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Appendix C
Analysis of coupled supermode perturbation equations

In sections 4.3 and 4.3.2, we have a pair of equations describing the perturbation from a coupled-pair soliton in the antisymmetric mode

∂ε ∂2ε ∂ε --y+ i(p2 + c2)--y2 + i(q – c0)εy + 2c1-y = 2iy2εy + iy2ε∗y (C.1) ∂ζ ∂τy ∂τy ∂ey+ i(p – c)∂2ey + i(q + c)e = 2iy2e + iy2e∗ (C.2) ∂ζ 2 2 ∂τ2y 0 y y y

and a further pair describing perturbation from a soliton in the symmetric mode

∂εx ∂2εx ∂εx 2 2∗ ∂ζ-+ i(p2 – c2)-∂τ2 +i(q+ c0)εx – 2c1∂τx = 2ix εx + ix εx (C.3) x 2 ∂ex+ i(p2 + c2)∂-ex2 + i(q – c0)ex = 2ix2ex + ix2e∗x (C.4) ∂ζ ∂τx

To these, we wish to find analytical solutions of the form εy = ˜ε y(τy)eλyζ, ey = ẽy(τy)elyζ, εx = ˜εx(τx)eλxζ and ex = ẽx(τx)elxζ. These can be formulated as eigenvalue problems, where the eigenfunctions ˜ε y(τy), ẽy(τy), ˜ε x(τx) and ẽx(τx) describe the perturbations, whilst their corresponding eigenvalues λy, ly, λx and lx determine whether the perturbations will grow and thus lead to instability.

C.1 Reduction to eigenvalue problem form

Antisymmetric mode

We split εy and ey into their real and imaginary components as εy = εy' + εy''i and ey = ey' + ey''i. Equations C.1 and C.2 can then be written as matrix equations of the form

[ ε' ] ∂ [ ε' ] ˆΛy y'' = --- y'' (C.5) [ εy ] ∂ζ[ εy ] e'y ∂ e'y ˆLy e'' = ∂ζ- e'' (C.6) y y
where the matrix operators ˆ Λy and ˆ Ly are defined by
⌊ ∂ ∂2 2 ⌋ ˆ ⌈ – 2c1 ∂τy- (p2 + c2)∂τy2+ q– c0 – y ⌉ Λy = – (p2 + c2) ∂∂2τ2– q+ c0 + 3y2 – 2c1∂∂τ- (C.7) ⌊ y y ⌋ 0 (p2 – c2) ∂∂2τ2+ q+ c0 – y2 ˆLy = ⌈ ∂2- 2 y ⌉ (C.8) – (p2 – c2)∂τ2y – q– c0 + 3y 0
Our trial functions are of the form
λyζ εy = ˜εy(τy)e (C.9) ey = ˜ey(τy)elyζ (C.10)
and we split the time dependent parts ˜ε y and ẽy up as
˜εy(τy) = ℜ [αy(τy)]+ ℜ [βy (τy)]i (C.11) ˜e (τ) = ℜ [a (τ)]+ ℜ [b (τ )]i (C.12) y y y y y y
Therefore, the vectors representing ey and εy can be rewritten as
[ ] ([ ] [ ] ) ε'y (ζ,τy) 1 αy (τy) λyζ α∗y (τy) λ∗yζ ε'y'(ζ,τy) = 2 βy (τy) e + β∗y (τy) e (C.13) [ ' ] ([ ] [ ∗ ] ) ey (ζ,τy) = 1 ay (τy) elyζ + ay(τy) el∗yζ (C.14) e'y'(ζ,τy) 2 by(τy) b∗y(τy)
Substituting these into equations C.19 and C.20 yields
[ ] [ ∗ ] [ ] [ ∗ ] Λˆy αy + ˆΛy αy = λy αy + λ∗y αy (C.15) βy β∗y βy β∗y [ a ] [ a∗ ] [ a ] [ a∗ ] ˆLy y + ˆLy y∗ = ly y + l∗y y∗ (C.16) by by by by
which can be reduced to eigenvalue problems of the form
[ ] [ ] αy αy Λˆy = λy (C.17) [ βy ] [ βy] ˆ ay ay Ly b = ly b (C.18) y y
as any solutions to equations C.17 and C.18 will automatically satisfy equations C.15 and C.16.

Symmetric mode

We split εx and ex into their real and imaginary components as εx = εx' + εx''i and ex = ex' + ex''i. Equations C.1 and C.2 can then be written as matrix equations of the form

[ ] [ ] ˆΛ ε'x = ∂-- ε'x (C.19) x ε''x ∂ζ ε''x [ ' ] [ ' ] ˆLx ex = ∂-- ex (C.20) e''x ∂ζ e'x'
where the matrix operators ˆΛx and Lˆx are defined by
[ -∂- ∂2- 2 ] ˆΛx = 2c21∂τx (p2 – c2)∂τx2+ q+ c0 – x (C.21) – (p2 – c2) ∂∂τx2– q– c0 + 3x2 2c1∂∂τx [ ∂2- 2 ] ˆLx = 20 (p2 + c2)∂τx2+ q– c0 – x (C.22) – (p2 + c2) ∂∂τx2– q+ c0 + 3x2 0
Our trial functions are of the form
εx = ˜εx(τx)eλxζ (C.23) lxζ ex = ˜ex(τx)e (C.24)
and we split the time dependent parts ˜ε x and ẽx up as
˜εx (τx) = ℜ [αx (τx)]+ ℜ [βx (τx)]i (C.25) ˜ex (τx) = ℜ [ax (τx)]+ ℜ [bx(τx)]i (C.26)
Therefore, the vectors representing εx and ex can be rewritten as
[ ] ([ ] [ ] ) ε'x(ζ,τx) 1 αx (τx) λxζ α ∗x(τx) λ∗ζ ε''(ζ,τ ) = 2 β (τ ) e + β ∗(τ ) e x (C.27) [ x x ] ([ x x ] [ x x ] ) e'x(ζ,τx) 1 ax (τx) lxζ a∗x(τx) l∗xζ e'x'(ζ,τx) = 2 bx(τx) e + b∗x(τx) e (C.28)
Substituting these into equations C.19 and C.20 yields
[ ] [ ] [ ] [ ] ˆ αx ˆ α∗x αx ∗ α∗x Λx βx + Λx β∗x = λx βx + λx β∗x (C.29) [ ] [ ∗ ] [ ] [ ∗ ] ˆL ax + ˆL ax = l ax + l∗ ax (C.30) x bx x b∗x x bx x b∗x
which can be reduced to eigenvalue problems of the form
[ ] [ ] Λˆx αx = λx αx (C.31) βx βx [ a ] [ a ] ˆLx x = lx x (C.32) bx bx
as any solutions to equations C.31 and C.32 will automatically satisfy equations C.29 and C.30.

C.2 Continuum spectrum of delocalised modes and its band-gap

If an eigenfunction is delocalised, it will (by definition) not tend towards zero at the extremities of time. By working at these extremities (at which the magnitude of the soliton tends towards zero) we can solve the equations of motion exactly, revealing an infinite set of sinusoidal eigenfunctions corresponding to a continuum of eigenvalues. In the locality of the soliton, the eigenfunctions will be pertubed, but the eigenvalues (which are time independent) will remain the same.

This continuum spectrum of delocalised-mode eigenvalues lies along the imaginary axis. This can be broken by a band-gap in which no delocalised-mode eigenvalues exist. The extent of this band-gap, and the conditions for its existence are derived below.

Antisymmetric mode

In the tails of the soliton, where y(τy) → 0, we choose trial functions of the form

[ ] [ ] αy ˜αyeiγyτy βy = β˜yeiγyτy (C.33) [ ] [ ] ay ˜ayeigyτy by = ˜byeigyτy (C.34)
where the constants ãy, ˜b y, α˜y, ˜β y define the complex arguments of εy and ey. The wavenumbers γy and gy must be real for a continuous wave solution. Substituting equations C.33 and C.34 into equations C.17 and C.18 shows the solutions to be valid, given the respective conditions
( 2 ) iλy = 2c1γy ± (p2 + c2)γy + c0 – q (C.35) ily = ± ((p2 – c2)g2y – c0 – q) (C.36)
If the condition that γy or gy must be real is violated, a band-gap will occur. We can search for band-gaps by solving equations C.35 and C.36 for γy and gy respectively to give
√ ------------------------ ∓2c1 ± ' c21 + (p2 +c2)(q– c0 ± iλy) γy = -------------p2 +-c2------------ (C.37) √ ---------- g = ± ' q+-c0 ±-ily (C.38) y p2 – c2
where the operators ± and ±' denote that the signs may be chosen independently. Non-real solutions (and hence a band-gap) will exist when the arguments in the square-root functions of equations C.37 and C.38 become negative. Therefore, about the point λy = 0, a band-gap will exist when c12 + (p2 + c2) (q– c0) < 0. Similarly, about the point ly = 0, a band-gap will exist when (q + c0)/(p2 – c2) < 0.

The edges of the bandgap occur when the arguments of the square-root functions become zero. Solving the resulting equations gives the extent of the band gaps as

| | | | || ---c21-- || || ---c21--|| –|q– c0 + p2 + c2 | < ℑ [λy] < |q– c0 + p2 + c2| (C.39) – |q+ c0| < ℑ[ly] < |q+ c0| (C.40)

Symmetric mode

In the tails of the soliton, where x(τx) → 0, we choose trial functions of the form

[ ] [ iγxτx ] αx = ˜αxe (C.41) βx β˜xeiγxτx [ a ] [ ˜a eigxτx ] x = ˜x igxτx (C.42) bx bxe
where the constants α˜x, ˜β x, ãx and ˜b x define the complex arguments of εx and ex. The wavenumbers gy and γy must be real for a continuous wave solution. Substituting equations C.41 and C.42 into equations C.31 and C.32 shows the solutions to be valid, given the respective conditions
iλx = – 2c1γx ± ((p2 – c2)γ2 – c0 – q) (C.43) ( 2 x ) ilx = ± (p2 + c2)gx + c0 – q (C.44)
If the condition that γx or gx must be real is violated, a band-gap will occur. We can search for band-gaps by solving equations C.43 and C.44 for γx and gx respectively to give
√ ------------------------ γ = ±2c1 ±-'-c21-+-(p2-– c2)(q+-c0 ±-iλx) (C.45) x p2 – c2 √ ---------- gx = ± ' q–-c0 ±-ily (C.46) p2 +c2
where the operators ± and ±' denote that the signs may be chosen independently. Non-real solutions (and hence a band-gap) will exist when the arguments in the square-root functions of equations C.45 and C.46 become negative. Therefore, about the point λx = 0, a band-gap will exist when c12 + (p2 – c2) (q+ c0) < 0. Similarly, about the point lx = 0, a band-gap will exist when (q – c0)/(p2 +c2) < 0.

The edges of the bandgap occur when the arguments of the square-root functions become zero. Solving the resulting equations gives the extent of the band gaps as

| 2 | | 2 | –||q+ c0 +---c1-- || < ℑ [λx] < ||q+ c0 +---c1-- || (C.47) | p2 – c2 | | p2 – c2 | – |q– c0| < ℑ[lx] < |q– c0| (C.48)

C.3 General numerical solutions

We can find approximate general solutions to equations C.17, C.18, C.31 and C.32 by discretising εy(τy), ey(τy), εx(τx) and ex(τx) into N-membered column vectors ˆε y, êy, ˆε x and êx.

In such a basis, differential operators become N × N matrices of the form

⌊ 0 1 ⌋ | | || – 1 0 1 || -∂- –→ ˆI' ≡ -1--|| – 1 0 ... || (C.49) ∂ τ 2Δτ || . . || |⌈ .. .. 1 |⌉ – 1 0 ⌊ ⌋ | – 2 1 | || 1 – 2 1 || ∂2-- ˆ'' --1-- || .. || ∂τ2 –→ I ≡ (Δτ)2 || 1 – 2 . || (C.50) |⌈ ... ... 1 |⌉ 1 – 2
where Δτ is the time step. Making the transformation gives ordinary (i.e. not containing differential operators) matrices of the form
[ – 2c ˆI' (p + c)Iˆ'' + qˆI – cIˆ– y2ˆI ] ˆΛy –→ ˆ'' 1ˆ ˆ 2ˆ 2 2 ˆ' 0 (C.51) [ – (p2 + c2)I – qI + c0I + 3y I – 2c1I ] 0 (p2 – c2)Iˆ'' + qˆI + c0Iˆ– y2ˆI ˆLy –→ ˆ'' ˆ ˆ 2ˆ (C.52) [ – (p2 – c2)I – qI – c0I + 3y I 0 ] ˆ 2c1ˆI' (p2 – c2)Iˆ'' + qˆI + c0ˆI – x2ˆI Λx –→ – (p – c)Iˆ'' – qˆI – c ˆI + 3x2ˆI 2cIˆ' (C.53) [ 2 2 0 1 ] ˆ 0 (p2 + c2)Iˆ'' + qˆI – c0ˆI – x2ˆI Lx –→ – (p2 + c2)Iˆ'' – qˆI + c0ˆI + 3x2ˆI 0 (C.54)
where Î is the N × N identity matrix. Finding the eigenvalues (λy, ly, λx and lx respectively) is a straightforward numerical task, for which the "eigs" function of MATLAB was used. Any eigenvalues lying outside the analytically predicted band structures were recorded, thus providing the results given in sections 4.3.1 and 4.3.2.

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