Pis'ma v ZhETF, vol. 101, iss. 7, pp. 550-555
© 2015 April 10
The effect of the <fiA kink's internal mode during scattering on
"PT-symmetric defect
D. Saadatmand1^, S. V. Dmitriev+*, D. I. BorisovxV, P. G. Kevrekidis0, M. A. Fatykhovv, K. Javidan Department of Physics, Ferdowsi University of Mashhad, 91775-1436 Mashhad, Iran + Institute for Metals Superplasticity Problems RAS, 450001 Ufa, Russia * National Research Tomsk State University, 634036 Tomsk, Russia x Institute of Mathematics CC USC RAS, 450008 Ufa, Russia ° Department of Mathematics and Statistics, University of Massachusetts, MA 01003 Amherst, USA v Bashkir State Pedagogical University, 450000 Ufa, Russia Submitted 29 December 2014
The effect of the <f>A kink's internal mode (IM) during the scattering from a PT-symmetric defect is investigated. It is demonstrated that if a <f>A kink hits the defect from the gain side, a noticeable IM is excited, while for the kink coming from the opposite direction the mode excitation is much weaker. In the case when the kink initially carries IM, the IM amplitude is affected by the defect if the kink moves from the gain side and it is not affected when the kink moves in the opposite direction. A two degree of freedom collective variable model is shown to be capable of reproducing principal findings of the present work.
DOI: 10.7868/S0370274X15070140
Introduction. Over the last fifteen years, Bender and co-authors have explored broad classes of non-Hermitian Hamiltonians possessing real spectra under the parity-time (VT) symmetry condition, where parity-time means spatial reflection and time reversal [1, 2]. This mathematical discovery has generated an intense interest in the consideration of open physical systems with balanced gain and loss and such setting have been realized experimentally in optics [3-8], electronic circuits [9-11], and mechanical systems [12].
The Klein-Gordon field theory with a VT-symmetric term, which describes a localized VT-symmetric defect, has been recently introduced by one of the authors [13]. A collective coordinate method for nonconservative systems was developed in that work to describe the kink interaction with the defect, see also [14-16]. It was shown that standing kinks in such models are stable (unstable) if they are centered at the loss side (gain side) of the defect [15], while standing breathers may exist only if centered exactly at the interface between gain and loss regions [16].
The interaction of the moving kinks and breathers with the spatially localized PT-symmetric perturbation was recently investigated in the realm of the sine-
e-mail: saadatmand.d@gmail.com
Gordon (SG) equation [17]. Several new soliton-defect interaction scenarios were observed such as the kink passing/trapping depending on whether the kink comes from the gain or loss side of the impurity, merger of the kink-antikink pair into a breather, and splitting of the breather into a kink-antikink pair. The kink phase shift as a result of interaction with the impurity and the threshold kink velocity to pass through the lossy side of the defect were successfully calculated with the help of the collective variable approach [13, 17].
It is well-known that in the integrable SG model, the kink does not support vibrational internal modes (IM), while the kinks in the non-integrable <f>4 model do support such a mode [18]. It is for that reason that the kink-antikink interactions are far richer in the case of the <j>4 model [19-21]. When a kink hits an impurity in a conservative model, a part of its energy is trapped towards the excitation of the impurity mode [22, 23] and another fraction leads to the emission of radiation bursts [24]. It is of particular interest to investigate the role of the kink's IM in the case when the kink interacts with the PT-symmetric impurity. This problem is addressed here for the <f>4 kinks.
The outline of the Letter is as follows. Firstly we introduce the spatially localized PT-symmetric inhomo-geneity into the <f>4 field and present the well-known <f>4
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kink solution and the kink's IM profile. Then a collective variable method is applied and analytically solved to reveal some features of the kink dynamics in this system. Next, we report on the numerical results for scattering of kinks on the "PT-symmetric defect. Finally, our conclusions and some future directions are presented.
The model. In this paper we study the modified <f>4 equation [13]
4>tt - 4>xx - 2</>(i - 4>2) = £7{x)4>u
(i)
where cj>(x,t) is the unknown scalar field, lower indices indicate partial derivatives with respect to the corresponding indices, and j(x) is introduced as
j(x) = tanh(/3x)sech(/3x)
(2)
which has the symmetry j(—x) = —j(x). The latter identity means that the right hand side in (1) is a VT-symmetric defect, i.e., it preserves its form under the change x — x and t —t. We also note that the same concerns the left hand side of equation (1). The physical meaning of equation (1) is that it describes an open system with gain and loss and the former balances the latter. The parameter /3 characterizes the defect inverse width.
As j(x) = 0, Eq. (1) is the non-integrable 4>4 equation with the following moving kink solution
4>k(x, t) = ±tanh{(5fc(x -x0 - Vkt)},
(3)
where Vk is the kink velocity, xq is the kink initial position and Sk = 1 / \J 1 — Vy.. Kink bearing IM can be approximately described as [20]
$K{x,t) = 4>K{x,t) + A£(x,i)sin(wi), (4)
£(x,t) = Y 2 tanh{4(x - x0 - Vkt)} x
x sech{(5fc(x - x0 - Vkt)}. (5)
The IM has amplitude A and frequency uj = a/3. This mode has been discussed by many authors due to its critical role in the collision phenomenology of the <f>4 model [19-21]. In this Letter we show that the kink's IM noticeably affects the kink dynamics during the interaction with the PT-symmetric defect.
Collective variable method. In [13] a two-degree of freedom collective variable model was offered and this model takes into account not only the kink's transla-tional mode but also the kink's IM. The <f>4 kink is effectively described by the two degree of freedom particle of mass M = 4/3, which is the mass of the standing kink. The kink coordinate X(t) (which in the unperturbed
case is given by xo + Vkt as a function of time t) and the kink's internal shape mode A(t) are solutions to the equations
MX
(4>'K + M!)Wk + M')x - Mh(x)dx, (6)
A = —uj A + e
Ç[-{<t>'K + + M]l{x)dx. (7)
The first equation describes the kink translational mode and the latter characterizes the amplitude of the internal shape mode of the 4>4 kink. These equations yield the general form of the nonconservative forcing including the coupling between the modes. Once A is small enough, we can neglect the terms of order O(A) and it simplifies the above equations:
MX = eX / [<f>'K(x - X)f1{x)dx,
A = - J1 A + e
[£(x - X)f1{x)dx.
(8)
(9)
Below we present the results of numerical solution for the two degree of freedom model (6) and (7).
Numerical results. To solve numerically Eq. (1) we introduce the mesh x = nh, where n = 0, ±1, ±2,... and h = 0.1 is the selected spacing. The accuracy of the finite difference approximation used is 0(h4). The resulting set of the ordinary differential equations is integrated numerically using the time step r = 0.005 in the numerical scheme with the accuracy of 0(t4). In the present study the simulations are carried out for fixed /3=1 (the impurity width is approximately equal to the kink width).
To solve numerically the collective variable equations of motion Eqs. (6), (7), the temporal variable is discretized, t = jr, where j = 0,1,2,.... The second-order central differences are used to replace X ~ (Xj-1 — -2Xj + Xj+\)/T2, X ~ {Xj+1 —Xj-i)/2t, and similarly for A and A. For the initial conditions to boost the kink with the initial velocity Vk and the initial IM amplitude A, we set X(j = 0) = X0, X(j = 1) = X0 + Vkr, A(j = 0) = A(j = 1) = A.
Kinks bearing no initial IM. Firstly we discuss the continuum model of Eq. (1). Fig. la and b) shows the kink kinetic energy as a function of time for the case when the kink, initially bearing no IM, interacts with the PT-symmetric defect with amplitude e = 0.15. In panel a the kink approaches the defect from the gain side and in panel b from the loss side. The kink initial
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50
100
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Fig. 1. Numerical results for the continuum model Eq. (1) showing the interaction of the kink initially bearing no IM with the PT-symmetric defect. In panels a and b shown is the kink kinetic energy as a function of time for the defect amplitude e = 0.15 and the kink initial velocity Vk = 0.4. The kink approaches the defect from the gain side (a) and from the loss side (b) of the defect. The figures reveal that in panel a kink's IM is excited after the interaction with the defect, while in panel b it is practically not excited, (c) - Amplitude of the kink kinetic energy after the interaction with the defect as a function of kink initial velocity for the case when kink hits the defect from the gain side (solid line) and loss side (dotted line) for e = 0.15 and 0.3
velocity is Vfc = 0.4 in both cases. As a result of interaction with the defect, the kink in panel a is firstly accelerated and then decelerated, while in panel b it is first decelerated and then accelerated. In both cases the kink's translational velocity after passing through the defect is practically identical to the initial velocity. Note that the kink without the IM excited has a constant in time kinetic energy, while the kinetic energy of the kink with the excited IM oscillates near the constant value with frequency 2 a/3, which i
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