Perturbation theory for the defocusing nonlinear Schrödinger equation

Perturbation theory for the defocusing nonlinear Schrödinger equation Theodoros P. Horikis University of Ioannina In collaboration with: M. J. Ablowitz, S. D. Nixon and D. J. Frantzeskakis

Outline What is a dark soliton? Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations The old theory Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations The old theory Something is missing Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations The old theory Something is missing The complete theory Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations The old theory Something is missing The complete theory An example Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations The old theory Something is missing The complete theory An example Special perturbations Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations The old theory Something is missing The complete theory An example Special perturbations Connection to other perturbed equations Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

Outline What is a dark soliton? Dark solitons as solutions of the defocusing NLS The effect of perturbations The old theory Something is missing The complete theory An example Special perturbations Connection to other perturbed equations References Theodoros P. Horikis Perturbation theory for the defocusing NLS 1 / 38

What is a Dark Soliton? Introduction Dark solitons are manifested as localized dips in intensity that decay off of a continuous wave background. They are termed black if the intensity dip goes to zero and grey otherwise. In terms of mathematics: Localized solutions of PDEs with non-zero boundary conditions and non-zero phase shift. Theodoros P. Horikis Perturbation theory for the defocusing NLS 2 / 38

Do we like dark solitons? Introduction Reasons to dislike: Theodoros P. Horikis Perturbation theory for the defocusing NLS 3 / 38

Introduction Do we like dark solitons? Reasons to dislike: Dark solitons have infinite energy due to the background (non-vanishing boundary conditions) Theodoros P. Horikis Perturbation theory for the defocusing NLS 3 / 38

Introduction Do we like dark solitons? Reasons to dislike: Dark solitons have infinite energy due to the background (non-vanishing boundary conditions) It is difficult to distinguish experimentally between the soliton and the background Special treatment of all integral quantities is required Reasons to like: Fundamental excitations of the universal defocusing NLS equation Theodoros P. Horikis Perturbation theory for the defocusing NLS 3 / 38

Introduction Do we like dark solitons? Reasons to dislike: Dark solitons have infinite energy due to the background (non-vanishing boundary conditions) It is difficult to distinguish experimentally between the soliton and the background Special treatment of all integral quantities is required Reasons to like: Fundamental excitations of the universal defocusing NLS equation Observed in many applications: liquids, discrete mechanical systems, thin magnetic films, optical media, BECs, etc Theodoros P. Horikis Perturbation theory for the defocusing NLS 3 / 38

Introduction Do we like dark solitons? Reasons to dislike: Dark solitons have infinite energy due to the background (non-vanishing boundary conditions) It is difficult to distinguish experimentally between the soliton and the background Special treatment of all integral quantities is required Reasons to like: Fundamental excitations of the universal defocusing NLS equation Observed in many applications: liquids, discrete mechanical systems, thin magnetic films, optical media, BECs, etc Conclusion: We don t like dark solitons! Theodoros P. Horikis Perturbation theory for the defocusing NLS 3 / 38

The Mathematical Properties Dark Solitons Solutions of the defocusing NLS Normalized 1D NLS: Dark soliton solution i u z 1 2 u 2 t 2 + u 2 u = 0 u(z, t) = n 0 exp( iµz)(btanhζ+ia) ζ µb ( t µa z ) A is the soliton velocity, B is the soliton depth A 2 + B 2 = 1, so hereafter A = sinφ, B = cosφ Black solitons: B = 1, gray solitons B < 1 u(0,t) 2 1 0.8 0.6 0.4 0.2 n 0 B 2 <1 black gray Density notch with phase jumps: [ ( ) A φ = 2 tan 1 π ] B 2 0-10 -5 0 5 10 t Theodoros P. Horikis Perturbation theory for the defocusing NLS 4 / 38

Integrals of motion The Mathematical Properties Dark Solitons The equation iu z 1 2 u tt + u 2 u = 0 also has the following conserved quantities: Energy: E = Momentum: I = + + Center of mass: R = Hamiltonian: H = + + ( u 2 u 2) dt i 2 (uu t u u t ) dt t ( u 2 u 2) dt 1 ( ut 2 +(u 2 2 u 2 ) 2) dt Theodoros P. Horikis Perturbation theory for the defocusing NLS 5 / 38

Perturbed Solitons What happens under perturbation Suppose now that we add a perturbation, namely iu z 1 2 u tt + u 2 u = ǫf[u] where ǫ 1. With u s (z, t) = (A + ibtanh[b(t Az t 0 )]) e iσ 0 we need to study the evolution of the soliton s parameters under F[u]. Theodoros P. Horikis Perturbation theory for the defocusing NLS 6 / 38

The Old Method The steps through an example Perturbation theory Consider the two photon absorption problem iu z + 1 2 u tt u 2 u = ik u 2 u Step 1: Distinguish soliton (dip) from background u(z, t) = u (z)v(z, t) so that i u z u 2 u = ik u 2 u Theodoros P. Horikis Perturbation theory for the defocusing NLS 7 / 38

The Old Method The steps through an example Perturbation theory Consider the two photon absorption problem iu z + 1 2 u tt u 2 u = ik u 2 u Step 2: Define new independent variables dζ = u 2 (z) dz and ξ = u (z) t so that iv ζ + 1 2 v ξξ ( v 2 1 ) v = ik ( v 2 1 ) v ( where v(ζ,ξ) = cosφtan T i sinφ, T = cosφ ξ ) + sinφ dζ Theodoros P. Horikis Perturbation theory for the defocusing NLS 8 / 38

The Old Method The steps through an example Perturbation theory Consider the two photon absorption problem iu z + 1 2 u tt u 2 u = ik u 2 u Step 3: Solve the modified conserved quantities dh dζ = ǫ + where F[v] = ik ( v 2 1 ) v so that dφ dζ = ( F[v]ṽ ζ F [v]v ζ ) dξ { ǫ + } 2 cos 2 φ sinφ Re F[v]v ζ dξ Theodoros P. Horikis Perturbation theory for the defocusing NLS 9 / 38

The results The Old Method Perturbation theory Finally, the background evolves according to u (z) = u (0) 1+2Ku 2 (0)z eiθ(z), θ(z) = z 0 u 2 (z ) dz and the soliton according to dφ dz = 1 3 Ku2 (z) sin(2φ) t 0 (z) = z 0 sin(z ) dz Theodoros P. Horikis Perturbation theory for the defocusing NLS 10 / 38

Analysis vs simulations The Old Method Perturbation theory Taken from Kivshar and Yang, PRE pp167, v49, 1994 Theodoros P. Horikis Perturbation theory for the defocusing NLS 11 / 38

Analysis vs simulations The Old Method Perturbation theory Something is missing? Theodoros P. Horikis Perturbation theory for the defocusing NLS 12 / 38

Analysis vs simulations The Old Method Perturbation theory Something is missing! Theodoros P. Horikis Perturbation theory for the defocusing NLS 13 / 38

Perturbation theory The true picture Core Soliton Moving Shelf u Outer Region Outer Region Outer Region Inner Region t Inner Region Outer Region phase Moving Shelf Core Soliton t Theodoros P. Horikis Perturbation theory for the defocusing NLS 14 / 38

Basic idea The Complete Theory Perturbation theory Go beyond adiabatic approximation Theodoros P. Horikis Perturbation theory for the defocusing NLS 15 / 38

The Complete Theory Perturbation theory Basic idea Go beyond adiabatic approximation The complete soliton consists of: core, background and shelf Theodoros P. Horikis Perturbation theory for the defocusing NLS 15 / 38

The Complete Theory Perturbation theory Basic idea Go beyond adiabatic approximation The complete soliton consists of: core, background and shelf The shelf is the discrepancy between the approximate soliton solution and the background To bridge the inner soliton and outer background we use a moving boundary layer Theodoros P. Horikis Perturbation theory for the defocusing NLS 15 / 38

The Complete Theory Perturbation theory Basic idea Go beyond adiabatic approximation The complete soliton consists of: core, background and shelf The shelf is the discrepancy between the approximate soliton solution and the background To bridge the inner soliton and outer background we use a moving boundary layer The shelf dynamics will be derived from the perturbed conservation laws Theodoros P. Horikis Perturbation theory for the defocusing NLS 15 / 38

The Complete Theory The Complete Theory Outline Consider the perturbed NLS in the form Perturbation theory iψ z 1 2 ψ tt + ψ 2 ψ = ǫf[ψ] and remove the background as usual with ψ = u(z, t) exp(i z 0 u dz ) iu z 1 2 u tt +( u 2 u 2 )u = ǫf[u] The additional steps are as follows: Break the problem into two regions Outer region consists of the background; inner region consists of soliton and shelf Theodoros P. Horikis Perturbation theory for the defocusing NLS 16 / 38

The Complete Theory The Complete Theory Outline Consider the perturbed NLS in the form Perturbation theory iψ z 1 2 ψ tt + ψ 2 ψ = ǫf[ψ] and remove the background as usual with ψ = u(z, t) exp(i z 0 u dz ) iu z 1 2 u tt +( u 2 u 2 )u = ǫf[u] The additional steps are as follows: Break the problem into two regions Outer region consists of the background; inner region consists of soliton and shelf Break u into phase and magnitude u = q exp(iφ) Theodoros P. Horikis Perturbation theory for the defocusing NLS 16 / 38

The Complete Theory The Complete Theory Outline Consider the perturbed NLS in the form Perturbation theory iψ z 1 2 ψ tt + ψ 2 ψ = ǫf[ψ] and remove the background as usual with ψ = u(z, t) exp(i z 0 u dz ) iu z 1 2 u tt +( u 2 u 2 )u = ǫf[u] The additional steps are as follows: Break the problem into two regions Outer region consists of the background; inner region consists of soliton and shelf Break u into phase and magnitude u = q exp(iφ) Expand in series of epsilon q = q 0 +ǫq 1 + O(ǫ 2 ), φ = φ 0 +ǫφ 1 + O(ǫ 2 ) Theodoros P. Horikis Perturbation theory for the defocusing NLS 16 / 38

The Complete Theory The Complete Theory Outline Consider the perturbed NLS in the form Perturbation theory iψ z 1 2 ψ tt + ψ 2 ψ = ǫf[ψ] and remove the background as usual with ψ = u(z, t) exp(i z 0 u dz ) iu z 1 2 u tt +( u 2 u 2 )u = ǫf[u] The additional steps are as follows: Break the problem into two regions Outer region consists of the background; inner region consists of soliton and shelf Break u into phase and magnitude u = q exp(iφ) Expand in series of epsilon q = q 0 +ǫq 1 + O(ǫ 2 ), φ = φ 0 +ǫφ 1 + O(ǫ 2 ) Note: The expansions are valid only in the inner region! Theodoros P. Horikis Perturbation theory for the defocusing NLS 16 / 38

The expansion At first order The Complete Theory Perturbation theory q 0 e iφ 0 = (A + ibtanh[b(t Az t 0 )]) e iσ 0 and A 2 + B 2 = u 2. Introduce a new scale Z = ǫz, then: At O(ǫ) the shape of the shelf is described by q ± 1 and φ± 1t where q 1 (Z, t) q ± 1 (Z) and φ 1t(Z, t) φ ± 1t (Z) as t ± Use the modified conserved quantities ( dh dz = ǫ E d dz u2 + 2Re de D dz = 2ǫIm + di dz = 2ǫRe dr dz = I+ 2ǫIm + + + ) F[u]u zdt F[u ]u F[u]u dt F[u]u t dt t (F[u ]u F[u]u ) dt Theodoros P. Horikis Perturbation theory for the defocusing NLS 17 / 38

The Complete Theory The evolution of the parameters Perturbation theory Then moving along the frame of reference T = t z 0 A(ǫs) ds t 0 d dz u = Im[F[u ]] 2B d ( + ) dz A = Re F[u 0 ]u 0T dt ( d + ) u dz σ 0 = B Z Im F[u ]u F[u 0 ]u0 dt + Re[F[u ]] q + 1 = 1 2 (σ 0Z + φ 0Z )/(u A) q 1 = 1 2 (σ 0Z φ 0Z )/(u + A) φ + 1T = 2q+ 1, φ 1T = 2q 1 B Z = (u u Z AA Z )/B φ 0Z = (2AB Z 2BA Z )/u 2 φ 0 = 2 tan 1 (A/B) Theodoros P. Horikis Perturbation theory for the defocusing NLS 18 / 38

Two photon absorption An example Perturbation theory As before we take F[u] = iγ u 2 u. Then: d dz A = γ d dz u = γu 3 ( 2 3 A 2 + 1 3 u2 ) A d dz φ 0 = 4 3 γab, φ 0 = 2 tan 1 (A/B) σ 0Z = γ B ( 2A 2 + 1 ) u 3 u2 ( q ± 1 = γ(u ± A) 2A 2 + 1 Bu 3 u2 ± 4 ) 3 Au ( φ + 1T = 2γ(u + A) 2A 2 + 1 Bu 3 u2 + 4 ) 3 Au ( φ 1T = 2γ(u A) 2A 2 + 1 Bu 3 u2 4 ) 3 Au Theodoros P. Horikis Perturbation theory for the defocusing NLS 19 / 38

Theory vs simulation Two photon absorption Perturbation theory 2 1.5 u (z) Numerics Asymptotics u 1 0.5 A(z) 0 0 5 10 15 20 25 30 z Theodoros P. Horikis Perturbation theory for the defocusing NLS 20 / 38

Two photon absorption Are we missing anything? Perturbation theory 30 25 Numerics Asymptotics 20 z 15 10 5 0-50 0 50 t Theodoros P. Horikis Perturbation theory for the defocusing NLS 21 / 38

Mode-locked lasers Special perturbations The Power-Energy Saturation (PES) equation To analyze dark solitons in mode-locked (ML) lasers we use a model expressed in the following dimensionless form, iψ z 1 2 ψ tt + ψ 2 ψ = ig 1+E/E 0 ψ+ iτ 1+E/E 0 ψ tt il 1+P/P 0 ψ where the complex electric field envelope ψ(z, t) is subject to the boundary conditions ψ(z, t) ψ as t. Here, E(z) = + ( ψ 2 ψ 2 ) dt is the dark-pulse energy, P(z, t) = ψ 2 ψ 2 is the instantaneous power, while E 0 and P 0 are related to the saturation energy and power, respectively. Furthermore, g, τ, and l are all positive, real constants, with the corresponding terms representing saturable gain, spectral filtering, and saturable loss. Theodoros P. Horikis Perturbation theory for the defocusing NLS 22 / 38

Interesting dynamics Mode-locked lasers Special perturbations Integrate the PES with initial profile with a π-phase jump. The gain parameter g is varied, while E 0 = P 0 = 1 and τ = l = 0.1. 5 g = 1.0 ψ 4 3 g = 0.7 2 g = 0.5 1 g = 0.2 g = 0.1 0 0 100 200 z 300 400 500 ψ 2 1 500 0 250 z 0 20 10 0 10 t 20 Locking onto stable dark solitons is only achieved when the gain term is sufficiently strong, i.e. the parameter g is large enough to counter balance the losses. Theodoros P. Horikis Perturbation theory for the defocusing NLS 23 / 38

Mode-locked lasers Interesting interactions Special perturbations Evolve the PES with initial condition ψ(0, t) = ψ tanh( ψ t + t 0 ) for t < 0 and ψ(0, t) = ψ tanh( ψ t t 0 ) for t > 0, where, ψ can be approximated by ψ /2; the dark-pulse energy is now twice that of the single soliton. z 100 80 60 40 1 0.8 0.6 0.4 20 0.2 0-20 -10 0 10 20 t The major difference with the bright case, where the interaction is logarithmically slow, is that pulses repel sooner here due to the shelf interactions. Theodoros P. Horikis Perturbation theory for the defocusing NLS 24 / 38

Mode-locked lasers Special perturbations Perturbation theory to the PES model To determine the evolution of the background wave in the framework of the PES, we assume that ψ(z) = ψ (z) exp[iθ(z)]. Separating real and imaginary parts, yields the following equation for the background amplitude ψ (z): dψ dz = g 1+2 ψ /E 0 ψ lψ. Here, an approximate solution in the form ψ(z, t) = ψ tanh(ψ t) is assumed, which gives E = 2 ψ. A stable equilibrium (attractor) exists and can be found setting dψ /dz = 0, namely, ψ = E ( 0 g ) 1 2 l This is the resulting background amplitude of the dark soliton and agrees with direct numerical simulation. Thus, dark solitons tend to an equilibrium (mode-lock) with constant energy and background. Theodoros P. Horikis Perturbation theory for the defocusing NLS 25 / 38

Mode-locked lasers Special perturbations Perturbation theory to the PES model To put the problem in the correct notation we set: ( ) g τ l ǫf[u] = i u+ u tt u 1+E/E sat 1+E/E sat 1+P/P sat where it is assumed that the small parameter ǫ is implicitly contained in the right hand side of the PES. Hence, { d } dz E(0) = 2ǫIm (F[u ]u F[u 0 ]u0 ) dt, ( d dz σ 0 = B Z 1 ) 2 E(0) Z /u, where E (0) is the first order approximation for the energy, i.e., E = E (0) +ǫe (1) + O(ǫ 2 ), and we have also used the fact that Re{F[u ]} = 0 for this perturbation. Notice that the energy E (0) at O(1), has contributions from both the core of the soliton and the shelf. Theodoros P. Horikis Perturbation theory for the defocusing NLS 26 / 38

Mode-locked lasers Special perturbations Perturbation theory to the PES model We find that the evolution of the above subset of soliton parameters is described by the following closed system of equations, d dz u = d dz A = d dz E(0) = g 1+E (0) /E sat u lu, g 1+E (0) /E sat A 4g 1+E (0) /E sat B + ( u 2 4l + P sat tanh 1 B 2 + P sat lp sat B B 2 + P sat tanh 1 2τ/3 1+E (0) /E sat B 3 ) B. B 2 + P sat ( B B 2 + P sat ) A, Note that the above equations are expressed in terms of z, since the small parameter ǫ is implicitly contained in the perturbation. Theodoros P. Horikis Perturbation theory for the defocusing NLS 27 / 38

Comments Mode-locked lasers Special perturbations There is a discrepancy between the soliton energy and the total energy and indicates that a shelf will develop around the soliton. The shelf height can be calculated as part of the perturbation analysis and, for the considered form of the perturbation, it turns out that it has a small size. Indeed, using typical parameter values, it can be seen that the shelf height is O(10 3 ), which is too small to be observed in a plot of the soliton. Theodoros P. Horikis Perturbation theory for the defocusing NLS 28 / 38

Mode-locked lasers Special perturbations Perturbation theory to the PES model 2 Energy E 1.5 1 Background u 0.5 Shelves begin Numerics interacting Asymptotics 0 0 10 20 z 30 40 50 Solid (blue) lines and dashed (red) lines correspond to the numerical results and the asymptotic analytical predictions, respectively. The vertical line indicates the spatial distance at which the shelves begin interacting. Here, the parameter values are g = 0.5, τ = 0.1, l = 0.1 and E sat = P sat = 1. Theodoros P. Horikis Perturbation theory for the defocusing NLS 29 / 38

Gray Solitons Mode-locked lasers Special perturbations It can be shown that any deviation from a purely stationary, black soliton state (i.e., any grey soliton) will eventually degenerate into a continuous wave with renormalized energy E = 0. u 4 3 2 1 0 20 0 20 40 0 Here, parameter values are g = 0.3, τ = 0.05, l = 0.1 and E sat = P sat = 1. t Theodoros P. Horikis Perturbation theory for the defocusing NLS 30 / 38 z 50

The NLS to KdV Connection Connection with other equations Consider the perturbed NLS i ψ t 1 2 ψ 2 x 2 + ψ 2 ψ = F[ψ], where F[ψ] is a general functional perturbation. Step 1: Write the solution in terms of a time dependent background function and a function u(t, x) such that ψ(t, x) = u (t)u(t, x) where the background and the function u(t, x) satisfy the system i u + u 2 u = F[u ] t i u t 1 2 u 2 x 2 + u 2 ( u 2 1)u = F[u u] F[u ]u u Theodoros P. Horikis Perturbation theory for the defocusing NLS 31 / 38

The NLS to KdV Connection Connection with other equations Consider the perturbed NLS i ψ t 1 2 ψ 2 x 2 + ψ 2 ψ = F[ψ], where F[ψ] is a general functional perturbation. Step 2: Change the independent variables dτ = u 2 dt and dξ = u dx, such that the above equation may be obtained from i u τ 1 2 u 2 ξ 2 +( u 2 1)u = F[u u] F[u ]u u u 2 Theodoros P. Horikis Perturbation theory for the defocusing NLS 32 / 38

The NLS to KdV Connection Connection with other equations Consider the perturbed NLS i ψ t 1 2 ψ 2 x 2 + ψ 2 ψ = F[ψ], where F[ψ] is a general functional perturbation. Step 3: Employ the so-called Madelung transformation u(τ, ξ) = ρ exp(iφ) (ρ and φ denote he amplitude and phase of u respectively) to reduce the NLS to the hydrodynamic equations. Step 4: Define new scales such that T = ε 3 τ, X = ε(ξ Cτ), where C is a constant to be determined. Theodoros P. Horikis Perturbation theory for the defocusing NLS 33 / 38

The NLS to KdV Connection Connection with other equations Consider the perturbed NLS i ψ t 1 2 ψ 2 x 2 + ψ 2 ψ = F[ψ], where F[ψ] is a general functional perturbation. Step 5: Expand amplitude and phase in powers of ε as follows: ρ = ρ 0 +ε 2 ρ 2 +ε 4 ρ 4 +ε 6 ρ 6 +..., φ = εφ 1 +ε 3 φ 3 +ε 5 φ 5 +ε 7 φ 7 +..., Finally: Match different orders of ε. Theodoros P. Horikis Perturbation theory for the defocusing NLS 34 / 38

Examples Connection with other equations Consider the pnls equation: i ψ t 1 2 ψ 2 x 2 + ψ 2 ψ = iν 2 ψ x 2, Write the solution as ψ(t, x) = u (t)u(t, x), where the background function satisfies u (t) = u 0 exp( i u 0 2 t) (where u 0 is a complex constant). Make the change τ = u 2 0t, ξ = u 0 x, Write u = ρ exp(iφ) and separate real and imaginary parts. Theodoros P. Horikis Perturbation theory for the defocusing NLS 35 / 38

Examples Connection with other equations Expand and match different orders of ε O(ε) : ρ 2 0 = 1, O(ε 2 ) : 2ρ 2 + C φ 1 X = 0, O(ε 3 ) : C ρ 2 X + 1 2 φ 1 2 X 2 = 0, O(ε 4 ) : 2ρ 4 + 3ρ 2 2 + φ 3 X 1 2 ρ 2 2 X 2 φ 1 T = ν ρ 2 X, O(ε 5 ) : ρ 4 X + 3ρ ρ 2 2 X 1 2 φ 3 2 X 2 + ρ 2 T = ρ 2 4νρ2 2 +ν 2 X 2. The compatibility conditions at O(ε 2 ) and O(ε 3 ) yields C = 1 and from the last two orders we obtain the perturbed KdV equation: ρ 2 T + 3ρ ρ 2 2 X 1 3 ρ 2 8 X 3 = ρ 2 2νρ2 2 +ν 2 X 2. Theodoros P. Horikis Perturbation theory for the defocusing NLS 36 / 38

Examples Connection with other equations Similarly from we get i ψ t 1 2 ψ 2 x 2 + ψ 2 ψ = iδ ψ 2 ψ. ρ 2 T + 3ρ ρ 2 2 X 1 3 ρ 2 8 X 3 = δρ 2. Theodoros P. Horikis Perturbation theory for the defocusing NLS 37 / 38

References M. J. Ablowitz, S. D. Nixon, T. P. Horikis, D. J. Frantzeskakis, Perturbations of dark solitons, Proc. Royal Soc. A 467, 2597-2621, 2011. M. J. Ablowitz, T. P. Horikis, S. D. Nixon, D. J. Frantzeskakis, Dark solitons in mode-locked lasers, Opt. Lett. 36, 793-795, 2011. M.J. Ablowitz, S.D. Nixon, T.P. Horikis, D.J. Frantzeskakis, Dark pulse generation in mode-locked lasers, J. Phys. A: Math. Theor. 46, 095201, 2013. T.P. Horikis, D.J. Frantzeskakis, On the NLS to KdV connection, Phys. Lett. A (submitted). Theodoros P. Horikis Perturbation theory for the defocusing NLS 38 / 38