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4Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 2, pp. 217-229. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220205
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 35C08, 35Q51, 65M06, 34K28
Changing the Dynamic Parameters of Localized Breather and Soliton Waves in the Sine-Gordon Model with Extended Impurity, External Force, and Decay in the Autoresonance Mode
E. G. Ekomasov, V. N. Nazarov, K. Yu. Samsonov
Possibility of changing the dynamic parameters of localized breather and soliton waves for the sine-Gordon equation in the model with extended impurity, variable external force and dissipation was investigated using the autoresonance method. The model of ferromagnetic structure consisting of two wide identical layers separated by a thin layer with modified values of magnetic anisotropy parameter was taken as a basis. Frequency of external field is a linear function of time. The sine-Gordon equation (SGE) was solved numerically using the finite differences method with explicit scheme of integration. For certain values of the extended impurity parameters a magnetic inhomogeneity in the form of magnetic breather is formed when domain wall passes through it with constant velocity. The numerical simulation showed that using special variable force and small amplitude it is possible to resonantly increase the amplitude of breather. For each case of the impurity parameters values, there is a threshold value of the magnetic field amplitude leading to resonance. Geometric parameters of thin layer also have influence on the resonance effect — for decreasing layer width the breather amplitude grows more slowly. For large layer width the translation mode of breather oscillations is also excited. For certain parameters of extended impurity, a soliton can form. For a special type of variable field with frequency linearly dependent on time, soliton is switched to antisoliton and vice versa.
Keywords: autoresonance, sine-Gordon equation, spatially modulated periodic potential, impurities, kink, breather, soliton
Received December 07, 2021 Accepted March 22, 2022
The studies by E. G. Ekomasov and K. Yu. Samsonov were supported by the Russian Foundation for Basic Research, project no. 20-31-90048. The studies by V. N. Nazarov were performed within the framework of state assignment no. AAAA-A19-119022290052-9.
Evgenii G. Ekomasov
EkomasovEG@gmail.com
Bashkir State University
ul. Zaki Walidi 32, 450076 Ufa, Russia
Vladimir N. Nazarov NazarovVN@gmail.com
Institute of Molecule and Crystal Physics Ufa Federal Research Centre of RAS prosp. Oktyabrya 151, 450075 Ufa, Russia
Kirill Yu. Samsonov k.y.samsonov@gmail.com
University of Tyumen
ul. Volodarskogo 6, 625003 Tyumen, Russia
1. Introduction
Nonlinear differential equations having a solution in the form of nonlinear localized waves (solitons) are often used in the models describing waves and oscillations in nonlinear media. For example, the sine-Gordon equation (SGE) is used for describing the dynamics of DNA in molecular biology, dynamics of domain walls in magnetics, dislocation in crystals and flux quanta in Josephson contacts and junctions (e.g., [1-4]). Researchers usually have to enter additional terms or account for the influence of perturbations in SGE for a more complete description of these processes. This leads to substantial changes in the structure and energy of solitons. The (3+1)-(2+1)-dimensional models are often studied (e.g., [5-9]). However, the (1+1)-dimensional models are the most studied. For some applications, it is necessary to account for dissipation and external force in the system (e. g., [1-3, 10-14]). Parameters of SGE are also often set as functions of coordinates and time [15-20]. In some cases (for small perturbations) the structure, static and dynamic properties of localized waves in such SGE models may be studied analytically [1-3, 1625]. However, it is more often necessary to use numerical methods [1-3, 13-15, 22, 23]. Many numerical methods for these equations have been developed [9, 13, 14, 26-30].
Many articles were devoted to spatial modulation (heterogeneity) of periodic potential, or the presence of an impurity in the system (e. g., [3, 13-16, 18-25]). Impurities acting on a kink as the attracting potential lead to the excitation of different impurity modes — localized oscillating state on an impurity. The importance of taking account of the presence of impurity modes in the system for kink dynamics was shown. The possibility of resonant interaction of kink with the excited localized mode for the cases of point and extended impurity was shown numerically and analytically [3, 14, 16, 18, 19, 21, 22]. For different applications of the sine-Gordon model it is important to account for the possibility of space modulation and other constants of the model [3, 13].
Using the autoresonance phenomenon, it is possible to control the structure and dynamics of solitons by applying an external variable force [31, 32] and accounting for the decay in the system. The autoresonant models of control allow one to substantially decrease the external influence on the system [31-35]. The phenomenon of autoresonance described by nonlinear partial differential equations (e.g., [36]) was also investigated for the case of SGE [32, 35]. However, the presence of solitons and breathers in the system was accounted for in these articles from the beginning. In this article we study the autoresonant model of control of the breather and soliton parameters in the sine-Gordon equation with extended impurities, variable external force, and small amplitude. The presence of dissipation in the system is also considered.
2. Basic equations and the solution method
The perturbed sine-Gordon equation for the model with one extended impurity in the presence of an external force and dissipation for a scalar field u(x, t) can be described as follows:
d2u d2u „, , 1 , . u du , .
+ /W smu + h(t) sin - + a — = 0, (2.1)
where f (x) is some function of the x coordinate, h(t) is the variable parameter of external force, and a is the constant of decay. Equation (2.1) can describe, for example, the viscous dynamics of magnetizability under the influence of an external magnetic field in multilayer ferromagnetic [37]. The unperturbed sine-Gordon equation obtained from (2.1) for f (x) = 1, a = h = 0 has the
well-known solutions of breather and kink types:
\/l —uJq sin UJ0t
u(x, t) = 4 arctan | v ^ ---j | , (2.2)
wo ch s/l - uj%(x - x0)
u(x, t) = 4 arctan ( exp ( ^ ) ) (2.3)
V W1"^//
where w0 is the frequency of breather oscillations, x0 is the position of the breather, and v0 is the parameter determining the velocity of kink movement. For the case of arbitrary f (x) and h(t) functions, the solution of Eq. (2.1) can be obtained only numerically.
For simplicity, in the one-dimensional case the f (x) function is often modeled by a rectangle [13-15]:
{W
1, N > T,
W (2.4)
K, M < y
where W, K are some constants. We consider the case of an attracting impurity, when it is a potential well for a kink. In this case, K < 1, and W ^ 0.
The finite difference method with the explicit scheme of integration is used for numerical solution of Eq. (2.1). Since the description of the applied numerical method was given in detail in [9, 13-15], we will give a brief description of its main points. Discretization of the equation was carried out using the standard five-point scheme of "cross" type which was used previously for simpler modifications of SGE (e.g., [13-15]). The uniform grid with step £ by coordinates x: {xi = £ ■ i, i = 0, ±1, ..., ±Nx} and with step t by time t: {tn = t ■ n, n = 0, 1, ..., Nt}, where Nx, Nt are the numbers of grid points, was used for calculation. Following the stability condition of the explicit, scheme | ^ 1, the u value was calculated at the following time points. The use of boundary conditions implies setting the fixed values at the grid edges. As shown by numerical simulation of this problem, the effect of waves reflection from grid edges occurs. This brings distortion of results for the long-continued process simulation. In order to exclude such a distortion of the results obtained, the idea of waves "absorption" in the boundary layer was used. The dissipative parameter a is set in the form of a piecewise constant function:
i1, x ^ xleft + Ddiss, x ^ xright — Ddiss,
(2.5)
a0, xleft + Ddiss < x < xright — Ddiss
where Ddiss is the width of the absorption area (as a rule, 3-5 % of the full simulated area width) and a0 is the value of the dissipative parameter in the main area. The use of the function (2.5) leads to an almost total decay of all waves that reached the grid edges.
3. The case of localized breather waves
At the initial time the u(x, 0) function was set in the form of the kink (2.3), which was located far from impurity. Further, the kink started to move with a stationary velocity sufficient to pass through the impurity area. It was shown previously that the localized high-amplitude nonlinear waves of soliton and breather types can be excited in the impurity area after a kink passes through it [13, 15, 19]. The amplitude of the excited-on-impurity breather (the u value
7 6 5 4
u 3 2 1 0
-1 0
_iL) x
Fig. 1. Passage of a kink with constant velocity 0.85 through an impurity with formation of localized breather waves. Here W = 2, K = —0.2, h = 0, a = 0
at x = 0) decreases with time, and its oscillations are accompanied by emission of the voluminous small-amplitude waves. First, we will consider values of the impurity parameters that correspond to formation of a localized breather-type wave for a kink passing through it at the constant velocity (Fig. 1). The numerical analysis shows that in the absence of an external force the breather amplitude is attenuated with time (Fig. 2). In this and the following figures we show the numerically obtained value of u at x = 0 (i.e., in the center of the impurity area) which will be further called the breather amplitude and the initial time corresponds to the moment of breather excitation in the impurity area. We will consider the case of a variable external force, when, using the resonance effects, it is possible to achieve the localized oscillations of sufficiently high amplitude. It is known [2, 12, 34] that the own frequency of a nonlinear system depends on its amplitude, which is why the resonance does not remain with an increase in amplitude. Nevertheless, the solutions with the increasing amplitude may appear under an appropriate change in the frequency of external force. As in [35], the h(t) function will be considered in the following form:
h = h0 cos wt, (3.1)
where w is a linear function of time:
w = w0 — ¡it. (3.2)
Here w0 is the natural frequency of the breather localized in the impurity area, calculated earlier in [13], and i is the small parameter. As was shown earlier analytically in [35], with this type of function, we can expect a sharp increase in the breather amplitude. The numerical simulation shows that the natural frequency of the breather weakly depends on the amplitude for the case of attracting impurity and becomes constant under the defined parameters. Therefore, in this
case, it is sufficient to take the frequency value in the h(t) function equal to the natural frequency of the localized oscillations of the breather. Then we will deal with a classic resonance in the linear system with constant frequency and the amplitude defined by the parameter of decay. There is also an alternative method [38], where the excitation frequency of "saw-toothed" type is used, allowing one to change the amplitude of the breather under the resonance frequency value varying from wmin to wmax.
t
Fig. 2. Dependence of the breather oscillations amplitude on time. Here W = 2, K = —0.2, a = 0.001
Fig. 3. Dependence of the breather oscillations amplitude on time under h0 = 0.1 a = 0.001 (red line), a = 0.0001 (blue line). Here W = 2, K = —0.2, force frequency of 0.63, parameter ¡i = 0.01
In principle, we can consider two cases for the frequency of external excitation: linear increase from wmin and linear decrease from wmax, assuming that i is small. Since both cases slightly differ, we will study only the linear decrease in frequency governed by the law (3.2). In the numerical experiment the initial frequency was set equal to the known natural frequency of the breather (Fig. 3). It is clear from Fig. 3 (red line) that the breather amplitude increases approximately twice under the dissipation parameter of a = 0.001 and the initial frequency of
breather oscillations of 0.63 for the time of t = 30. If the dissipation parameter decreases ten times (a = 0.0001), while the frequency and time remain the same, then the breather amplitude increases approximately three times (Fig. 3, blue line). Let us note that the breather emits waves during oscillations. The radiation is getting stronger with an increase in the breather amplitude. The emission of waves leads to the slowdown of growth in the breather amplitude and its transition to some stationary value (for the u ^ 6 case shown in the figure).
Fig. 4. Dependence of the breather oscillations amplitude on time: h0 = 0.1 (a), h0 = 0.01 (b), and dependence of breather oscillations the envelope of amplitude on time h0 = 0.05 (c). Here a = 0.001, W = 2, K = —0.2, force frequency of 0.63, parameter j = 0.01
We will consider the influence of the external force amplitude. We assume that the excitation amplitude is small: h0 ^ 1. It is clear from Fig. 4a that h0 = 0.1 corresponds to the case of resonant increase in the breather amplitude, while a decrease in the excitation amplitude up to h0 = 0.05 leads to a small increase in the breather amplitude at first, followed by its gradual
decrease at t > 100 and transition to a stationary mode with the amplitude of u < 3 (Fig. 4c). Due to a large time scale of t < 500 the lines describing the period of breather oscillations will be too close to each other, which is why we show the envelope of amplitudes in Fig. 4c. For a small excitation amplitude of h0 = 0.01 the resonance effects are not observed, and a gradual decay of breather occurs (Fig. 4b). That is, for each case of the W and K values, there is a threshold value of the minimum initial amplitude leading to a resonance.
u
30
60
90
(a)
\J
120 150
t
100
2.0 1.5 1.0 0.5 0
-0.5 -1.0 -1.5 -2.0
30
60
90
(c)
120 150
t
Fig. 5. Dependence of the breather oscillations amplitude on time: h0 =0.1 and W = 3, force frequency of 0.44 (a), W = 2 (b), W =1, force frequency of 0.58 (c). Here a = 0.001, W = 2, ¡i = 0.01
Figure 5 shows the evolution of the breather amplitude for different values of W. We take into account the fact that the natural frequency of the breather strongly depends on the values of these parameters [13]. For example, Figs. 5b and 5c correspond to the breather's natural frequency of 0.58 and 0.86, respectively. It is clear that the amplitude of breather oscillations can be increased faster (for the same value of the h0 amplitude) with an increase in the parameter W (Figs. 5b, 5c). For example, if W decreases twice, the increase in the breather
90
(c)
120
150
Fig. 6. Dependence of the breather oscillations amplitude on time in the variable field of amplitude h0 = = 0.1 for different layer depth: K = —0.2, force frequency of 0.63 (a), K = —0.1, force frequency of 0.68 (b), K = —0.3, force frequency of 0.58 (c). Here a = 0.001, W = 2, ¡j, = 0.01
amplitude is half as slow in time. However, for W > 2 it is not possible to increase the breather amplitude using the resonant effects (Fig. 5a). For a large width of the impurity area, the center of the oscillating breather does not remain in the area's geometric center and the breather's translational oscillations begin to be excited as well.
A similar pattern is observed for a change in the K value (Fig. 6). For a decrease in the K value, the increase in the amplitude of breather oscillations occurs faster. But for a big change in its value compared to 1 (K = —0.5) it is also not possible to enhance the breather amplitude using the resonant effects (Fig. 6c). Though for a negative K the breather oscillations become bigger, but the amplitude of the emitted waves increases, leading to the bigger energy loss of the breather. But contrary to the case of the extended impurity discussed above, the energy loss may be compensated by decreasing K. It is also clear from Figs. 6a, 6b that for a decrease in K it is possible to increase (at the same value of the h0 amplitude) the amplitude of breather faster (Figs. 6a, 6b).
4. The case of localized soliton waves
Fig. 7. Passage of a kink with constant velocity of 0.85 through an impurity with formation of localized waves of antisoliton type, where W = 2, K = —1.4, h = 0, a = 0
It was shown earlier [13] that when the W and K parameters increase to a certain value, a localized wave of soliton or antisoliton type is excited in the impurity area (Fig. 7). We call a localized wave a soliton when u > 0 and an antisoliton when u < 0. Its amplitude (the u value at x = 0) oscillates, maintaining its sign, and approaches a constant value with time. It
-1.4 -1.8 u -2.2 -2.6
50 100 150 200
t
Fig. 8. Dependence of the amplitude envelope of soliton oscillations on time without a field (red line) and of antisoliton oscillations in a variable field (blue line), with h0 = 0.1, the parameter j = 0.01, force frequency 0.83, and the well parameters W = 2, K = —1.4
can exist for a long time. If we apply a constant external force of a certain sign, then we can expect a change in sign of the soliton amplitude for some critical value of the external force. For example, for the case of W = 2, K = —2 at a high amplitude of the field exceeding the value of h = 0.5, the transition from soliton to antisoliton is observed in numerical simulations. We will consider the case of a variable external force and the use of the autoresonance phenomenon for dynamic control of the soliton amplitude by a weaker external force. As in the case of a breather, the external force will be described by the time-varying function (3.1), where the frequency is a linear function of time w = w0 + jt. For example, in the case K = —1.4 and W = 2 the traveling kink generates a localized soliton wave (Fig. 8, red line). For h0 = 0.1, j = 0.01 and the initial frequency of 0.83 we will observe the generation of an antisoliton. If for the case of h = = 0 the oscillations of the soliton amplitude decay with time, then for the case of external force of a certain frequency (associated with the natural frequency of the soliton amplitude's oscillations on impurity), the antisoliton oscillations amplitude increases twice (Fig. 8, blue line). But further increase in the amplitude of antisoliton oscillations is limited by wave radiation. It is important to note that in a variable field the soliton-antisoliton transition occurs for a small amplitude of the field. For example, for the considered case of K = —1.4, W = 2, for such a transition, a threshold value of external force's amplitude is h0 = 0.04, which is an order of magnitude less than for the case of a constant force.
For decreasing K a similar situation takes place — the amplitude of antisoliton oscillations will increase twice, but with a changed frequency, because the frequency of antisoliton oscillations depends on the impurity parameters. Such a restriction on the increase in the oscillation amplitude is caused by the fact that the center of the antisoliton no longer remains in the center of impurity, and the translational mode of antisoliton oscillations along the x coordinate is excited, accompanied by a strong radiation of waves. However, for the case of sufficiently small impurity width, it is possible to implement the case leading to the disappearance of the translational mode of soliton amplitude's oscillations. Here it is possible to achieve a bigger value of the soliton's oscillations amplitude (almost by one order of magnitude) compared to the case of h = 0.
5. Conclusion
In this paper we have investigated the possibility of governing the dynamic parameters of breather and soliton waves in the sine-Gordon equation using autoresonance for a model with extended impurity, variable external force and dissipation. It is shown numerically that, using
a special type of variable force and small amplitude, it is possible to resonantly increase the breather oscillations amplitude and to switch soliton to antisoliton and vice versa. By changing the parameters of extended impurity, it is possible to increase or decrease the breather's and soliton's oscillations amplitude, as well as time required for its significant growth. It is found that, for each set of the parameters of extended impurity, there is a threshold amplitude of external force, leading to an autoresonant increase in the breather amplitude.
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