1. Literature Summary: Instabilities, solitons, and rogue
waves in PT-coupled nonlinear waveguides
J. Schoenfeld
Southern Methodist University,
Department of Mathematics
December 10, 2013
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 1 / 12

2. Goals
Study rogue waves in PT-symmetric optical models based on
dual-core couplers
Examine how the presence of balanced dissipation and gain affects
the modulational instability of the background and possibly the
creation of waves localized in space and time
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 2 / 12

3. Outline
Introduction to PT-symmetry
The model
CW solutions
Modulational instability
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 3 / 12

4. Introduction: PT-Symmetry in quantum mechanics
ˆP: ˆx → −ˆx, ˆp → −ˆp, ˆT: ˆx → ˆx, ˆp → −ˆp, ˆi → −ˆi
ˆH = ˆp2
2 + V (ˆx) is PT-symmetric if [ˆH, ˆP ˆT] = 0, which implies that
VR(−ˆx) = VR(ˆx), and VI (−ˆx) = −VI (ˆx).
Proof.
Assume (ˆH ˆP ˆT)f = (ˆP ˆT ˆH)f , for any f. Then observe
ˆTf (x) = f ∗
(x), ˆHf (x) = [
ˆp2
2
+ V (ˆx)]f (x),
ˆP ˆTf (x) = f ∗
(−x), ˆT ˆHf (x) = [
ˆp2
2
+ V ∗
(ˆx)]f ∗
(x),
ˆH ˆP ˆTf (x) = [
ˆp2
2
+ V (ˆx)]f ∗
(−x). ˆP ˆT ˆHf (x) = [
ˆp2
2
+ V ∗
( ˆ−x)]f ∗
(−x).
Therefore, V (ˆx) = V ∗
(−ˆx).
Thus, ˆH and ˆP ˆT share eigenfunctions.
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 4 / 12

5. The model
Consider the following system of linearly coupled NLSEs for ψ1 and ψ2:
i
∂ψ1
∂z
= −
∂2ψ1
∂x2
+ χ1 |ψ1|2
+ χ |ψ2|2
ψ1 + iγψ1 − ψ2, (1)
i
∂ψ2
∂z
= −
∂2ψ2
∂x2
+ χ |ψ1|2
+ χ1 |ψ2|2
ψ2 − iγψ2 − ψ1. (2)
Describes a set of two parallel planar waveguides, where z and x are
the dimensionless propagation and transverse coordinates.
Initial condition: optical beam shone into waveguides input at z = zi .
Also, describes a dual-core fiber coupler, where here x represents the
temporal variable.
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 5 / 12

6. The model continued
i
∂ψ1
∂z
= −
∂2ψ1
∂x2
+ χ1 |ψ1|2
+ χ |ψ2|2
ψ1 + iγψ1 − ψ2,
i
∂ψ2
∂z
= −
∂2ψ2
∂x2
+ χ |ψ1|2
+ χ1 |ψ2|2
ψ2 − iγψ2 − ψ1.
The two equations are coupled
nonlinearly by the cross-phase modulation (XPM) ≈ χ,
linearly by the last term. (Here, the coupling constant is scaled to be
equal to 1).
γ > 0 describes the mutual balance in gain in Eq. (1) and dissipation
in Eq. (2) , i.e. the PT symmetry.
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 6 / 12

7. Model: Parametrization
In optics, this situation can be achieved with two lossy waveguides
coupled in parallel, where one is being pumped by the external source
of gain-providing atoms.
Though the first core carries the gain, the linear coupling between this
core and its lossy partner cause the zero state to be neutrally stable,
resulting in the propagation of linear waves.
In this case, modes do not arise spontaneously, but can be excited by
input beams.
This situation occurs when the gain/loss term is small compared to
the linear coupling through which the core with gain transfers energy
to the lossy one.
Here, this occurs when γ ≤ 1.
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 7 / 12

8. Model: Parametrization
We introduce the following convenient parameterization,
γ = sin(δ), 0 < δ <
π
2
. (3)
We seek PT-symmetric and antisymmtric solutions to Eqs. (1) and (2)
such that
ψ2(x, z) = ±e±iδ
ψ1(x, z). (4)
Then ψ1 is such that
i
∂ψ1
∂z
= −
∂2ψ1
∂x2
+ (χ1 + χ) |ψ1|2
ψ1 + iγψ1 − ψ2
= −
∂2ψ1
∂x2
+ (χ1 + χ) |ψ1|2
ψ1 + isin(δ) e±iδ
ψ1
= −
∂2ψ1
∂x2
+ (χ1 + χ) |ψ1|2
ψ1 cos(δ)ψ1
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 8 / 12

9. Model: Parameterization
To ensure conventional symmetry is not broken, we do the following:
If ψ1(x, z), ψ2(x, z), is a solution to Eq. (1) and (2), then so is
ψ2(x, −z), ψ1(x, −z) .
This is equivalent to δ → π − δ. So now, 0 < δ < π.
The values δ and π − δ correspond to the two different solutions with
the same gain and dissipation.
PT-symmetric solutions : 0 ≤ δ ≤ π
2
PT-antisymmetric solutions : π
2 ≤ δ ≤ π
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 9 / 12

10. CW Solutions
We are looking for a solution of the form
ψ1 = ρeikx−iΩz
. (5)
Thus,
∂ψ1
∂z
= iΩρeikx−iΩz
,
∂ψ1
∂x
= ikρeikx−iΩz
,
∂2ψ1
∂x2
= (ik)2
ρeikx−iΩz
, ...
→ Ω = k2
+ (χ1 + χ)ρ2
cos(δ). (6)
So generalizing this, the CW solutions (up to a trivial phase shift) to Eqs.
(1) and (2) are
ψ
(cw)
j = ρeikx−ibz+i(−1)j δ/2
, (7)
where k is the background current and b = k2 + (χ1 + χ)ρ2 − cos(δ).
Note, both cores have equal amplitudes of the fields.
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 10 / 12

11. Modulational instability
Ansatz:
ψj = ρ ei(−1)j δ/2
+ ηj e−i(βz−κx)
+ νj ei(βz−κz)
eikx−ibz
, (8)
for j = 1, 2 and |ηj | / |νj | 1.
The branches, β = β1,2(k), of the dispersion relation for the stability
eigenvalues are as follows:
β1(k) = 2kκ ± κ κ2 + 2ρ2(χ1 + χ), (9)
β2(k) = 2kκ ± (κ2 + 2cosδ)(κ2 + 2cosδ + 2ρ2(χ1 + χ)) (10)
Note: Due to the Galilean invariance of Eqs. (1) and (2), the instability is
not affected by boost k.
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 11 / 12

12. Modulational instability
Observe, there are three sources of MI:
χ1 + χ < 0
Stems from β1(k) due to long-wavelengths excitations.
Not influenced by gain and dissipation.
cos(δ) < max[0, ρ2(χ − χ1)]
Stems from β2(k) due to the linear coupling between NLSEs.
Gain and dissipation (δ = 0, π) is very different than a conservative
system (δ = 0orδ = π).
Occurs only due to imbalance of gain and loss
Results in nearly homogeneous grow/ decay of the field in the
waveguide with gain/ dissipation.
J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 12 / 12