u2x u2−uxx
u
"
x
with initial data not depending on ε
m(x, 0) = A2(x), n(x, 0) = S′(x)
The modulational instability comes from ε → 0.
2.3 Conservation laws
For now we shall consider the following dimensionless nonlinear Schr¨odinger equation to model rogue wave in deep water
iεut+ε2 2uxx−!
f (|u|2) + V (x)"
u = 0. (4)
where f is a real-valued smooth function and V (x) is a given real-valued electrostatic potential and u(x, t) is the wave function. The small free parameter ϵ with 0 < ϵ ≪ 1 denotes the modulation scale. The wave function can be used to compute primary physical quantities such as the mass density
ρ(x, t) = |u(x, t)|2 the momentum density
J(x, t) = εIm(u∗(x, t)∇u(x, t)) = ε
2i(u∗∇u − u∇u∗) and energy density
E(x, t) =ε2
2|∇u(x, t)|2+ V (x)|u(x, t)|2
where the “*” denotes complex conjugation. The solution of the self-focussing nonlinear Schr¨odinger equation (4) will be sought later in the semiclassical limit. Our interest stems from the existence of modulationally unstable solution behavior.
In this section we shall obtain the conservation laws of mass density and momentum density in one dimension. This will help us to analyze and understand the mechanism of the solution. We start by considering simpler case where the potential V (x) is zero
iεut+ε2
2uxx− U′(|u|2)u = 0. (5)
We have also assumed f (|u|2) = U′(|u|2) to be a twice differentiable nonlinear real-valued function.
The local conservation laws are then
∂ρ
To obtain the first conservation law (6), we first eliminate U′(|u|2)u by multiplying the complex conjugate of u to (5). On the other hand, we take the complex conjugate of (5) and multiply −u to it. These both give
iεu∗ut+ε2
2u∗uxx− U′(|u|2)u∗u = 0 iεuu∗t−ε2
2uu∗xx+ U′(|u|2)u∗u = 0 Adding them together we will have
iε!
where J(x, t) = −iε2!
u∗ux− uu∗x
"
in one dimension.
To obtain the second conservation law (7), we first take the derivative of J(x, t) with respect to time.
∂
∂tJ(x, t) = −iε 2
!u∗tux+ u∗uxt− utu∗x− uu∗xt
"
Since iεut+ε22uxx− U′(ρ)u = 0, we have
iεut= −ε2
2uxx+ U′(ρ)u
−iεu∗t = −ε2
2u∗xx+ U′(ρ)u∗ iεutx= −ε2
2uxxx+ U′′(ρ)ρxu + U′(ρ)ux
iεu∗tx= ε2
2u∗xxx− U′′(ρ)ρxu∗− U′(ρ)u∗x Plugging them into the time derivative of J(x, t) we see that
∂
∂tJ(x, t) + ∂
∂xP (ρ) = −ε2
4Q(u, ux, uxx) (8)
where P (ρ) = ρU′(ρ) − U(ρ) and Q(u, ux, uxx) = 2ux+ u∗x− uu∗xx− u∗uxx. On the other hand,
ρ = uu∗ ρx= uxu∗+ uu∗x (ρx)2= (uxu∗+ uu∗x)2
= (uxu∗− uu∗x)2+ 4uu∗uxu∗x
∴(uxu∗− uu∗x)2= (ρx)2− 4ρuxu∗x Adding (8) to the equation above, we obtain
∂J
These two conservation equations have the form of a perturbation of the compressible Euler equa-tions of fluid dynamics with the pressure P (ρ). If the Euler part of these equaequa-tions is to be hyper-bolic, then the pressure P (ρ) must be a strictly increasing function of ρ, that is P′(ρ) = ρU′′(ρ) > 0.
This implies U must be a strictly convex function of ρ and this corresponds to a defocusing NLS case.
On the other hand, for the focusing NLS case, we must have P′(ρ) = ρU′′(ρ) < 0. This means that when the mass density ρ increases, the pressure P (ρ) decreases. Thus there will be a phenomenon leading to the development of mass concentrations.
Also on the right hand side, we have the O (ε2) dispersive term. If the mass density ρ(x, t) and momentum density J(x, t) have a singularity, then the small dispersive term can no longer be negligible since it will develop a small wavelength oscillations.
3 Derivation of coupled NLS equation from birefringent op-tical fibers
We shall follow the derivation in [20] and start from Maxwell’s wave equation for plane waves
∂2E
∂z2 − 1 c2
∂2D
∂t2 = 0 (9)
where E is the electric field, D is the dielectric response, c is the speed of light, z and t are the propagation distance and time. The relationship of dielectric response and electric field is given by
D= E + 4πP
where P is the polarizability. Note that E and D are observed fields which means E, D, P are real.
The next task is to decide the relationship of P and E. We shall consider the linear response and nonlinear response separately and combine them into Maxwell’s wave equation in order to derive the equations describing pulse propagation in birefringent optical fibers.
3.1 Linear Response
For the linear response, we shall assume that the medium is non-isotropic. This means that the medium is birefringent along the z-direction. Thus, PLand E are related through a tensor χ
PL(z, t) = ' t
−∞χ(t − ˜t) · E(z, ˜t)d˜t (10)
We now consider the Fourier transforms of E, PLand χ. In general, f (z, ω) =ˆ
' ∞
−∞
f (z, t)eiωtdt
and
f (z, t) = 1 2π
' ∞
−∞
f (z, ω)eˆ −iωtdω. (11)
We shall define
fˆ+(z, ω) =
⎧
⎪⎪
⎨
⎪⎪
⎩
f (z, ω),ˆ ω > 0
0, ω < 0
(12)
and ˆf−(z, ω) = ˆf (z, ω) − ˆf+(z, ω). The corresponding quantities f+(z, t) and f−(z, t) are then defined by (11). Following this, we have defined the terms E±(z, ω), ˆE±(z, ω) and so on. Hence we can rewrite the linear response
Pˆ+(z, ω) = ˆχ+(ω) · ˆE+(z, ω) (13)
Here we shall assume that the nonzero contribution to E and D comes from a small region in ω space surrounding some carrier frequency ω0 and another small region surrounding its opposite
−ω0. Since E(z, t) is real, we always have ˆE(z, −ω) = ˆE∗(z, ω). This implies we can simply consider the positive frequency components in the small region around ω = ω0 and restore the negative frequency components at the end of calculation.
Next, we shall write
Eˆ+= ˆE+1e1+ ˆE2+e2 (14) Pˆ+= ˆP1+e1+ ˆP2+e2 (15)
where e1, e2 are the two orthonormal eigenvectors of ˆχ+ at any arbitrary frequency ω with the corresponding eigenvalues χ1and χ2. This gives us
Pˆ1+= ˆP+e1=#Eˆ+· ˆχ+$e1= χ1Eˆ+e1= χ1Eˆ1+ (16) Pˆ2+= ˆP+e2=#Eˆ+· ˆχ+$e2= χ2Eˆ+e2= χ2Eˆ2+ (17)
where e1· e1∗= e2· e2∗= 1 and e1· e2∗= 0. We may further assume that in the positive small region in ω space, e1(ω) = e1(ω0) and e2(ω) = e2(ω0) so as to ignore the coupling of linear mode.
We then have
P1+(z, t) = ' t
−∞
χ1(t − ˜t)E1+(z, ˜t)d˜t (18)
P2+(z, t) = ' t
−∞
χ2(t − ˜t)E2+(z, ˜t)d˜t (19) If we let
P1+(z, t) = ρ(z, t)eik0z−iω0t (20) E+1(z, t) = U (z, t)eik0z−iω0t (21)
where ρ and U are the envelopes of P1+ and E1+ with k0 = k(ω0) given by the linear dispersion relations corresponding to the eigenmodes of ˆχ+
k(ω) = ω
Substitute (20) and (21) into (18), we obtain ρ(z, t) =
Till now, we have reduced all relationship to U , in order to obtain the equation linearly describing pulse propagation in birefringent optical fibers, we first assemble D+1 by substituting ρ(z, t) into P1+
where ˆϵ1= 1 + 4π ˆχ1has been previously defined in the linear dispersion relation. Hence plugging D+1 and E+1 into the Maxwell’s wave equation where we have omitted the third order and fourth order time derivatives. Since k(ω) = ωc,ˆϵ1(ω) we see that
This yields (24) can be simplified
i!∂U
Since U is slowly varying in time and space, we require ∂U∂z and ∂U∂t to be in the same order, that is
∂U
∂z + k′∂U
∂t ≡ 0
This gives
∂2U
∂z2 = (k′)2∂2U
∂t2 Thus, (25) can further be simplified as
i!∂U
Similarly, if we let
E2+(z, t) = V (z, t)eil0z−iω0t
For the nonlinear response, we shall consider third order nonlinearity due to Kerr effect for which the polarizability is in the form
PNL(z, t) =
We now make the assumption that the nonlinearity is rapid compared to ω−10 or the period of the light. Thus, we have
¯
The equation (28) can now be written
PNL(z, t) = ¯χNL/E(z, t) · E(z, t)0E(z, t) (29)
As in linear response, we shall consider the the contribution of PNL in positive small region con-centrated near ω = ω0, by taking P+
P+(z, t) = ¯χNL/2E+(z, t) · E−(z, t)0E+(z, t) + ¯χNL/E+(z, t) · E+(z, t)0E−(z, t)