Nonlinear Beam Propagation in a Class of Complex Non-$\mathcal {PT}$-Symmetric Potentials

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.

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

$$\displaystyle \begin{aligned} m(\boldsymbol{\delta})=\frac{1}{2}\mathbf{F}(\mathbf{u})^{\mathrm{T}}\mathbf{F}(\mathbf{u})+ \boldsymbol{\delta}^{\mathrm{T}}\mathbf{J}(\mathbf{u})^{\mathrm{T}}\mathbf{F}(\mathbf{u})+ \boldsymbol{\delta}^{\mathrm{T}}\mathbf{J}(\mathbf{u})^{\mathrm{T}}\mathbf{J}(\mathbf{u})\boldsymbol{\delta}.\end{aligned} $$

(26)

In addition, δ must fulfill the relation

$$\displaystyle \begin{aligned} ||\mathbf{D}\cdot\boldsymbol{\delta}||<\varDelta, \end{aligned} $$

(27)

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:

$$\displaystyle \begin{aligned} (\mathbf{J}({\mathbf{u}}_j)^{\mathrm{T}}\mathbf{J}({\mathbf{u}}_j)+\lambda_j\mathbf{D})\boldsymbol{\delta}_j= -\mathbf{J}({\mathbf{u}}_j)^{\mathrm{T}}\mathbf{F}({\mathbf{u}}_j) \end{aligned} $$

(28)

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.