A variational approach to Schrödinger equation with parity-time symmetry Gaussian complex potential
A variational technique is established to deal with the Schrödinger equation with parity-time () symmetric Gaussian complex potential. The method is extended to the linear and self-focusing and defocusing nonlinear cases. Some unusual properties in PT systems such as transverse power flow and PT breaking points can be analyzed by this method. Following numerical simulations, the analytical results are in good agreement with the numerical results.
Laboratory of Photonic Information Technology, South China Normal University,
Guangzhou 510631, P. R. China \address Dept of physics, Guangdong University of Petrochemical Technology,
Maoming 525000, P. R. China \addressCorresponding author:firstname.lastname@example.org
INTRODUCTION: People have paid much attention to optical properties in parity-time () symmetric system during recent years both theoretically and experimentally[1, 2, 3, 4, 5, 6]. Bender et al found that the non-Hermitian Hamiltonian with symmetry can exhibit entirely real spectrum .In optics, symmetric potentials can be also constructed by a complex gain/loss refractive-index distribution into the waveguide, which make the refractive index distribution is symmetry[1, 2, 3]. Many unusual features stemming from symmetry have been found, such as power oscillation , absorption enhanced transmission , nonlinear switching structure , and unidirectional invisibility .
But the properties of symmetry mentioned above are mostly discussed by numerical simulations and experimental results. In order to get a good understanding of the properties of symmetry solitons, it is essential to present an analytic solution, even an approximate one. Musslimani et al have present closed form solutions to a certain class of nonlinear Schrodinger equations involving symmetry potential. R.EI-Ganainy et al have developed a formalism for coupled optical parity-time symmetric systems by Lagranian principles. It would be very desirable to obtain approximate analytical results for the general symmetry Schrödinger equations. For the nonlinear Schrödinger equations, the approximate solutions can provide a better physical understanding of the interplay between symmetry potential and nonlinear effects in connection with solitons propagation. The variational method has been widely applied to obtain approximated solutions for problems concerning beam propagation within the framework of the NLSE. Motivated by the previous works in which it was studied the dissipative solitons solutions by the variational approach[14, 15]. In this paper, we will employ a variational approach based on the dissipative system to deal with the Schrödinger equation with PT symmetric Gaussian complex potential, and some unusual properties in system can be analyzed by this method.
MODEL: We consider the (1+1)-dimensional evolution equation along the longitudinal direction in linear or Kerr-nonlinear media with complex PT potentials. The propagation of the beam is governed by the Schrödinger equation, i.e.
Here is the slowly varying complex field envelop, is the transverse coordinate. represents the self-focusing propagation, represents the self-defocusing propagation, and represents the linear situation. and are the real and the imaginary parts of symmetry complex potentials respectively. The normalization relation is given in the reference .
If the solution of stationary wave modes for Eq. (1) has the form like , where is the propagation constant, and are real functions standing for the amplitude and phase of light, then
It is noteworthy that Eq. (3) is a specialized form of Poynting’s theorem for electromagnetic fields in dispersive media with losses, where the transverse component of Poynting vector can be written as (only the transverse differential remains for stationary modes). The term represents the absorption(or gain) in the medium. Equation (3) can be applicable to the general dissipative system. At the center of the system, we have
That means the energy-flow at the center must equal to the total loss (or gain) at each side. As a result, the intensity at the center can not be zero for any stable solution and the dark solitons can not exist in any system.
The symmetric gaussian complex potentials are used here as and ,where and represent the depth of the real and the imaginary parts of potentials respectively. We suppose the trial function as,
where , , , and stand for the amplitude, width, chirp and the phase of the beam, , and stand for the amplitude and the width of wavefront lean . By substituting the trial solution into Eq.(5) and the averaged conservative Lagrangian can be obtained,
The Euler-Lagrange equations for parameter is
Equations (9) is a trial result which means the power of a stationary modes is unchange during propagation. From Eqs.(10) and (11), we find that the shape of the phase function is independent of the beam width. Numerical simulations show that even for unstable solution, the shape of the phase function is very similar to the error function, and its amplitude () is mainly determined by . We also find that these properties are identical to the multimode solitons and other systems such as Scarff II potential.
Introducing , we arrive at the evolution equations,
LINEAR MODES: For the case of stable solution, the derivatives of beam width and curvature in the evolution equations are vanished. In linear propagation, i.e. , we can get , ,
From Eqs. (16), for a large value of , we have a maximum value of , which can be regarded as the -breaking point. For a small value of , there doesn’t exist a maximum value of , however, when , , and can be regarded as the -breaking point. Figures (1)(a) and (b) shows the relation to the eigenvalues with for different in numerical and variational method. We can see that for a large (=10), the obtained by the variational method is in good agreement with the -breaking point in the numerical method, and for a small (), the is close to the -breaking point. The -breaking point with different in numerical method and the relation of the or the with in analysis are shown in the fig. (1)(d). One can see that the or the is in good agreement with the -breaking point. Figure(1)(c) shows that the amplitude and the nontrivial phase of the eigenfunction are in good agreement with the numerical results too.
SOLITONS: For the nonlinear propagation, i.e. , we can get and for the solitons. For a given value of and , the solitons obey
Without losing of generality, we take and in this subsection.
Figures (2) (a) and (b) are the relations between the power and the beam width of solitons with the propagation constants, respectively. The critical propagation constant at zero power is the eigenvalue for the linear mode. When propagation constant , it is corresponding to the solitons in the self-focusing nonlinear media, otherwise it represents the solitons in the self-defocusing media. We can see that in the self-defocusing media, the power of the bright solitons decreases with increasing of the propagation constants, however, it is contrary to the bright solitons in the self-focusing media. However, the beam width decreases with increasing of the propagation constants in the whole region. The amplitude and the nontrivial phase of the solitons in numerical method and variational approach are shown in Figs. (2)(c) and (d), which are corresponding to the cases marked as circle symbols in Figs. 2(a) and (b). One can see that the analytical results are in excellent agreement with the numerical results for both self-focusing and self-defocusing cases.
Stability of the solitons: From the evolution equations, Eq. (18), we define
The equation associated with the the Jacoby determinant corresponding to the stability criterion is
When , or , solitons is stable. Substituting and to Eq. (19), one has
CONCLUSION: In conclusion, we have analyzed the properties of the beam in PT symmetric Gaussian complex potentials by the variation method. According to the analysis, the transverse power-flow density is associated with the wavefront lean of the beam. For the linear case, the -breaking points are obtained by the variational approach, and the eigenvalue obtained by the variational approach are in agreement with the ones by the numerical method. For the nonlinear case, there exists a critical propagation constant for bright soliton existing in the self-focusing and self-defocusing media. The relations between the power of solitons with the propagation constants and the beam width are gotten analytically and confirmed by numerical simulations. The amplitude and the nontrivial phase of the solitons in both the self-focusing and self-defocusing media are agreement with the numerical results.
ACKNOWLEDGMENTS This research was supported by the National Natural Science Foundation of China (Grant Nos. 10804033 and 11174090), the Program for Innovative Research No. 06CXTD005), and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.200805740002).
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