THE VARIED NONLINEAR INTERFACE WAVES IN A SCALE-LIMITED TWO-MEDIUM OSCILLATORY SYSTEM
X. Q. Huanga'h* X. H. Cuic, J. Y. Huanga'h
" School of Biomedical Engineering, Capital Medical University 100069, Beijing, China
bBeijing Key Laboratory of Fundamental Research on Biomechanics in Clinical Application, Capital Medical University
100069, Beijing, China
'"School of Mathematics and Physics, North China Electric Power University 102206, Beijing, China
Received January 5, 2014
We investigate spontaneously generated waves around the interfaces between two different media in a system where the domain scales are limited. These two media are carefully selected such that there exists a theoretical interface wave with the frequency and wave number that can be predicted according to the control parameters. We present the rules of how the frequency and wave number vary with reducing the scales of media domains. We find that the frequency decreases with reducing the scale of antiwave (AW) media, but increases with reducing the scale of normal wave (NW) media in both one-dimensional and two-dimensional systems. The wave number always decreases with reducing scales of either NW or AW media. The least scale to generate the theoretical wave is the predicted wavelength. These special phenomena around the interfaces may be applied to detect the limited scale of a system.
DOI: 10.7868/S0044451014070190
1. INTRODUCTION
The formation, propagation, and interaction of nonlinear oscillation around the interface between different media have long been interesting topics. The results have attracted much attention in recent decades in the area of reaction diffusion systems, optics, ultrasonic, biological systems, and so on. Numerous characteristic features and complex phenomena have been investigated fl 12]. Together with the normal wave (NW) with positive phase velocity, the recently found antiwave (AW) with negative phase velocity provides even more interesting dynamical behaviors and pattern formations [8 12].
The method for generating an antiwave in a homogeneous oscillatory medium has no essential difference from that for normal waves. A proper initial condition or an external pacing can both produce the desired wave. Researches have mentioned that the control parameters of media determine the kind or kinds of wave
E-mail: xiaoqingh'fflccmu.edu.cn
that can be produced in it. Pacing can generate NW in NW media, AW in AW media, and different pacings can generate either NW or AW in N AW media due to the dispersion relation determined by the control parameters [13,14]. One interesting phenomenon that involves both an NW and an AW is the generation of an interface-selected wave (ISW), first noted in [15]. When a system is constructed by two linked domains of one NW medium and one AW medium, and the dispersion relations of these two media have an intersection point, the ISW can be generated spontaneously from the interface of the two media. The ISW has the frequency and the wave number that can be theoretically calculated by the dispersion relations of two media. Once an ISW is generated, it eventually occupies the whole two-medium system.
Intuitively, the ISW is the result of an interplay of two different media at the interface. The geometrical shape and scale of each medium should have no effect on the dynamical behavior. Also, a system on which a theoretical experiment carried out is normally large enough to avoid the effect of a boundary, if the system can be considered a continuous medium, not a series of
discreto grids. But by reducing the scale properly, it might become possible to observe the detailed behavior occurring exactly on the interface and reveal the generation process of interface waves.
We first study the interface waves by reducing the scales of media in a one-dimensional (ID) two-medium system. It is surprising that the frequency and wave number of the generated ISW both vary continuously with the scale of the media. Analyzing the results, we find that there is a necessary condition for the generated wave to be the theoretically predicted one, i.e., have the same frequency and wave number. That is, the length scale of each participant medium should be at least equal to half the predicted wavelength. Once the condition is fulfilled, the frequency and wave number of the survivor wave are equal to the predicted ones. Otherwise, the two-medium system can still enter a homogeneous dynamics, but with a different oscillating frequency and wave number, which are related to the geometrical scale of the system. Numerical studies in a two-dimensional (2D) patched system yield similar results. The variation is the same. However, irrespective of how large each participant domain is, the generated frequency can never reach the theoretical value.
This paper is organized as follows. In Sec. 2, we explain the selected model and our motivation. In Sec. 3, we present series of results for ID and 2D two-medium oscillatory systems. The alterations of frequency and wave number are specified in different situations. Section 4 contains a discussion and analysis of the phenomena. Section 5 is our conclusion.
2. MOTIVATION
We construct our system using the complex Ginz-burg Landau equation, which is the commonly used model describing extended systems in the vicinity of a Hopf bifurcation from a homogeneous stationary-state [16 19]:
F> 4
= .4 - (1 + la)\AfA + (1 + 13)V2A. (1)
For a general reaction diffusion system, the dynamics of oscillations with the amplitude and phase are scaled to one complex order parameter ,4. When time is scaled by the characteristic reaction time, space by the characteristic diffusion length, and the modulus of the amplitude by the radius of the limit cycle, the remaining two control parameters a: and 3 govern the universal dynamics around the bifurcation [19].
If the frequency ui of the external pacing is close to
the natural frequency u>o = « of the media, the dispersion relation
u> = u>o + fik2 = a + (3 ~ a)k2
determines the characteristics of generated waves in the following way under the conditions of a 1:1 pacing-reaction region and uiuio >0 [14]:
AWs for u;0/i = a(3 - a) < 0, (2a)
NWs for u;0/i = a(3 - a) > 0. (2b)
We have |u>| < |u>o| for AWs and |u>| > |u>o| for NWs. For two media whose dispersion relation lines have an intersection at one's AW region and one's NW region, ISW trains emerge at the interface of these two media. This means that the ISW trains are always normal waves in the NW domain and antiwaves in the AW domain. Because the frequencies used in this paper are always positive, the wave numbers for different waves then have different signs. For simplicity and convenience, we focus on the absolute value and the square of the wave number in different regions.
Theoretically, the control parameters of two media determine the frequency and wave number of the generated ISW as
a23i ~ «1A , 10! =---— , (3a)
a.2 - «1 + 0i — 32
1 ~ a2 - «1 + /?i - 3'2 ' In previous studies, the results are perfectly well consistent with the theoretical values in ID systems [15,20]. But there are some inconsistent situations in 2D systems, for example, in the patching system where one medium is surrounded by another. In that case, the frequency is always slightly different from the theoretical value [20 23]. This has not yet been given a clear explanation.
If the structure of media can alter the dynamical behavior, what happens when the geometrical scale is changed? If the ISW is generated exactly on the interface, the geometrical scale should have no effect on the properties of the waves. However, from another standpoint, if the generation of an ISW requires a range of the medium, a very small system may not be able to produce ISWs. Studies in this paper aim to answer these questions. By reducing the geometrical scale of a two-medium system in several ways and by comparing the results in ID and 2D systems, we reveal the exact dynamics occurring around the interface.
In the next section, we first present our experimental results for a ID two-medium system. The results for a 2D patched two-medium system are then shown.
3. RESULTS OF MANIPULATING THE SCALES OF MEDIA IN A TWO-MEDIUM SYSTEM
3.1. Reducing one medium in a ID two-medium system
We construct a one-dimensional two-medium system as follows:
= Ai -(l+i«i) 1^112 Ax +(l+«/3i) V2 Ax,
o < X < Li,
dA
„ A2-(l+ia2)\A2\2A2+(l+i/52)\72A2, at (4b)
Li < x < Li + L2 + 1,
A -A ^(J) - M2(/) ( .
Al(I)-Am, (4c)
The system is divided into two domains. We let the left domain Mi be an AW media of length Li, and the right domain M2 be an NW media of length L2. Equation (4c) is the continuity condition, where I means the value on the interface and dA^/dn is the gradient to the normal direction. No-flux boundary conditions are used on all outer boundaries, such that the inner dynamics is not affected. We then set ^4(0) = ^4(1) and A(LX + L2 + 1) = A(LX + L2), and [l,Li + L2] is the area that we calculated. The system is integrated using second order Runge-Kutta (RK2) method and the standard three-point approximation for the Laplace operator.
We carefully choose the control parameters such that the interface select waves can be generated. For the AW media olx = 0.4 and /?i = -1.0
(ai(/?i - ai) < 0). For the NW media M2, a2 = 0.2 and /?2 = 2.0 (a2(/?2 — a2) >0). According to Eqs. (3a) and (3b), the frequency and wave number of the ISW can be theoretically predicted as oji = 0.3125 and k2 = = 0.0625, whence \kj\ = 0.25. As shown in Fig. la, the dispersion relation curves of these two media have an intersection point with the coordinates oji = 0.3125 and k2j = 0.0625. That implies that if these two media are connected together, the ISWs emerge from the interface spontaneously, with the frequency and wave number equal to the coordinates of the intersection point. In Fig. 1 we show a spatiotemporal pattern of the two-medium system in which the AW me
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