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1、arXiv:patt-sol/9905011v1 27 May 1999 Soliton structure dynamics in inhomogeneous media L. E. Guerrero Departamento de F sica, Universidad Sim on Bol var, Apartado Postal 89000, Caracas 1080-A, Venezuela A. Bellor n Departamento de F sica, Universidad Sim on Bol var, Apartado Postal 89000, Carac as 1
2、080-A, Venezuela J. A. Gonz alez Centro de F sica, Instituto Venezolano de Investigaciones Cient fi cas, Apartado Postal 21827, Caracas 1020-A, Venezuela (February 7, 2008) We show that soliton interaction with fi nite-width inhomogeneities can activate a great number of soliton internal modes. We o
3、btain the exact stationary soliton solution in the presence of inho- mogeneities and solve exactly the stability problem. We present a Karhunen-Lo eve analysis of the soliton structure dynamics as a time-dependent force pumps energy into the traslational mode of the kink. We show the importance of t
4、he internal modes of the soliton as they can generate shape chaos for the soliton as well as cases in which the fi rst shape mode leads the dynamics. I. INTRODUCTION The propagation of solitons in the presence of inhomogeneities concerns a wide variety of condensed matter systems. The traditional ap
5、proach considers structureless solitons and delta-function-like impurities. Real scenarios involve fi nite-width impurities and under certain circunstances, the extended character of the soliton must be considered 14.For instance, the lenght scale competition between the width of inhomogeneities, th
6、e distance between them and the width of the kink-soliton leads to interesting phenomena like soliton explosions 2. In this paper we take into account the extended character of both the soliton and the impurity and show that these considerations lead to the existence of a fi nite number of soliton i
7、nternal modes that underlies a rich spatiotemporal dynamics. We present a model for which the exact stationary soliton solution in the presence of inhomogeneities can be obtained and the stability problem can be solved exactly. We use the Karhunen-Lo eve (KL) decomposition to relate the excitation o
8、f soliton internal modes with the sequence of bifurcations obtained as the amplitude of a space-time-dependent driving force (fi tted to the shape of the translational mode) is increased. II. THE MODEL The topological solitons studied in the present paper possess important applications in condensed
9、matter physics. For instance, in solid state physics, they describe domain walls in ferromagnets or ferroelectric materials, dislocations in crystals, charge-density waves, interphase boundaries in metal alloys, fl uxons in long Josephson junctions and Josephson transmission lines, etc. 5,6 Although
10、 some of the above mentioned systems are described by the 4-model and others by the sine-Gordon equation (and these equations, in their unperturbed versions, present diff erences like the fact that the sine-Gordon equation is completely integrable whereas the 4-model is not) the properties of the so
11、litons supported by sine- Gordon and 4equations are very similar. In fact, these equations are topologically equivalent and very often the result obtained for one of them can be applied to the other 5. Here we consider the 4equation in the presence of inhomogeneities and damping: xx tt t+ 1 2 ? 3? =
12、 N(x) F(x),(1) Corresponding author. Fax: +582-9063601; e-mail: lguerreusb.ve 1 where F(x) is a function with (at least) one zero and N(x) is a bell-shaped function that rapidly decays to zero for x . An impurity of the kind N(x), but using delta functions, has been presented in Ref. 7. In ferroelec
13、tric materials is the displacement of the ions from their equilibrium position in the lattice, 1 2 ? 3? is the force due to the anharmonic crystalline potential, F(x) is an applied electric fi eld, and N(x) describes an impurity in one of the anharmonic oscillators of the lattice 8. In Josephson jun
14、ctions, is the phase diff erence of the superconducting electrons across the junction, F(x) is the external current, and N(x) can describe a microshort or a microresistor 9. In a Josephson transmission line it is possible to apply nonuniformly distributed current sources (F(x) and to create inhomoge
15、neities of type N(x) using diff erent electronic circuits in some specifi c elements of the chain 6,10. In the present paper the functions F(x) and N(x) will be defi ned as, F(x) = 1 2A(A 2 1)tanh(Bx),(2) N(x) = 1 2 (4B2 A2) cosh2(Bx) .(3) The case F = const. has been studied in many papers (see e.g
16、. 5). Here Eq. (2) represents an external fi eld (or a source current in a Josephson junction) that is almost constant in most part of the chain but changes its sign in x = 0 (this is very important in order to have soliton pinning 1). Microshorts, microresistors or impurities in atomic chains 9 are
17、 usually described by Diracs delta functions (x) where the width of the impurity is neglected. The function N(x) is topologically equivalent to a (x) but it allows us to consider the infl uence of the width of the impurity. III. STABILITY ANALYSIS Suppose the existence of a static kink solution k(x)
18、 corresponding to a soliton placed in a stable equilibrium state created by the inhomogeneities of Eq. (1). We analyze the small amplitude oscillations around the kink solution (x,t) = k(x) + (x,t). We get for the function (x,t) the following equation: xx tt t+ 1 2(1 3 2 k+ 2N(x) = 0. (4) The study
19、of the stability of the equilibrium solution k(x) leads to the following eigenvalue problem (we introduce (x,t) = f(x)exp(t) into Eq. (4): fxx+ 1 2(3 2 k 1 2N(x)f = f, (5) where 2 . For the functions F(x) and N(x) (defi ned as Eqs.(2-3) the exact solution describing the static soliton can be written
20、: k(x) = Atanh(Bx).The spectral problem (Eq. (5) brings the following eigenvalues for the discrete spectrum: n= 1 2A 2 1 2 + B2( + 2n n2 2); here is defi ned as, ( + 1) = A2 B2 + 2. The integer part of () defi nes the number of modes of the discrete spectrum. The stability condition for the translat
21、ional mode is, 16B4+2B2(57A2)+(1A2)2 1, then there will exist three equilibrium points for the soliton: two stable (at points x = x1 0 and x = x2 0) and one unstable at point x = 0. This is because for huge values of |x| the leading inhomogeneity is F(x), which is non-local and not zero at infi nity
22、. This inhomogeneity acts as a restoring force that pushes the soliton towards the point x = 0. As a result of the competition between the local instability induced by N(x) at point x = 0 and the non-local inhomogeneity F(x), an eff ective double-well potential is created. This is equivalent to a pi
23、tchfork bifurcation. We should make some remarks about the stability investigation. Writing down Eq. (4) we are making an approx- imation because the terms 2and 3are considered zero. Under this assumption the solutions of Eq. (4) can be used as an approximation for the kink dynamics only for small p
24、erturbation of the static soliton solution. However, the stability conditions obtained for the diff erent modes are exact. In fact, when we say that the traslational mode is stable for some set of values of the parameters, this means that in a neighborhood of this equilibrium point the eff ective po
25、tential for the soliton center of mass is a well (a minimum). On the contrary, when the parameters are 2 changed such that the stability condition does not hold anymore, then a small deviation in the initial condition of the soliton center of mass will cause the soliton to move away from the equilib
26、rium position. The same is valid for the stability of the shape modes. For example, if the stability condition for the fi rst shape mode is not satisfi ed, then small perturbation of the soliton profi le will cause the soliton to explode. This has been checked numerically 11. In general, the stabili
27、ty problems for perturbed soliton equations are very hard 9. This is because in order to solve it, we fi rst should have an exact solution of the equilibrium problem (which is rarely the case), and then one should solve the eigenvalue problem which usually has no solution in terms of elementary func
28、tions. The investigation we have performed includes several steps. First, we have to solve an inverse problem in order to have external perturbations with the “shapes” that are relevant to the physical situations we want to discuss; second, we assure that the exact solutions will be known to us, and
29、 third, we should be able to solve exactly the stability problem. This last condition is fulfi lled because Eq. (5) is a Schr odinger equation with a P oschl-Teller potential 13. The solution of this spectral problem can be found in Ref. 12. In our case we were lucky enough to obtain exact solutions
30、 to perturbations that are generic and topologically equivalent to well-known perturbation models (e.g. the pitchfork bifurcation). IV. KARHUNEN-LOEVE ANALYSIS Let us consider a space-time-dependent force G(x,t) beside the space-dependent forces F(x) and N(x).In a previous work 1, Gonz alez and Ho l
31、yst found that if G(x,t) has a spatial shape such that it coincides with one of the eigenfunctions of the stability operator of the soliton, then it is possible to get resonance if the frequency of the force also coincides with the resonant frequency of the considered mode. Therefore we can pump ene
32、rgy only into the traslational mode of the kink selecting a space-time-dependent force of the form G(x,t) = cos(t) ? 1 cosh(B(x x1) + 1 cosh(B(x + x1) ? .(6) In Fig. 1(a) we present a sequence of bifurcations of the soliton center-of-mass coordinate Xc.m.= R l/2 l/2 x2 xdx R l/2 l/2 2 xdx (sampled a
33、t times equal to multiples of the period of the driving force) as the driving amplitude is increased and other parameters remain fi xed (A = 1.22, B = 0.32, = 1.22, x1= 2.5 and = 0.3). For these values of A and B the stability condition for the translational mode is fulfi lled, the soliton moves in
34、a single-well potential and the system is in a regime with three discrete modes ( = 3). Previous articles have studied the bistable case as well as the single-well case created by inhomogeneities of the type F(x) 1,2. In this article we want to stress the complexity of the internal dynamics of the s
35、oliton when, besides F(x), there is an impurity of the type N(x). We have integrated the equation using a standard implicit fi nite diff erence method with open boundary conditions x(0,t) = x(l,t) = 0 and a system length l = 80. We use a kink-soliton with zero velocity as initial condition. FIG. 1.
36、(a) Bifurcation diagram for the position of the center of mass of the soliton. (b) Relative weight of the highest KL eigenvalue. (c) Number of KL modes that contains 99.9% of the dynamics. 3 Poincar e maps for the soliton center-of-mass coordinate have revealed quasiperiodic and chaotic attractors f
37、or the non-periodic solutions of Fig. 1(a): period one solutions precede a window of quasiperiodic bifurcations (the torus entangles as the amplitude of the time-dependent driving force increases). At a certain value a period two window appears and is followed by quasiperiodic (two-tori) bifurcation
38、s. For = 0.55 the Poincar e maps reveal high-dimensional chaotic motion followed by period one solutions. The KL decomposition 13,14 allows to describe the dynamics in terms of an adequate basis of orthonormal functions or modes. The eigenvalues ncan be regarded as the weight of the mode n. Figure 1
39、(b) presents the the greater eigenvalue normalized by the weight, W = P n, whereas Fig. 1(c) presents the number of modes that contains 99.9% of the weight. FIG. 2. KL spectra for the sequence of bifurcations presented in Fig. 1. The inset shows the fi rst mode of the KL spectrum for = 0.20 and = 0.
40、60. Figure 2 reveals the increasing excitation of the KL modes as the amplitude of the space-time-dependent force increases. Note the sudden changes of the spectra when periodic motion is regained (period-two for = 0.40 and period-one for = 0.60). For these solutions the amplitude of the oscillation
41、s around the point x = 0 diminishes even though the amplitude of the driving force has increased. This agrees with the higher contribution to the dynamics of the few modes of shape whereas all the rest of the modes decreased their contribution. Furthermore, for = 0.60 the fi rst shape mode replaces
42、the translational mode as the leading mode of the dynamics. The inset of the Figure 2 presents the leading KL eigenmodes for the period-one solutions that initiates and ends the sequence of bifurcations considered in this section. The eigenmode for = 0.20 appears to be the superposition of a pair of
43、 translational modes centered at the equilibrium points for the soliton. Similar situation occurs for = 0.60 but the eigenvalue appears to be the superposition of a pair of shape modes. This work has been partially supported by Consejo Nacional de Investigaciones Cient ifi cas y Tecnol ogicas (CONIC
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45、linger, Physica D 1 (1980) 1. 6 J. A. Gonz alez, L. E. Guerrero, and A. Bellor n, Phys. Rev. E 54 (1996) 1265. 7 Y. S. Kivshar, A. S anchez, and L. V azquez in Nonlinear Coherent Structures in Physics and Biology, M. Remoissenet and M. Peyrard, eds., Springer-Verlag, Berlin, 1991. 8 M. A. Collins, A
46、. Blumen, J. F. Currie, and J. Ross, Phys. Rev. B 19 (1979) 3630. 9 Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61 (1989) 763. 10 R. Landauer, J. Appl. Phys. 51 (1980) 5594. 11 L. E. Guerrero, A. Bellor n, J. R. Carb o and J. A. Gonz alez, to appear in Chaos, Soliton and Fractals. 4 12 S. Fl ug
47、ge, Practical Quantum Mechanics, Springer-Verlag, Berlin, 1971. 13 K. Karhunen, Ann. Acad. Sci. Fennicae, Ser A1, Math. Phys. 37 (1946); M. M. Lo eve, Probability Theory, Van Nostrand, Princeton, 1955. 14 S. Ciliberto and B. Nicolaenko, Europhys. Lett. 14 (1991) 303. 5 0.20.30.40.50.6 2.0 6.0 10.0 N(99.9%) 0.6 0.7 0.8 max 10 6 2 2 Xc.m.(nT) (a) (b) (c) 110100 n 10 8 10 6 10 4 10 2 10 0 10 2 n =0.20 =0.35 =0.40 =0.50 =0.55 =0.60 402002040 x 0.5 0.0 0.5 1.0 f1(x)