considering Eq. (7) for m = 2 and m = 0 yields
k¼
1 3ak L
ð13aÞ
wðx; sÞ ¼ Un ðxÞWðsÞ;
g 2 ðxÞ ¼ a5 e
a11 ¼
a11 2 ðða2 þ 4k Þ cosða6 LÞ ða26 þ a29 Þ sinða9 LÞ 6
a10 ¼
whose solution is
ða6 xÞ
a26 þ 4k ; a26 þ a29
ð33Þ
;
q ð22Þ
d# N fn ¼ q3 rq þ cos #: ds X 2X
ð34Þ
The Eqs. (33) and (34) can be found in standard books about perturbation methods [6–21], where the primary resonance is carefully studied. Note that from Eqs. (11) and (32) we can express the field to the second approximation as
uðx; tÞ ¼ 2Un ðxÞqðtÞ cosðXt #Þ 2bq2 ðtÞðg 0 ðxÞ ð23Þ
þ g 2 ðxÞ cosð2Xt 2#ÞÞ; where q and # are given by Eqs. (33) and (34).
ð35Þ
60
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
The validity of the approximate solution should be expected to be restricted on bounded intervals of the s-variable and on time-scale t = O(1/e2). If one wishes to construct solutions on intervals such that s = O(1/e), then the higher terms must be included, because they will in general affect the solution. Periodic solutions of the complete system described by Eq. (7) correspond to the fixed points of Eqs. (33) and (34), which are obtained by the conditions dq/ds = d#/ds = 0. We find the equilibrium points (steady state solutions) q0, #0,
q30 þ
#0 ¼ 0 or #0 ¼ p;
rX N
q20
fn ¼ 0; 2N
ð36Þ
where the minus sign corresponds to #0 = 0 and the plus sign to #0 = p. In order to establish the stability of steady state solutions, we superpose small perturbations in the amplitudes and the phases on the steady state solutions and the resulting equations are then linearized. Subsequently we consider the eigenvalues of the corresponding system of first order differential equations with constant coefficients (the Jacobian matrix). The eigenvalue equation is 2
k þ Q ¼ 0:
ð37Þ
Fig. 1. Amplitude of the response (q) as function of the external excitation (f) of the first mode. Solid lines stand for stable solutions.
if F n < 3p4 ffiffi3, then we obtain three equilibrium points, given by (Fig. 2)
pffiffiffi p u 3 ðq1E ; #1E Þ ¼ 2 R cos ;p 3
where
Q¼
rþ
3Nq20
X
rþ
N q20
and then only saddle (Q < 0) or elliptic (Q > 0) points occur for the model system (33) and (34). We observe that the modulation Eqs. (33) and (34) have only two independent parameters, because trough the rescaling
q ! Lq; s ! T s;
Lfn fn ! ; T
pffiffiffi u 3 ;0 ; ðq2E ; #2E Þ ¼ 2 R cos 3 pffiffiffi u 4 3 ðq3E ; #3E Þ ¼ 2 R cos þ p ;0 ; 3 3
ð38Þ
X
sffiffiffiffiffiffiffiffiffiffi jrjX L¼ ; jNj
T¼
1 jrj
;
ð42Þ ð43Þ
ð44Þ
where
pffiffiffi 3 3F n cos u ¼ ; 4
1 R ¼ pffiffiffi : 3 3
ð45Þ
ð39aÞ we can always set
N
X
¼ 1;
f
r ¼ 1; F n ¼ n :
ð39bÞ
X
We distinguish four cases: (i) N/X = 1, r = 1: there is only an elliptic equilibrium point, given by (Fig. 1)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u u u u 2 3 3 F n 1 tF n F 2n 1 C B tF n ðqE ; #E Þ ¼ @ þ þ þ þ ; pA 4 16 27 4 16 27 ð40Þ 4 ffiffi p 3 3
(ii) N/X = 1, r = 1: if F n > there is only an elliptic equilibrium point, given by (Fig. 2)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u u u u 2 3 3 F n 1 tF n F 2n 1 C B tF n ðqE ; #E Þ ¼ @ þ þ ; pA 4 16 27 4 16 27
ð41Þ
Fig. 2. Amplitude of the response (q) as function of the external excitation (f) of the first mode. Solid lines stand for stable solutions, dashed lines for unstable solutions.
61
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
The equilibrium point given by Eq. (43) is a saddle point, while the other two equilibrium points given by Eqs. (42) and (44) are elliptic points. In Fig. 2, we observe an interesting behavior, because if we consider the smaller elliptic point and increase the amplitude of the external force, suddenly in correspondence with the critical value F n ¼ 3p4 ffiffi3, the solution jumps into the second elliptic point. (iii) N/X = 1, r = 1: there is only an elliptic equilibrium point, given by Eq. (40) but with #E = 0. (iv) N/X = 1, r = 1: if F n > 3p4 ffiffi3 there is only an elliptic point, given by Eq. (41) but with #E = 0; if F n < 3p4 ffiffi3 there are three equilibrium points, the first (saddle) and the second (elliptic) given by Eqs. (43) and (44) but with #E = p, and the third, which is always elliptic, given by Eq. (42) but with #E = 0. The last two situations are connected by a simple temporal inversion (and the transformation # ? p + #) to the first two cases. A jump phenomenon is clearly observable in Fig. 2 because for increasing values of the external excitation we observe a discontinuous transition from the low amplitude solution to the high amplitude steady-state periodic response. When f < 0.78, the response depends on the initial conditions and the system tends asymptotically to oscillate around the smaller steady-state periodic response or to the larger one. If we want to observe the jump phenomenon we must vary f from zero and the response will be the low amplitude solution if the initial conditions correspond to an amplitude q near the low amplitude and until f reaches 0.78. A small increase in f beyond 0.78 causes an upward jump to the high amplitude steady-state response. As f increases further, the solution amplitude continues to increase slowly (hysteresis effect). The frequency–response equation follows from Eq. (34), 2
r¼
Nq
X
fn : 2Xq
ð46Þ
In Figs. 3 and 4 we show the frequency–response curves (as usual, solid lines stand for stable solutions and dashed lines for unstable solutions). We see in Fig. 3 that only a steady-state periodic response is possible, on the contrary one, two or three solutions are possible in Fig. 4, depending on the value of r. If r < 0.010, only a stable solution is possible. If 0.010 < r < 0.003, there are three possible solutions: an intermediate unstable solution and tow stable, low and high amplitude, solutions. For r > 0.003 only the stable high amplitude solution is possible. 3. Modulated motion Using the global analysis [11–18] we investigate the model system (33) and (34) and demonstrate the existence of closed orbits, which correspond to two-period quasiperiodic solutions of the Boussinesq equation (7). We find the following integral of motion
3 Nq fn Hðq; #Þ ¼ q þ rq cos # : 2X X
ð47Þ
Fig. 3. The frequency–response curve for the fundamental resonance of the first mode, i.e., the response (q) as function of the detuning (r). Solid line stands for stable solutions.
Fig. 4. The frequency–response curve for the fundamental resonance of the first mode, i.e., the response (q) as function of the detuning (r). Solid line stands for stable solutions, dashed lines for unstable solutions.
As a consequence the energy-like function H(q, #) is constant along the solution curves. In Figs. 5 and 6 we show different solution curves in the plane (q, #) and along each solution curve the energy-like function H(q, #) of the Eqs. (33) and (34) is constant. The case F n < 3p4 ffiffi3 is represented in Fig. 5 and the case F n > 3p4 ffiffi3 in Fig. 6. The frequency of the small oscillations around the elliptic point in the plane (q, #) is
X1 ¼
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3Nq20 N q2 rþ rþ 0 :
X
X
ð48Þ
The periodic motion around the elliptic point for the Eqs. (33) and (34) corresponds to a two period quasi-periodic motion for the Boussinesq equation (7). If we perform a linearization of Eqs. (33) and (34) near the elliptic point, we can write explicitly the solution for the small oscillations
qðtÞ ¼ qE þ ðqI qE Þ cosðX1 tÞ þ sinðX1 tÞ;
fn ð#I #E Þ cos #E 2XX1 ð49Þ
62
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
ut ðx; 0Þ ¼ u_ 0 ðxÞ ¼ 2XUn ðxÞqI sinð#I Þ 4Xbq2I g 2 ðxÞ sinð2#I Þ;
ð52Þ
The solution is then characterized by two frequencies and only if qI = qE, #I = #E, the solution is simply periodic and corresponds to the elliptic equilibrium point. Another interesting possibility which can be easily treated is the case of the libration
N 3 q þ rq fn ; I X I 2X
ð53Þ
because we find that the Eqs. (33) and (34) yield
e t; #ðtÞ ¼ #I X
qðtÞ ¼ qI
ð54Þ
e t þ #I ÞÞ fn ðcos #I cosð X ; e 2X X
ð55Þ
where
N
e ¼ q2 þ r : X X I Fig. 5. Phase space representation of solutions for the system of Eqs. (33) and (34) in the case (iii). Only an elliptic points is present.
ð56Þ
e The solution is quasi-periodic with the frequencies 1 and X and is always given by the Eq. (35). The transition from oscillation around the elliptic point and the libration occurs through an infinite-period bifurcation. For very low value of the external excitation we have demonstrated the existence of a modulated motion with frequency (56). A better approximation for the modulation b can be obtained directly from Eq. (34), after frequency X substituting q with qI (being the modulation of q(t) very small, see Eq. (55)),
2p Te ¼ ¼ e X
Z 2p 0
2XqI d# fn2 cos # 2XqI r 2Nq3i
4pXqI ffi: ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 4qi Xr þ Nq2i fn2
ð57Þ
The period increases with fn and diverges as fn approaches
fC ¼ 2qI Xr þ N q2I . We can estimate the order of divergence by noting that
pffiffiffi p 2 2XqI e T pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; fC fC fn
Fig. 6. Phase space representation pffiffiffiof solutions for the system of Eqs. (33) and (34) in the case (iv), F < 4=3 3. Two elliptic points and a saddle point are present.
#ðtÞ ¼ #E þ ð#I #E Þ cosðX1 tÞ
ð3Nq2E þ XrÞðqI qE Þ
qE XX1
sinðX1 tÞ;
ð50Þ
where qI, #I are connected to the initial conditions trough the following relations
uðx; 0Þ ¼ u0 ðxÞ ¼ 2Un ðxÞqI cosð#I Þ 2bq2I ðg 0 ðxÞ þ g 2 ðxÞ cosð2#I ÞÞ;
ð51Þ
ð58Þ
e blows up following a square-root scalwhich shows that T ing law while the amplitude of motion remains finite (Fig. 7). For the Boussinesq equation (7) we observe a modulated motion with the carrier frequency and the modulae ¼ 2p= T e . When fn is slightly less than fC, tion frequency X the fixed points of Eqs. (33) and (34) make themselves felt because they slow the passage through the ‘bottleneck’ near them. If f continues to increase then an infinite-period bifurcation occurs, because the modulation period lengthens and becomes infinite. After this point, the asymptotic motion is governed by the steady state response given by an oscillation around the elliptic point of the system (33) and (34) while the asymptotic motion corresponds to a two-period quasiperiodic behavior for the Boussinesq equation (7).
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
63
The solution of the Boussinesq equation (7) is then for large times
uðx; tÞ ¼ 2Un ðxÞq2E cosðXt #2E Þ 2bq22E ðg 0 ðxÞ þ g 2 ðxÞ cosð2Xt 2#2E ÞÞ;
ð64Þ
i.e. a phase-locked periodic solution corresponding to the saddle point. Another particular case is the perfect resonance with the external excitation (r = 0). The rescaling (39) cannot be introduced and we simply consider the Eqs. (33) and (34) with r = 0. We find only an elliptic equilibrium point, given by
rffiffiffiffiffiffiffi 3 fn ; qE ¼ 2N
#E ¼ 0;
ð65Þ
while the frequency of the small oscillations is
Fig. 7. The effective potential corresponding to a solutions along the separatrix. Note that the maximum value is zero. A material point with energy equal to zero reaches the maximum in an infinite time.
Global analysis of Eqs. (33) and (34) can be also performed through the study of the motion of a material point in an appropriately chosen potential. In fact, Eqs. (33), (34) and (47) yield after a lengthy calculation
ðRs Þ2 H2 þ VðRÞ ¼ ; 2 2
R ¼ q2 ;
ð59Þ
i.e. the temporal evolution of R is equivalent to that of a material point (mass = 1) with energy given by H2/2 in the potential
f2 HN R2 NrR3 VðRÞ ¼ rH þ n 2 R þ r2 þ X 2 2X 2X þ
N 2 R4
ð60Þ
8X2
and taking into account Eqs. (39) we find
VðRÞ ¼ s2 H þ
! F 2n R2 s1 s2 R3 R4 R þ ð1 s1 HÞ þ þ ; 2 2 2 8
ð61Þ
where
Fn ¼
fn
X
;
s1 ¼ sgnðN=XÞ;
s2 ¼ sgnðrÞ:
ð62Þ
We consider now the important case of the motion along the separatrix of Eqs. (33) and (34), in this case Eq. (61) yields
5 1 Rs 3 2 VðRÞ ¼ Rs 1 Rs þ 2R2s R þ þ Rs 2 2 2 4 R2
R3 R4 þ ; 2 8
Rs ¼ q22E ;
ð63Þ
and q2E is given by Eq. (43). The equivalent energy H2/2 is equal to the potential of the saddle point and if the initial conditions are chosen along the separatrix then in the limit of large times the motion arrive at the saddle point.
X1 ¼
pffiffiffi 3Nq20
X
:
ð66Þ
Note that it coincides with the case r = 0 of the Eq. (48). The approximate analytic solution (49) and (50) with the relations (51) and (52) for the initial conditions is also valid, after taking r = 0. 4. Conclusion We have investigated the properties of the approximate analytic solutions to the externally excited Boussinesq equation in an infinite wall. No assumptions are made a priori on the spatial dependence of the motion. We have employed the asymptotic perturbation method that is able to calculate with excellent precision the behavior of a weakly nonlinear partial differential equation, in resonance or in quasi-resonance with an external force. A system of nonlinear model equations describing the modulation of the amplitude and of the phase of the oscillation has been derived and the position of the equilibrium points and their dependence on the external force can be easily deduced. It is also worthwhile to notice that the AP method furnishes a way to go beyond the first approximation and to calculate the solution with an arbitrarily large increase of precision. We have shown external force-response curves and observed the presence of jump phenomena. We have demonstrated that in certain cases (oscillations around the elliptic point or librations) a second low frequency appears in addition to the forcing frequency and then stable two period quasi-periodic motions are present with amplitudes depending on the initial conditions. The value of the low frequency depends on the initial conditions. Librations are also possible with a frequency depending on the external periodic excitation. Moreover, an infinite-period bifurcation is possible when the external excitation is increased and the initial conditions are appropriately chosen, because the modulation period of the libration lengthens and becomes infinite, while the attractor settles down into a periodic motion (oscillation around the elliptic point) for the system model Eqs. (33) and (34).
64
A. Maccari / Chaos, Solitons & Fractals 54 (2013) 57–64
Possible extensions and generalizations can be accomplished in many different ways. We can study the damping effects that has been not considered in the Boussinesq equation (6) and is the main cause of the quasi-periodic observed behavior. Finally, other nonlinear equations can be analyzed, for example the Kadomtsev–Petviashvili or Davey–Stewartson equation. References [1] Boussinesq J. Theorie des ondes et de remous qui se propagent le long d’un canal rectangulaire horizontal . . .. J Math Pures Appl 1872;17(Series 2):55–108. [2] Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press; 1991. [3] Guenther RB, Lee JW. Partial Differential Equations of Mathematical Physics and Integral Equations. NJ: Prentice Hall; 1988. [4] Maccari A. The Boussinesq equation as a source of equations integrable by the spectral transform. Nonlinearity 1997;10(4):849–56. [5] Maccari A. Periodic behavior for the parametrically excited Boussinesq equation. Chaos Solitons Fractals 2004;22(1):25–34. [6] Nayfeh A. Perturbation Methods. New York: Wiley-Interscience; 1973. [7] Maccari A. Non-resonant interacting ion acoustic waves in a magnetized plasma. Journal of Physics 1999;32A(4):693–700.
[8] Maccari A. Approximate particle-like solutions for a nonlinear relativistic scalar complex field model in 3 + 1 dimensions. Physics Letters 2000;276A:79–90. [9] Maccari A. Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Liénard systems. Int J NonLinear Mech 2000;35:239–62. [10] Maccari A. Non-resonant interacting waves for the nonlinear Klein– Gordon equation in three-dimensional space. Physica D 2000;135(3–4):331–44. [11] Maccari A. Non-resonant interacting water waves in 2 + 1 dimensions. Chaos Solitons Fractals 2002;14(1):105–16. [12] Gorbatova NK, Kiselev OM, Glebov SG. Resonant excitations of nonlinear waves. AIP Conf Proc 2006;834:196–205. [13] Hurewicz W. Lectures on Ordinary Differential Equations. Cambridge, MA: MIT Press; 1958. [14] Cesari L. Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. New York: Academic; 1963. [15] Chowand S-N, Hale JR. Methods of Bifurcation Theory. New York: Springer-Verlag; 1982. [16] Jordan DW, Smith P. Nonlinear Ordinary Differential Equations. second ed. Oxford: Oxford University Press; 1987. [17] Grimshaw R. Nonlinear Ordinary Differential Equations. Oxford: Blackwell; 1990. [18] Perko L. Differential Equations and Dynamical Systems. New York: Springer; 1991. [19] Strogatz SH. Nonlinear Dynamics and Chaos. Reading, MA: AddisonWesley; 1994. [20] Nayfeh AH, Balachandran B. Applied Nonlinear Dynamics. New York: John Wiley; 1995. [21] Bogoliubov NN, Mitropolski YA. Asymptotic Methods in the Theory of Non-Linear Oscillations. New York: Gordon and Breach; 1961.