Abstract
The subject of \(\mathcal {PT}\)-symmetry and its areas of application have been blossoming over the past decade. Here, we consider a nonlinear Schrödinger model with a complex potential that can be tuned controllably away from being \(\mathcal {PT}\)-symmetric, as it might be the case in realistic applications. We utilize two parameters: the first one breaks \(\mathcal {PT}\)-symmetry but retains a proportionality between the imaginary and the derivative of the real part of the potential; the second one, detunes from this latter proportionality. It is shown that the departure of the potential from the \(\mathcal {PT}\)-symmetric form does not allow for the numerical identification of exact stationary solutions. Nevertheless, it is of crucial importance to consider the dynamical evolution of initial beam profiles. In that light, we define a suitable notion of optimization and find that even for non \(\mathcal {PT}\)-symmetric cases, the beam dynamics, both in 1D and 2D – although prone to weak growth or decay– suggests that the optimized profiles do not change significantly under propagation for specific parameter regimes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Below, we use the term “soliton” in a loose sense, without implying complete integrability [21].
References
Bender, C.M., Boettcher, S.: Real spectra in non-hermitian Hamiltonians having \(\mathcal {PT}\) symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)
Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)
Ruter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192–195 (2010)
Peng, B., Ozdemir, S.K., Lei, F., Monifi, F., Gianfreda, M., Long, G.L., Fan, S., Nori, F., Bender, C.M., Yang, L.: Parity-time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394–398 (2014)
Peng, B., Ozdemir, S.K., Rotter, S., Yilmaz, H., Liertzer, M., Monifi, F., Bender, C.M., Nori, F., Yang, L.: Loss-induced suppression and revival of lasing. Science 346, 328–332 (2014)
Wimmer, M., Regensburger A., Miri, M.-A., Bersch, C., Christodoulides, D.N., Peschel, U.: Observation of optical solitons in PT-symmetric lattices. Nat. Commun. 6, 7782 (2015)
Suchkov, S.V., Sukhorukov, A.A., Huang, J., Dmitriev, S.V., Lee, C., Kivshar, Yu.S.: Nonlinear switching and solitons in PT-symmetric photonic systems. Laser Photon. Rev. 10, 177–213 (2016)
Konotop, V.V., Yang, J., Zezyulin, D.A.: Nonlinear waves in \(\mathcal {PT}\)-symmetric systems. Rev. Mod. Phys. 88, 035002 (2016)
Schindler, J., Li, A., Zheng, M.C., Ellis, F.M., Kottos, T.: Experimental study of active LRC circuits with \(\mathcal {PT}\) symmetries. Phys. Rev. A 84, 040101 (2011)
Schindler, J., Lin, Z., Lee, J.M., Ramezani, H., Ellis, F.M., Kottos, T.: \(\mathcal {PT}\)-symmetric electronics. J. Phys. A Math. Theor. 45, 444029 (2012)
Bender, N., Factor, S., Bodyfelt, J.D., Ramezani, H., Christodoulides, D.N., Ellis, F.M., Kottos, T.: Observation of asymmetric transport in structures with active nonlinearities. Phys. Rev. Lett. 110, 234101 (2013)
Bender, C.M., Berntson, B., Parker, D., Samuel, E.: Observation of \(\mathcal {PT}\) phase transition in a simple mechanical system. Am. J. Phys. 81, 173–179 (2013)
Cuevas-Maraver, J., Kevrekidis, P.G., Saxena, A., Cooper, F., Khare, A., Comech, A., Bender, C.M.: Solitary waves of a \(\mathcal {PT}\)-symmetric nonlinear dirac equation. IEEE J. Select. Top. Quant. Electron. 22, 5000109 (2016)
Demirkaya, A., Frantzeskakis, D.J., Kevrekidis, P.G., Saxena, A., Stefanov, A.: Effects of parity-time symmetry in nonlinear Klein-Gordon models and their stationary kinks. Phys. Rev. E 88, 023203 (2013); see also: Demirkaya, A., Kapitula, T., Kevrekidis, P.G., Stanislavova M., Stefanov, A: On the spectral stability of kinks in some PT-symmetric variants of the classical Klein-Gordon field theories. Stud. Appl. Math. 133, 298–317 (2014)
Tsoy, E.N., Allayarov, I.M., Abdullaev, F.Kh.: Stable localized modes in asymmetric waveguides with gain and loss. Opt. Lett. 39, 4215–4218 (2014)
Konotop, V.V., Zezyulin, D.A.: Families of stationary modes in complex potentials. Opt. Lett. 39, 5535–5538 (2014)
Yang, J.: Partially \(\mathcal {PT}\)-symmetric optical potentials with all-real spectra and soliton families in multi-dimensions. Opt. Lett. 39, 1133–1136 (2014)
D’Ambroise, J., Kevrekidis, P.G.: Existence, stability and dynamics of nonlinear modes in a 2D partially \(\mathcal {PT}\)-symmetric potential. Appl. Sci. 7, 223 1–10 (2017)
Kominis, Y.: Dynamic power balance for nonlinear waves in unbalanced gain and loss landscapes. Phys. Rev. A 92, 063849 (2015)
Kominis, Y.: Soliton dynamics in symmetric and non-symmetric complex potentials. Opt. Commun. 334, 265–272 (2015)
Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Nixon, S.D., Yang, J.: Bifurcation of soliton families from linear modes in non-\(\mathcal {PT}\)-symmetric complex potentials. Stud. Appl. Math. 136, 459 (2016)
Dutykh, D., Clamond, D., Durán, Á.: Efficient computation of capillary-gravity generalised solitary waves. Wave Motion 65, 1 (2016)
Yang, J.: Symmetry breaking of solitons in two-dimensional complex potentials. Phys. Rev. E 91, 023201 (2015)
Chen, H., Hu, S.: The asymmetric solitons in two-dimensional parity-time symmetric potentials. Phys. Lett. A 380, 162 (2016)
Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2, 164 (1944)
Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431 (1963)
Cuevas, J., Rodrigues, A.S., Carretero-González, R., Kevrekidis, P.G., Frantzeskakis, D.J.: Nonlinear excitations, stability inversions, and dissipative dynamics in quasi-one-dimensional polariton condensates. Phys. Rev. B 83, 245140 (2011)
Rodrigues, A.S., Kevrekidis, P.G., Cuevas, J., Carretero-González, R., Frantzeskakis, D.J.: Symmetry-breaking effects for polariton condensates in double-well potentials. In: Malomed, B.A. (ed.) Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, pp. 509–529. Springer, Berlin/Heidelberg (2013)
Rodrigues, A.S., Kevrekidis, P.G., Carretero-González, R., Cuevas-Maraver, J., Frantzeskakis, D.J., Palmero, F.: From nodeless clouds and vortices to gray ring solitons and symmetry-broken states in two-dimensional polariton condensates. J. Phys. Condens. Matter 26, 155801 (2014)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes. The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)
Acknowledgements
J.C.-M. thanks financial support from MAT2016-79866-R project (AEI/FEDER, UE). P.G.K. gratefully acknowledges the support of NSF-PHY-1602994, the Alexander von Humboldt Foundation, the Stavros Niarchos Foundation via the Greek Diaspora Fellowship Program, and the ERC under FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-605096). The authors gratefully acknowledge numerous valuable discussions with and input from Professor Jianke Yang during the course of this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix: The Levenberg–Marquardt Algorithm
Appendix: The Levenberg–Marquardt Algorithm
Classical fixed-point methods like Newton–Raphson cannot be used for solving the problem F[u(x)] = 0 in the setting considered in the context of this Chapter, essentially because there might not exist a u(x) that fulfils this relation (to arbitrarily prescribed accuracy). However, it is possible to find a function u(x) that can minimize F[u(x)]. To this aim, an efficient method is the Levenberg–Marquardt algorithm (LMA, for short), which is also known as the damped least-square method. This method is also used to solve nonlinear least squares curve fitting [26, 27]. LMA is implemented as a black box in the Optimization Toolbox of Matlab TM and in MINPACK library for Fortran, and can be considered as an interpolation between the Gauss-Newton algorithm and the steepest-descent method or viewed as a damped Gauss-Newton method using a trust region approach. Notice that LMA can find exact solutions, in case that they exist, as it is the case of the results presented, e.g., in Ref. [23].
Prior to applying LMA, we need to discretize our Eq. (10). Thus, we take a grid x n = −L∕2 + nh with n = 0, 1, 2…M and L being the domain length, and denote u n ≡ u(x n) and F n ≡ F[u(x n)]. With this definition u xx can be cast as (u n+1 + u n−1 − 2u n)∕h 2. In order to simplify the notation in what follows, let us call \(\mathbf {u}\equiv \{u_n\}_{n=1}^M\) and \(\mathbf {F}(\mathbf {u})\equiv \{F_n\}_{n=1}^M\). We will also need to define the Jacobian matrix \(\mathbf {J}(\mathbf {u})\equiv \{J_{n,m}\}_{n,m=1}^M\) with \(J_{n,m}=\partial _{u_m}F_n\). In the presently considered optimization framework, F(u) is also knows as the residue vector.
Let us recall that fixed point methods typically seek a solution by performing the iteration u j+1 = u j + δ j from the seed u 0 until the residue norm ||F(u)|| is below the prescribed tolerance; here δ j is dubbed as the search direction. In the Newton–Raphson method, the search direction is the solution of the equation system J(u j)δ j = −F(u j). If the Jacobian is non-singular, the equation system can be easily solved (as a linear system); however, if this is not the case, one must look for alternatives like the linear least square algorithm. It was successfully used for some of the authors for solving the complex Gross–Pitaevskii equation that describes the dynamics of exciton-polariton condensates [28,29,30]. This technique also allowed us to find optimized beams in the present problem, but presented poor convergence rates, as we were unable to decrease the residue norm controllably below the order of unity.
As fixed point methods are unable to give a reasonably small residue norm, we decided to use a trust-region reflective optimization method. Such methods consist of finding the search direction that minimizes the so called merit function
In addition, δ must fulfill the relation
where D is a scaling matrix and Δ is the radius of the trust region where the problem is constrained to ensure convergence. There are several trust-region reflective methods, with the LMA being the one that has given us the best results for the problem at hand. This is a relatively simple method for finding the search direction δ by means of a Gauss-Newton algorithm (which is mainly used for nonlinear least squares fitting) with a scalar damping parameter λ > 0 according to:
with D being the scaling matrix introduced in Eq. (27). There are several possibilities for choosing such a matrix. In the present work, we have taken the simplest option, that is D = I (the identity matrix), so (27) simplifies to ||δ j|| < Δ. Notice that for λ j = 0, (28) transforms into the Gauss-Newton equation, while for λ j →∞ the equation turns into the steepest descent method. Consequently, the LMA interpolates between the two methods. Notice also the subscript in λ j: this is because the damping parameter must be changed in each iteration, with the choice of a suitable λ j constituting the main difficulty of the algorithm.
The scheme of the LMA is described in a quite easy way in the Numerical Recipes book [31, Chapter 15.5.2] and is summarized below:
-
1.
Take a seed u 0 and compute ||F(u 0)||
-
2.
Choose a value for λ 0. In our particular problem, we have taken λ 0 = 0.1.
-
3.
Solve the equation system (28) in order to get δ 0 and compute ||F(u 0 + δ 0)||
-
4.
-
If ||F(u 0 + δ 0)||≥||F(u 0)||, then take λ 1 = 10λ 0 and u 1 = u 0, as with this choice of λ 0 the residue norm has not decreased.
-
If ||F(u 0 + δ 0)|| < ||F(u 0)||, then take λ 1 = λ 0∕10 and u 1 = u 0 + δ 0, as with this choice of λ 0 has succeeded in decreasing the residue norm.
-
-
5.
Go back to step 3 doing λ 0 = λ 1 and u 0 = u 1
This algorithm is repeated while ||F(u)|| is above the prescribed tolerance.
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Cuevas-Maraver, J., Kevrekidis, P.G., Frantzeskakis, D.J., Kominis, Y. (2018). Nonlinear Beam Propagation in a Class of Complex Non-\(\mathcal {PT}\)-Symmetric Potentials. In: Christodoulides, D., Yang, J. (eds) Parity-time Symmetry and Its Applications. Springer Tracts in Modern Physics, vol 280. Springer, Singapore. https://doi.org/10.1007/978-981-13-1247-2_20
Download citation
DOI: https://doi.org/10.1007/978-981-13-1247-2_20
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-1246-5
Online ISBN: 978-981-13-1247-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)