Polychromatic solitary waves
in a periodic and nonlinear Maxwell system
Dmitry Pelinovsky
Department of Mathematics, McMaster University, Hamilton ON, Canada
CRM Montreal, March, 2014
with Gideon Simpson - Drexel University
Michael I. Weinstein - Columbia University
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
1 / 26
Introduction
Shocks and Spatial Periodicity
Spatially Homogeneous Quasilinear Hyperbolic System
∂t v + ∂x f(v) = 0
Smooth data generates typically a shock wave in finite time (Lax 64)
Spatially Periodic Quasilinear Hyperbolic System
∂t v + ∂x f(x, v) = 0,
f(x + 2π, v) = f(x, v)
Can spatial periodicity stabilize shock formation?
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
2 / 26
Introduction
Regularizing Shocks
Diffusive regularization:
vt + vvx = µvxx
Dispersive regularization:
vt + vvx = αvxxx .
Dispersion from spatial periodicity (Maxwell Model):
∂t2 n2 (z)E + χE 3 = ∂z2 E ,
where n(z + 2π) = n(z) is the refractive index of the periodic media.
Does this model display wave breaking (shocks)?
Does this model admit stable localized states (solitons)?
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
3 / 26
Coupled Mode Equations
Maxwell & Coupled Mode Equations
Periodic Nonlinear Maxwell Equation
where
∂t2 n2 (z)E + χE 3 = ∂z2 E
n2 (z) = 1 + ǫ
X
p∈Z\{0}
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
Np e ipz , ǫ ≪ 1.
CRM Montreal, March, 2014
4 / 26
Coupled Mode Equations
Maxwell & Coupled Mode Equations
Periodic Nonlinear Maxwell Equation
where
∂t2 n2 (z)E + χE 3 = ∂z2 E
n2 (z) = 1 + ǫ
X
p∈Z\{0}
Np e ipz , ǫ ≪ 1.
Two-wave approximation of small-amplitude resonant waves
E ≈ ǫ1/2 E + (ǫz, ǫt)e i(z−t) + E − (ǫz, ǫt)e −i(z+t)
yields the Nonlinear Coupled Mode Equations (NLCME) for E ± (Z , T ) in
slow variables Z = ǫz and T = ǫt.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
4 / 26
Coupled Mode Equations
Properties of the NLCME
The Nonlinear Coupled Mode Equations (NLCME)
2 2
∂T E + + ∂Z E + = iN2 E − + iΓ E + + 2 E − E + ,
¯ 2 E + + iΓ E − 2 + 2 E + 2 E −
∂T E − − ∂Z E − = i N
where Γ = 3χ/2.
Dispersive: E ± e i(KZ −ΩT ) with Ω2 = K 2 + |N2 |2 ,
Possess explicit solitary wave solutions (Aceves–Wabnitz 89),
Globally well-posed in H 1 (R) (Goodman et al. 01), but
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
5 / 26
Coupled Mode Equations
Properties of the NLCME
The Nonlinear Coupled Mode Equations (NLCME)
2 2
∂T E + + ∂Z E + = iN2 E − + iΓ E + + 2 E − E + ,
¯ 2 E + + iΓ E − 2 + 2 E + 2 E −
∂T E − − ∂Z E − = i N
where Γ = 3χ/2.
Dispersive: E ± e i(KZ −ΩT ) with Ω2 = K 2 + |N2 |2 ,
Possess explicit solitary wave solutions (Aceves–Wabnitz 89),
Globally well-posed in H 1 (R) (Goodman et al. 01), but
˜
Mathematically inconsistent, because the correction term E,
3
3
∂t2 − ∂z2 E˜ = E + e 3i(z−t) + E − e −3i(z+t) + . . . ,
grow linearly in t.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
5 / 26
Coupled Mode Equations
Numerics with Soliton Data
Seed NLCME Soliton (E + , E − ) into Maxwell equations,
E (z, t) = ǫ1/2 E + (ǫz, ǫt)e i(z−t) + E − (ǫz, ǫt)e −i(z+t) .
No periodic potential:
∂t2 E + χE 3 = ∂z2 E
Small cos-periodic potential:
∂t2 E + ǫ cos(z)E + χE 3 = ∂z2 E
Side pulses are absent in the NLCME.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
6 / 26
Extended Coupled Mode Equations
Revised Asymptotic Expansion
Simpson–Weinstein’ 2011
Nonlinear and Periodic Maxwell Equation
2
∂t2 E + ǫN(z)E + χ |E | E = ∂z2 E .
Generalized Ansatz
E = ǫ1/2 E + (z − t, Z , T ) + E − (z + t, Z , T ) + ǫE (1) (z, t) + . . . .
Constraint on the Growth of the Correction Term
Z
1 T
(1) lim
E dt = 0.
T →∞ T 0
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
7 / 26
Extended Coupled Mode Equations
Integro-Differential equations for E ± (φ, Z , T )
Let N(z) = N(z + 2π). The correction term is bounded if
(∂T + ∂Z )E
+
(∂T − ∂Z )E −
Z π
1
−
N(φ + θ)E (Z , T , φ + 2θ)dθ
= ∂φ
2π −π
Z π
Γ
1
+ 3
−
2
+
+ ∂φ (E ) + 3
|E (Z , T , θ)| dθ E ,
3
2π −π
Z π
1
+
N(φ − θ)E (Z , T , φ − 2θ)dθ
= −∂φ
2π −π
Z π
1
Γ
|E + (Z , T , θ)|2 dθ E − ,
− ∂φ (E − )3 + 3
3
2π −π
where Γ ≡ 23 χ.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
8 / 26
Extended Coupled Mode Equations
Extended Nonlinear Coupled Mode Equations (xNLCMEs)
Periodically Varying Index of Refraction
N(z) = N(z + 2π)
⇒
N(z) =
X
Np e ipz ,
N0 = 0
Fourier Decomposition
E ± (φ, Z , T ) =
Dmitry Pelinovsky (McMaster University)
X
Ep± (Z , T )e ipφ .
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
9 / 26
Extended Coupled Mode Equations
Extended Nonlinear Coupled Mode Equations (xNLCMEs)
Periodically Varying Index of Refraction
N(z) = N(z + 2π)
⇒
N(z) =
X
Np e ipz ,
N0 = 0
Fourier Decomposition
E ± (φ, Z , T ) =
X
Ep± (Z , T )e ipφ .
Fourier amplitudes satisfy the infinite-dimensional NLCMEs:
X i
ip hX + + +
Eq− 2 Ep+
Eq Er Ep−q−r + 3
3
hX
X i
ip
−
+
Eq+ 2 Ep−
¯ 2p Ep +
Eq− Er− Ep−q−r
+3
= ip N
3
∂T Ep+ + ∂Z Ep+ = ipN2p Ep− +
∂T Ep− − ∂Z Ep−
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
9 / 26
Extended Coupled Mode Equations
Numerics with Soliton Data
±
Inclusion of third harmonic (E±3
), resolves side pulses
Questions:
Do the xNLCMEs admit localized stationary states (solitons)?
Are localized states robust in the dynamics of the xNLCMEs?
Simplifications:
1
We reduce the system of xNLCMEs near band edges to a system of
coupled nonlinear Schr¨
odinger equations.
2
We use the Gaussian trial functions and variational approximations.
3
We truncate the system of equations and perform numerical
continuations.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
10 / 26
Localized states
Localized stationary solutions
Stationary decomposition
−ipΩT
Ep± (Z , T ) = A±
,
p (Z )e
Amplitude equations (xNLCMEs)

Γ
iA′p (Z ) + pΩAp + pN2p Bp + p 3Ap
3
X
q∈Z
|Bq |2 +
X
q,r ∈Z

Aq Ar Ap−q−r  = 0,


X
X
Γ
¯ 2p Ap + p 3Bp
Bq Br Bp−q−r  = 0,
|Aq |2 +
−iBp′ (Z ) + pΩBp + p N
3
q∈Z
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
q,r ∈Z
CRM Montreal, March, 2014
11 / 26
Localized states
Band Edge Approximation
Linear approximation
√
−|p||Z | |N2p |2 −Ω2
.
A±
p (Z ) ∼ e
Exponential decay is provided by the assumption
N2p = 1 for all p and Ω ∈ (−1, 1).
(Therefore, N(z) is a periodic sequence of Dirac delta-distributions.)
Localized states near a band edge
Ω
2
A±
p (Z ) = ±µUp (µZ ) + O(µ ), Ω =
p
1 − µ2 .
K
This expansion allows us to derive coupled nonlinear
Schr¨
odinger equations.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
12 / 26
Localized states
The coupled NLS equations
Coupled Stationary Nonlinear Schr¨
odinger Equation
X
X
|Uq |2 +
Uq Ur Up−q−r = 0,
Up′′ (ζ) − p 2 Up + 23 p 2 3Up
where ζ = µZ .
With the Fourier series,
U(θ, ζ) =
X
Up (ζ)e ipθ
p∈Zodd
the system is converted to a scalar equation
Z π
2 2
1
2
2
2
3
(∂ζ + ∂φ )U = ∂φ U + 3
|U(ζ, θ)| dθ U .
3
2π −π
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
13 / 26
Localized states
Justification theorem
Theorem
Assume the existence of a localized state U ∈ X s of the NLS equations,
¯ φ) = U(ζ, φ), , s > 1,
X s ≡ U(ζ, φ) ∈ H s (R × T) : U(ζ,
satisfying the symmetry Up (ζ) = U−p (−ζ).pThere exists µ0 > 0 such that
for any |µ| < µ0 , the xNLCMEs with Ω = 1 − µ2 admit a unique
localized state A± ∈ X s satisfying the bound
∃C > 0 :
kA± ∓ µU(µ·, ·)kX s ≤ C µ2 .
H s (R × T) is a Banach algebra with respect to the pointwise
multiplication for any s > 1.
If U ∈ X s for s > 1, then U ∈ L∞ (R × T) and
lim U(ζ, φ) = 0,
|ζ|→∞
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
∀φ ∈ T.
CRM Montreal, March, 2014
14 / 26
Localized states
Existence of localized stationary states
Coupled NLS equations
X
X
|Uq |2 +
Uq Ur Up−q−r = 0
Up′′ (ζ) − p 2 Up + 23 p 2 3Up
The main question is to establish the existence of a localized state U ∈ X s
for s > 1 satisfying the symmetry Up (ζ) = U−p (−ζ).
For a scalar NLS equation at p = ±1, we have the NLS soliton
1
U±1 = √ sech(ζ).
3
Will this solution persist in the system of coupled NLS equations?
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
15 / 26
Localized states
Energy arguments
Energy functional is well defined in X s for any s ≥ 1.


2
X
X
X 1
1
¯ p Uq Ur U
¯ q+r −p  dζ

U
|Up′ |2 − 
|Up |2  −
H=
2
p
3
R
Z

p∈Z
Dmitry Pelinovsky (McMaster University)
p∈Z
p,q,r ∈Z
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
16 / 26
Localized states
Energy arguments
Energy functional is well defined in X s for any s ≥ 1.


2
X
X
X 1
1
¯ p Uq Ur U
¯ q+r −p  dζ

U
|Up′ |2 − 
|Up |2  −
H=
2
p
3
R
Z

p∈Z
p∈Z
p,q,r ∈Z
Constrained variational problem
minimize H subject to fixed N =
Z X
R
|Up |2 dζ.
However, H is unbounded from below, even under the constraint. Let
1/2
Up (ζ) = λn W (λn ζ) (δp,n + δp,−n ) ,
p ∈ Z,
where W ∈ H 1 (R) is fixed and λn = n, n ∈ N. Then,
H = kW ′ k2L2 − 6nkBk4L4 → −∞ as n → ∞.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
16 / 26
Rayleigh–Ritz Approximations
Rayleigh–Ritz Approximation
Gaussian Ansatz
2
Up (ζ) = ap e −bp ζ ,
p ∈ Zodd ,
Reduced Energy
HG =
X
p 2
√
ap2 aq2
b p ap
ap2
2ap aq ar ap−q−r
− p
.
+p −p
2
p
bp
bp + bq
3 bp + bq + br + bp−q−r
Euler–Lagrange Equations
∇a HG (a, b) = 0,
Dmitry Pelinovsky (McMaster University)
∇b HG (a, b) = 0.
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
17 / 26
Rayleigh–Ritz Approximations
Rayleigh–Ritz Approximation, Results
Truncated Solutions of Euler–Lagrange Equations:
No. of Modes
1
2
3
a1
0.56060
0.56321
0.56329
Dmitry Pelinovsky (McMaster University)
b1
0.33333
0.33148
0.33189
a3
-0.13918
-0.14585
b3
3.9413
3.6287
Nonlinear Maxwell Equations
a5
0.062822
b5
8.5577
CRM Montreal, March, 2014
18 / 26
Rayleigh–Ritz Approximations
Rayleigh–Ritz Approximation, Results
Truncated Solutions of Euler–Lagrange Equations:
No. of Modes
1
2
3
a1
0.56060
0.56321
0.56329
b1
0.33333
0.33148
0.33189
a3
-0.13918
-0.14585
b3
3.9413
3.6287
a5
0.062822
b5
8.5577
Questions:
Does the solution converge to a localized state with finite energy H?
Is the alternating sign between the modes important?
Does the alternating sign persist with the number of modes?
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
18 / 26
Rayleigh–Ritz Approximations
Reduced Rayleigh–Ritz Approximation
Simplified Gaussian Ansatz
2
Up (ζ) = ap e −bp ζ ,
with
p ∈ Zodd ,
ap = A(−1)(|p|−1)/2 |p|−γ ,
bp =
p2
3
Two Parameter Energy
HG ≡ hG (γ, A) = A2 f (γ) − A4 g (γ)
At a critical point, this expression simplifies to
hG (γ, A(γ)) =
Dmitry Pelinovsky (McMaster University)
f 2 (γ)
4g (γ)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
19 / 26
Rayleigh–Ritz Approximations
Reduced Rayleigh–Ritz Approximation, Results
0.9
The Gaussian ansatz
0.8
p2 2
|Up (ζ)| ∼ |p|−γ e − 3 ζ ,
0.7
with
0.6
γ⋆ ∼ 1.26
produces
0.5
˜1
h
˜2
h
˜3
h
˜4
h
˜5
h
˜6
h
0.4
0.3
0.2
0
0.5
1
1.5
γ
Dmitry Pelinovsky (McMaster University)
2
2.5
U ∈ X s,
s < γ⋆
Numerical verification of the
condition of the theorem.
3
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
20 / 26
Rayleigh–Ritz Approximations
Ansatz without Alternating Signs
For the ansatz
p2
2
Up (ζ) = A |p|−γ e − 3 ζ ,
p ∈ Zodd ,
no extrema points occur in the reduced energy dependence on γ.
0.8
0.7
0.6
0.5
0.4
˜1
h
˜2
h
˜3
h
˜4
h
˜5
h
˜6
h
0.3
0.2
0.1
0
Dmitry Pelinovsky (McMaster University)
0.5
1
1.5
γ
2
Nonlinear Maxwell Equations
2.5
3
CRM Montreal, March, 2014
21 / 26
Stable localized states
Direct Numerical Solution of Truncated NLS System
NLS System
X
X
|Uq |2 +
Uq Ur Up−q−r = 0
Up′′ (ζ) − p 2 Up + 23 p 2 3Up
For up to 12 modes, the structure of sign alternations persists:
!’(
)
ï1
ï3
ï5
ï7
ï9
ï 11
!’%
!’#
!
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)
!
Dmitry Pelinovsky (McMaster University)
"
#
$
Nonlinear Maxwell Equations
%
&
CRM Montreal, March, 2014
22 / 26
Stable localized states
Equivalent integro-differential equation
Elliptic equation
(∂ζ2
+
∂φ2 )U
Z π
2 2
1
2
3
= ∂φ U + 3
|U(ζ, θ)| dθ U ,
3
2π −π
where
U(θ, ζ) =
X
Up (ζ)e ipθ .
p∈Zodd
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
23 / 26
Stable localized states
Time evolution of NLS Solitons in xNLCMEs
i
X ip hX + + +
Eq− 2 Ep+
Eq Er Ep−q−r + 3
3
i
hX
X ip
−
¯ 2p Ep+ +
Eq+ 2 Ep− .
= ip N
Eq− Er− Ep−q−r
+3
3
∂T Ep+ + ∂Z Ep+ = ipN2p Ep− +
µ = 0.4
Ep± (Z , 0) = ±µUp (µZ ), p = 1
µ = 0.1
40
40
0.55
0.5
35
0.55
0.5
35
0.45
0.4
25
0.35
t
0.3
20
0.25
30
0.4
25
0.35
0.3
t
30
+
E (ζ )/µ 1
0.45
20
0.25
15
0.2
15
0.2
10
0.15
10
0.15
0.1
0.1
5
5
0
ï5
0.05
0.05
0
ζ
Dmitry Pelinovsky (McMaster University)
5
+
E (ζ )/µ 1
∂T Ep− − ∂Z Ep−
0
ï5
Nonlinear Maxwell Equations
0
ζ
5
CRM Montreal, March, 2014
24 / 26
Stable localized states
Time evolution of NLS Solitons in xNLCMEs
i
X ip hX + + +
Eq− 2 Ep+
Eq Er Ep−q−r + 3
3
i
hX
X ip
−
+
¯ 2p Ep +
Eq+ 2 Ep− .
= ip N
Eq− Er− Ep−q−r
+3
3
∂T Ep+ + ∂Z Ep+ = ipN2p Ep− +
∂T Ep− − ∂Z Ep−
p=3
µ = 0.1
40
0.14
40
0.14
35
0.12
35
0.12
30
0.06
15
0.04
10
0.02
5
0
ζ
Dmitry Pelinovsky (McMaster University)
5
0.1
25
0.08
t
t
0.08
20
+
E (ζ )/µ 3
0.1
25
0
ï5
30
20
0.06
15
0.04
10
0.02
5
0
ï5
Nonlinear Maxwell Equations
+
E (ζ )/µ 3
µ = 0.4
0
ζ
5
CRM Montreal, March, 2014
24 / 26
Stable localized states
Time evolution of NLS Solitons in xNLCMEs
i
X ip hX + + +
Eq− 2 Ep+
Eq Er Ep−q−r + 3
3
i
hX
X ip
−
+
¯ 2p Ep +
Eq+ 2 Ep− .
= ip N
Eq− Er− Ep−q−r
+3
3
∂T Ep+ + ∂Z Ep+ = ipN2p Ep− +
∂T Ep− − ∂Z Ep−
p=5
40
0.08
35
0.07
35
0.07
30
0.06
30
0.06
25
0.05
25
0.05
20
0.04
20
0.04
15
0.03
15
0.03
10
0.02
10
0.02
5
0.01
5
0.01
0
ï5
0
ζ
Dmitry Pelinovsky (McMaster University)
5
t
0.08
0
ï5
Nonlinear Maxwell Equations
0
ζ
+
E (ζ )/µ 5
µ = 0.1
40
+
E (ζ )/µ 5
t
µ = 0.4
5
CRM Montreal, March, 2014
24 / 26
Conclusion
Conclusion
Summary:
Our results suggest that the localized states are robust for the nonlinear
periodic Maxwell model. Existence of such states do not eliminate a
possibility of shocks for large amplitudes.
Further directions
Prove the existence of localized solutions in the coupled NLS
equations (or in the equivalent elliptic problem)
Justify the coupled NLS equations in the original Maxwell system
with periodic Dirac delta-distributions
Consider localized solutions in the Maxwell system with bounded
refractive index.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
25 / 26
Conclusion
References
G. Simpson and M.I. Weinstein, “Coherent structures and carrier
shocks in the nonlinear Maxwell equations”, Multiscale Model Simul.
9 (2011), 955–990.
D.E. Pelinovsky, G. Simpson, and M.I. Weinstein, “Polychromatic
solitary waves in a periodic and nonlinear Maxwell system”, SIAM J.
Appl. Dynam. Syst. 11 (2012), 478–506.
D.E. Pelinovsky and D.V. Ponomarev, “Justification of a nonlinear
Schr¨
odinger model for laser beams in photopolymers”, Zeitschrift f¨
ur
angewandte Mathematik und Physik (2014), in print.
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
26 / 26
Conclusion
References
G. Simpson and M.I. Weinstein, “Coherent structures and carrier
shocks in the nonlinear Maxwell equations”, Multiscale Model Simul.
9 (2011), 955–990.
D.E. Pelinovsky, G. Simpson, and M.I. Weinstein, “Polychromatic
solitary waves in a periodic and nonlinear Maxwell system”, SIAM J.
Appl. Dynam. Syst. 11 (2012), 478–506.
D.E. Pelinovsky and D.V. Ponomarev, “Justification of a nonlinear
Schr¨
odinger model for laser beams in photopolymers”, Zeitschrift f¨
ur
angewandte Mathematik und Physik (2014), in print.
Merci beaucoup pour votre attention!
Dmitry Pelinovsky (McMaster University)
Nonlinear Maxwell Equations
CRM Montreal, March, 2014
26 / 26

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