From 766d05aac6f701fa85b7ceed1ce3a473a62cae55 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sun, 22 Sep 2024 18:09:57 +0200 Subject: Expand knowledge base --- .../nonlinear-schrodinger-equation/index.md | 91 ++-- .../know/concept/optical-soliton/bright-full.png | Bin 0 -> 85508 bytes .../know/concept/optical-soliton/bright-half.avif | Bin 0 -> 13232 bytes .../know/concept/optical-soliton/bright-half.jpg | Bin 0 -> 72712 bytes .../know/concept/optical-soliton/bright-half.png | Bin 0 -> 72999 bytes .../know/concept/optical-soliton/bright-half.webp | Bin 0 -> 31602 bytes source/know/concept/optical-soliton/dark-full.png | Bin 0 -> 164793 bytes source/know/concept/optical-soliton/dark-half.avif | Bin 0 -> 25523 bytes source/know/concept/optical-soliton/dark-half.jpg | Bin 0 -> 125295 bytes source/know/concept/optical-soliton/dark-half.png | Bin 0 -> 125292 bytes source/know/concept/optical-soliton/dark-half.webp | Bin 0 -> 57198 bytes source/know/concept/optical-soliton/index.md | 576 +++++++++++++++++++++ 12 files changed, 632 insertions(+), 35 deletions(-) create mode 100644 source/know/concept/optical-soliton/bright-full.png create mode 100644 source/know/concept/optical-soliton/bright-half.avif create mode 100644 source/know/concept/optical-soliton/bright-half.jpg create mode 100644 source/know/concept/optical-soliton/bright-half.png create mode 100644 source/know/concept/optical-soliton/bright-half.webp create mode 100644 source/know/concept/optical-soliton/dark-full.png create mode 100644 source/know/concept/optical-soliton/dark-half.avif create mode 100644 source/know/concept/optical-soliton/dark-half.jpg create mode 100644 source/know/concept/optical-soliton/dark-half.png create mode 100644 source/know/concept/optical-soliton/dark-half.webp create mode 100644 source/know/concept/optical-soliton/index.md (limited to 'source/know/concept') diff --git a/source/know/concept/nonlinear-schrodinger-equation/index.md b/source/know/concept/nonlinear-schrodinger-equation/index.md index 506600e..2ea1b23 100644 --- a/source/know/concept/nonlinear-schrodinger-equation/index.md +++ b/source/know/concept/nonlinear-schrodinger-equation/index.md @@ -13,27 +13,39 @@ layout: "concept" The **nonlinear Schrödinger (NLS) equation** is a nonlinear 1+1D partial differential equation that appears in many areas of physics. -It is used to describe pulses in fiber optics (as derived below), -waves over deep water, local opening of DNA chains, and more. -It is often given as: +It is often given in its dimensionless form, +where it governs the envelope $$u(z, t)$$ +of an underlying carrier wave, +with $$t$$ the transverse coordinate, +and $$r = \pm 1$$ a parameter determining +which of two regimes the equation is intended for: $$\begin{aligned} \boxed{ - i \pdv{u}{z} + \pdvn{2}{u}{t} + |u|^2 u + i \pdv{u}{z} + \pdvn{2}{u}{t} + r |u|^2 u = 0 } \end{aligned}$$ -Which is its dimensionless form, -governing the envelope $$u(z, t)$$ -of an underlying carrier wave, -with $$t$$ being the transverse coordinate. -Notably, the NLS equation has **soliton** solutions, -where $$u$$ maintains its shape over great distances. - - - -## Derivation +Many variants exist, depending on the conventions used by authors. +The NLS equation is used to describe pulses in fiber optics (as derived below), +waves over deep water, local opening of DNA chains, and much more. +Very roughly speaking, it is a valid description of +"all" weakly nonlinear, slowly modulated waves in physics. + +It exhibits an incredible range of behaviors, +from "simple" effects such as +[dispersive broadening](/know/concept/dispersive-broadening/), +[self-phase modulation](/know/concept/self-phase-modulation/) +and [first-order solitons](/know/concept/optical-soliton/), +to weirder and more complicated phenomena like +[modulational instability](/know/concept/modulational-instability/), +[optical wave breaking](/know/concept/optical-wave-breaking/) +and periodic *higher-order solitons*. +It is also often modified to include additional physics, +further enriching its results with e.g. +[self-steepening](/know/concept/self-steepening/) +and *soliton self-frequency shifting*. We only consider fiber optics here; the NLS equation can be derived in many other ways. @@ -174,7 +186,7 @@ $$\begin{aligned} \nabla^2 E - \mu_0 \varepsilon_0 \pdvn{2}{E}{t} - \mu_0 \pdvn{2}{P_\mathrm{L}}{t} - \mu_0 \pdvn{2}{P_\mathrm{NL}}{t} \bigg) e^{-i \omega_0 t} \\ - &= \bigg( + &\approx \bigg( \nabla^2 E - \Big( 1 + \chi^{(1)}_{xx} + \frac{3}{4} \chi^{(3)}_{xxxx} |E|^2 \Big) \mu_0 \varepsilon_0 \pdvn{2}{E}{t} \bigg) e^{-i \omega_0 t} \end{aligned}$$ @@ -622,46 +634,55 @@ In other words, we demand: $$\begin{aligned} \frac{\beta_2 Z_c}{2 T_c^2} - = \mp 1 + = -1 \qquad\qquad \gamma_0 A_c^2 Z_c - = 1 + = r \end{aligned}$$ -Where the choice of $$\mp$$ will be explained shortly. -Note that we only have two equations for three unknowns +Where $$r \equiv \pm 1$$, whose sign choice will be explained shortly. +Note that we have two equations for three unknowns ($$A_c$$, $$Z_c$$ and $$T_c$$), so one of the parameters needs to fixed manually. -For example, we could choose $$Z_c = 1\:\mathrm{m}$$, and then: +For example, we could choose our "input power" +$$A_c \equiv \sqrt{1\:\mathrm{W}}$$, and then: $$\begin{aligned} - A_c - = \frac{1}{\sqrt{\gamma Z_c}} - \qquad\qquad + Z_c + = - \frac{2 T_c^2}{\beta_2} + \qquad + T_c^2 + = -\frac{r \beta_2}{2 \gamma_0 A_c^2} + \qquad\implies\qquad + Z_c + = \frac{r}{\gamma_0 A_c^2} + \qquad T_c - = \sqrt{\frac{\mp \beta_2 Z_c}{2}} + = \sqrt{ -\frac{r \beta_2}{2 \gamma_0 A_c^2} } \end{aligned}$$ -Note that this requires that $$\gamma_0 > 0$$, -which is true for the vast majority of materials, -and that we choose the sign $$\mp$$ such that $$\mp \beta_2 > 0$$. +Because $$T_c$$ must be real, +we should choose $$r \equiv - \sgn(\gamma_0 \beta_2)$$. We thus arrive at: $$\begin{aligned} \boxed{ 0 = i \pdv{\tilde{A}}{\tilde{Z}} - \pm \pdvn{2}{\tilde{A}}{\tilde{T}} - + \big|\tilde{A}\big|^2 \tilde{A} + + \pdvn{2}{\tilde{A}}{\tilde{T}} + + r \big|\tilde{A}\big|^2 \tilde{A} } \end{aligned}$$ -Because soliton solutions only exist -in the *anomalous dispersion* regime $$\beta_2 < 0$$, -most authors just write $$+$$. -There are still plenty of interesting effects -in the *normal dispersion* regime $$\beta_2 > 0$$, -hence we write $$\pm$$ for the sake of completeness. +In fiber optics, $$\gamma_0 > 0$$ for all materials, +meaning $$r$$ represents the dispersion regime, +so $$r = 1$$ is called *anomalous dispersion* +and $$r = -1$$ *normal dispersion*. +In some other fields, where $$\beta_2 < 0$$ always, +$$r = 1$$ is called a *focusing nonlinearity* +and $$r = -1$$ a *defocusing nonlinearity*. +The famous bright solitons only exist for $$r = 1$$, +so many authors only show that case. diff --git 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+They are only explainable by nonlinear physics, +but many (often unrelated) nonlinear equations give rise to solitons: +the [Boussinesq equations](/know/concept/boussinesq-wave-theory/), +the [Korteweg-de Vries equation](/know/concept/korteweg-de-vries-equation/), +the [nonlinear Schrödinger (NLS) equation](/know/concept/nonlinear-schrodinger-equation/), +and more. +Here we consider waveguide optics, +which is governed by the NLS equation, +given in dimensionless form by: + +$$\begin{aligned} + i u_z + u_{tt} + r |u|^2 u + = 0 +\end{aligned}$$ + +Where $$r = \pm 1$$ determines the dispersion regime, +and subscripts denote differentiation. +We start by making the most general ansatz +for the pulse envelope $$u(z, t)$$, namely: + +$$\begin{aligned} + u(z, t) + = \phi(z, t) \: e^{i \theta(z, t)} +\end{aligned}$$ + +With $$\phi$$ and $$\theta$$ both real. +Note that no generality has been lost yet: +we have simply split a single complex function +into two real ones. +The derivatives of $$u$$ thus become: + +$$\begin{aligned} + u_z + &= (\phi_z + i \phi \theta_z) \: e^{i \theta} + \\ + u_t + &= (\phi_t + i \phi \theta_t) \: e^{i \theta} + \\ + u_{tt} + &= (\phi_{tt} + 2 i \phi_t \theta_t + i \phi \theta_{tt} - \phi \theta_t^2) \: e^{i \theta} +\end{aligned}$$ + +Inserting $$u_z$$ and $$u_{tt}$$ into the NLS equation leads us to: + +$$\begin{aligned} + 0 + &= i \phi_z - \phi \theta_z + \phi_{tt} + 2 i \phi_t \theta_t + i \phi \theta_{tt} - \phi \theta_t^2 + r \phi^3 + \\ + &= \phi_{tt} - \phi \theta_t^2 - \phi \theta_z + r \phi^3 + i (\phi \theta_{tt} + 2 \phi_t \theta_t + \phi_z) +\end{aligned}$$ + +Since $$\phi$$ and $$\theta$$ are both real, +we can split this equation into its real and imaginary parts: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + 0 + &= \phi_{tt} - \phi \theta_t^2 - \phi \theta_z + r \phi^3 + \\ + 0 + &= \phi \theta_{tt} + 2 \phi_t \theta_t + \phi_z + \end{aligned} + } +\end{aligned}$$ + +Still no generality has been lost so far: +these coupled equation are totally equivalent to the NLS equation. +But now it is time make a more specific ansatz, +namely that $$\phi$$ and $$\theta$$ both have a fixed shape +but move at a group velocity $$v$$ +and phase velocity $$w$$, respectively: + +$$\begin{aligned} + \phi(z, t) + &= \phi(t - v z) + \\ + \theta(z, t) + &= \theta(t - w z) +\end{aligned}$$ + +Meaning $$\phi_z = -v \phi_t$$ and $$\theta_z = -w \theta_t$$. +Now the coupled equations are given by: + +$$\begin{aligned} + 0 + &= \phi_{tt} - \phi \theta_t^2 + w \phi \theta_t + r \phi^3 + \\ + 0 + &= \phi \theta_{tt} + 2 \phi_t \theta_t - v \phi_t +\end{aligned}$$ + +We multiply the imaginary part's equation by $$\phi$$ and take its indefinite integral, +which can then be evaluated by recognizing the product rule of differentiation: + +$$\begin{aligned} + 0 + &= \int \Big( \phi^2 \theta_{tt} + 2 \phi \phi_t \theta_t - v \phi \phi_t \Big) \dd{t} + \\ + &= \phi^2 \theta_t - \frac{v}{2} \phi^2 +\end{aligned}$$ + +Where the integration constant has been set to zero. +This implies $$\theta_t = v/2$$, which we insert into the real part's equation, giving: + +$$\begin{aligned} + 0 + &= \phi_{tt} + \frac{v}{4} (2 w - v) \phi + r \phi^3 +\end{aligned}$$ + +Defining $$B \equiv v (v - 2 w) / 4$$, +multiplying by $$2 \phi_t$$, and integrating in the same way: + +$$\begin{aligned} + 0 + &= \int \Big( 2 \phi_t \phi_{tt} - 2 B \phi \phi_t + 2 r \phi^3 \phi_t \Big) \dd{t} + \\ + &= \phi_t^2 - B \phi^2 + \frac{r}{2} \phi^4 - C +\end{aligned}$$ + +Where $$C$$ is an integration constant. +Rearranging this yields a powerful equation, +which can be interpreted as a "pseudoparticle" +with kinetic energy $$\phi_t^2$$ moving in a potential $$-P(\phi)$$: + +$$\begin{aligned} + \boxed{ + \phi_t^2 + = P(\phi) + \equiv -\frac{r}{2} \phi^4 + B \phi^2 + C + } +\end{aligned}$$ + +We further restrict the set of acceptable solutions +by demanding that $$\phi(t)$$ is localized, +meaning $$\phi \to \phi_\infty$$ when $$t \to \pm \infty$$, +for a finite constant $$\phi_\infty$$. +This implies $$\phi_t \to 0$$ and $$\phi_{tt} \to 0$$: +the former clearly requires $$P(\phi_\infty) = 0$$. +Regarding the latter, we differentiate +the pseudoparticle equation with respect to $$t$$, +which tells us for $$t \to \pm \infty$$: + +$$\begin{aligned} + 0 + = \phi_{tt} + &= \frac{1}{2} P'(\phi_\infty) + = (B - r \phi_\infty^2) \phi_\infty +\end{aligned}$$ + +Here we have two options: +the "bright" case $$\phi_\infty = 0$$, +and the "dark" case $$\phi_\infty^2 = r B$$. +Before we investigate those further, +let us finish finding $$\theta$$: +we know that $$\theta_t = v/2$$, so: + +$$\begin{aligned} + \theta(t - w z) + = \int \theta_t \dd{(t - w v)} + = \frac{v}{2} (t - w v) +\end{aligned}$$ + +Where we can ignore the integration constant +because the NLS equation has *Gauge symmetry*, +i.e. it is invariant under a transformation +of the form $$u \to u e^{i a}$$ with constant $$a$$. +Finally, we rewrite this result to eliminate $$w$$ in favor of $$B$$: + +$$\begin{aligned} + \theta(z, t) + = \frac{v}{2} t - \bigg( \frac{v^2}{4} - B \bigg) z +\end{aligned}$$ + + + +## Bright solitons + +First we consider the "bright" option $$\phi_\infty = 0$$, +where our requirement that $$P(\theta_\infty) = 0$$ +clearly means that we must set $$C = 0$$. +We are therefore left with: + +$$\begin{aligned} + \phi_t^2 + = P(\phi) + = -\frac{r}{2} \phi^4 + B \phi^2 +\end{aligned}$$ + +We must consider $$r = 1$$ and $$r = -1$$, and the sign of $$B$$; +the possible forms of $$P(\phi)$$ are shown in the sketch below. +Because $$\phi_t$$ is real by definition, +valid solutions can only exist in the shaded regions where $$P(\phi) \ge 0$$: + +{% include image.html file="bright-full.png" width="75%" + alt="Sketch of candidate potentials for bright solitons" %} + +However, in order to have *stable* solutions +where $$\phi$$ does not grow uncontrolably, +we must restrict ourselves to shaded regions with a finite area. +Otherwise, if they are infinite (as for $$r = -1$$), +then a positive feedback loop arises: +$$\phi_t^2$$ grows, so $$|\phi|$$ increases, +then according to the sketch $$\phi_t^2$$ grows even more, etc. +While mathematically correct, that would be physically unacceptable, +so the only valid case here is $$r = 1$$ with $$B > 0$$. + +Armed with this knowledge, +we are now ready to integrate the pseudoparticle integration. +First, we rewrite it as follows, defining $$x \equiv t - vz$$: + +$$\begin{aligned} + \phi_t + = \pdv{\phi}{x} + = \pm \sqrt{P(\phi)} + = \pm \phi \sqrt{B - \phi^2 / 2} +\end{aligned}$$ + +This can be rearranged such that the differential elements +$$\dd{x}$$ and $$\dd{\phi}$$ are on opposite sides, +which can then each be wrapped in an integral, like so: + +$$\begin{aligned} + \dd{x} + = \pm \frac{\sqrt{2}}{\phi \sqrt{2 B - \phi^2}} \dd{\phi} + \qquad\implies\qquad + \int_{x_0}^{x} \dd{\xi} + = \pm \sqrt{2} \int_{\phi_0}^{\phi} \frac{1}{\psi \sqrt{2 B - \psi^2}} \dd{\psi} +\end{aligned}$$ + +Note that these are *indefinite* integrals, +which have been written as *definite* integrals +by placing the constants $$x_0$$ and $$\phi_0$$ +and target variables $$x$$ and $$\phi$$ in the limits. + +In order to integrate by substitution, +we define the new variable $$f \equiv \psi / \sqrt{2 B}$$ +and update the limits accordingly +to $$F \equiv \phi / \sqrt{2 B}$$ +and $$F_0 \equiv \phi_0 / \sqrt{2 B}$$: + +$$\begin{aligned} + x - x_0 + &= \pm \sqrt{2} \int_{F_0}^{F} \frac{\sqrt{2 B}}{f \sqrt{2 B} \sqrt{2 B - 2 B f^2}} \dd{f} + \\ + &= \pm \frac{1}{\sqrt{B}} \int_{F_0}^{F} \frac{1}{f \sqrt{1 - f^2}} \dd{f} +\end{aligned}$$ + +We look up this integrand, and discover that it is in fact the derivative +of the inverse $$\sech^{-1}$$ of the hyperbolic secant function, so we arrive at: + +$$\begin{aligned} + x - x_0 + &= \pm \frac{1}{\sqrt{B}} \int_{F_0}^{F} \dv{}{f} \Big( \sech^{-1}(f) \Big) \dd{f} + \\ + &= \pm \frac{1}{\sqrt{B}} \sech^{-1}(F) \mp \frac{1}{\sqrt{B}} \sech^{-1}(F_0) +\end{aligned}$$ + +Rearranging and combining the integration constants +$$x_0$$ and $$F_0$$ into a single $$t_0$$, we get: + +$$\begin{aligned} + \sech^{-1}(F) + = \pm \sqrt{B} (x - t_0) + \qquad\qquad + t_0 + \equiv x_0 \mp \frac{1}{\sqrt{B}} \sech^{-1}(F_0) +\end{aligned}$$ + +Then, wrapping everything in $$\sech$$ +(which is an even function, so we can discard the $$\pm$$) +and using $$F \equiv \phi / \sqrt{2 B}$$, +we finally arrive at the desired solution for $$\phi$$: + +$$\begin{aligned} + \phi(x) + = \sqrt{2 B} \sech\!\Big( \sqrt{B} (x - t_0) \Big) +\end{aligned}$$ + +Combining this result with our earlier solution for $$\theta$$, +we find that the full so-called **bright soliton** $$u$$ +is as follows, controlled by two real parameters +$$B > 0$$ and $$v$$: + +$$\begin{aligned} + \boxed{ + u(z, t) + = \sqrt{2 B} \sech\!\bigg( \sqrt{B} (t - v z - t_0) \bigg) + \exp\!\bigg( i \frac{v}{2} t - i \Big( \frac{v^2}{4} - B \Big) z \bigg) + } +\end{aligned}$$ + +It is always possible to transform the NLS equation +into a new moving coordinate system such that $$v = 0$$, +yielding a stationary soliton given by: + +$$\begin{aligned} + \boxed{ + u(z, t) + = \sqrt{2 B} \sech\!\Big( \sqrt{B} (t - t_0) \Big) \exp(i B z) + } +\end{aligned}$$ + +You may be wondering how we can set $$v = 0$$ without affecting $$B$$; +a more correct way of saying it would be that +we take the limits $$v \to 0$$ and $$w \to -\infty$$. + +That was for the dimensionless form of the NLS equation; +let us specialize this to its usual form in fiber optics. +We thus make a transformation $$u \to U/U_c$$, +$$t \to T/T_c$$ and $$z \to Z/Z_c$$: + +$$\begin{aligned} + \frac{U(Z, T)}{U_c} + &= \sqrt{2 B} \sech\!\bigg( \sqrt{B} \: \frac{T - T_0}{T_c} \bigg) + \exp\!\bigg( i B \frac{Z}{Z_c} \bigg) +\end{aligned}$$ + +Where $$U_c$$, $$T_c$$ and $$Z_c$$ are scale constants +determined during non-dimensionalization +to obey the relations below. +We only have two relations, so we can choose one value freely, +say, $$U_c$$: + +$$\begin{aligned} + Z_c + = \frac{1}{\gamma_0 U_c^2} + \qquad\qquad + T_c + = \sqrt{\frac{- \beta_2}{2 \gamma_0 U_c^2}} +\end{aligned}$$ + +Note that $$r = 1$$ implies $$\beta_2 < 0$$ assuming $$\gamma_0 > 0$$. +In other words, bright solitons only exist +in the anomalous dispersion regime of an optical fiber. +Inserting these relations into the expression +and defining the peak power $$P_0 \equiv 2 B U_c^2$$ yields: + +$$\begin{aligned} + U(Z, T) + &= \sqrt{P_0} + \sech\!\Bigg( \sqrt{\frac{\gamma_0 P_0}{- \beta_2}} (T - T_0) \Bigg) + \exp\!\bigg( i \frac{\gamma_0 P_0}{2} Z \bigg) +\end{aligned}$$ + +In practice, most authors write this as follows, +where $$T_\mathrm{w}$$ determines the width of the pulse: + +$$\begin{aligned} + \boxed{ + U(Z, T) + = \sqrt{P_0} \sech\!\bigg( \frac{T - T_0}{T_\mathrm{w}} \bigg) \exp\!\bigg( i \frac{\gamma_0 P_0}{2} Z \bigg) + } +\end{aligned}$$ + +Clearly, for this to be a valid solution of the NLS equation, +$$T_\mathrm{w}$$ must be subject to a constraint +involving the so-called **soliton number** $$N_\mathrm{sol}$$: + +$$\begin{aligned} + \boxed{ + N_\mathrm{sol}^2 + \equiv \frac{L_D}{L_N} + = \frac{\gamma_0 P_0 T_\mathrm{w}^2}{|\beta_2|} + = 1 + } +\end{aligned}$$ + +Where $$L_D \equiv T_0 / |\beta_2|$$ is the linear length scale +of [dispersive broadening](/know/concept/dispersive-broadening/), +and $$L_N \equiv 1 / (\gamma_0 P_0)$$ is the nonlinear length scale +of [self-phase modulation](/know/concept/self-phase-modulation/). +A *first-order* soliton has $$N_\mathrm{sol} = 1$$ +and simply maintains its shape, +whereas higher-order solitons have complicated periodic dynamics. + + + +## Dark solitons + +The other option to satisfy $$P'(\phi_\infty) = 0$$ +is $$\phi_\infty^2 = r B$$, which implies $$r B > 0$$ +such that $$\phi_\infty$$ is real. +With this in mind, we again sketch all remaining candidates for $$P(\phi)$$: + +{% include image.html file="dark-full.png" width="75%" + alt="Sketch of candidate potentials for dark solitons" %} + +At a glance, there are plenty of solutions here, even stable ones! +However, as explained earlier, our localization requirement +means that we need $$P(\phi_\infty) = 0$$ and $$P'(\phi_\infty) = 0$$. +The latter is only satisfied by the solid curve above, +so we must limit ourselves to $$r = -1$$ and $$B < 0$$, +with $$C = C_0$$ for some positive $$C_0$$. +The next step is to find $$C_0$$. + +We notice that the target curve has two double roots +at $$\pm \phi_\infty$$, so we can rewrite: + +$$\begin{aligned} + P(\phi) + &= \frac{1}{2} \Big( \phi^4 + 2 B \phi^2 + 2 C \Big) + \\ + &= \frac{1}{2} \Big( \phi^4 + 2 B \phi^2 + B^2 - B^2 + 2 C \Big) + \\ + &= \frac{1}{2} \big( \phi^2 + B \big)^2 - \frac{1}{2} \big( B^2 - 2 C \big) +\end{aligned}$$ + +Here we see that $$P(\phi_\infty)$$ can only have a double root +when $$C = C_0 = B^2 / 2$$, in which case the root is clearly $$\phi_\infty = \pm \sqrt{-B}$$. +We are therefore left with: + +$$\begin{aligned} + \phi_t^2 + = P(\phi) + = \frac{1}{2} \big( \phi^2 + B \big)^2 +\end{aligned}$$ + +Now we are ready to integrate this equation. +Taking the square root with $$x \equiv t - v z$$: + +$$\begin{aligned} + \phi_t + = \pdv{\phi}{x} + = \pm \sqrt{P(\phi)} + = \pm \frac{1}{\sqrt{2}} (\phi^2 + B) +\end{aligned}$$ + +We put the differential elements $$\dd{\phi}$$ and $$\dd{x}$$ +on opposite sides and take the integrals: + +$$\begin{aligned} + \dd{x} + = \pm \frac{\sqrt{2}}{\phi^2 + B} \dd{\phi} + \qquad\implies\qquad + \int_{x_0}^{x} \dd{\xi} + = \pm \sqrt{2} \int_{\phi_0}^{\phi} \frac{1}{\psi^2 + B} \dd{\psi} +\end{aligned}$$ + +Then we define $$f \equiv \psi / \sqrt{-B}$$, +and update the limits to +$$F = \phi / \sqrt{-B}$$ and $$F_0 = \phi_0 / \sqrt{-B}$$, +in order to integrate by substitution: + +$$\begin{aligned} + x - x_0 + &= \pm \sqrt{2} \int_{F_0}^{F} \frac{\sqrt{-B}}{- B f^2 + B} \dd{f} + \\ + &= \pm \sqrt{-\frac{2}{B}} \int_{F_0}^{F} \frac{1}{1 - f^2} \dd{f} +\end{aligned}$$ + +The integrand can be looked up: +it turns out be the derivative of $$\tanh^{-1}$$, +the inverse hyperbolic tangent function, +so we arrive at: + +$$\begin{aligned} + x - x_0 + &= \pm \sqrt{-\frac{2}{B}} \int_{F_0}^{F} \dv{}{f} \Big( \tanh^{-1}(f) \Big) \dd{f} + \\ + &= \pm \sqrt{-\frac{2}{B}} \tanh^{-1}(F) \mp \sqrt{-\frac{2}{B}} \tanh^{-1}(F_0) +\end{aligned}$$ + +Rearranging, and combining the integration constants +$$x_0$$ and $$F_0$$ into a single $$t_0$$, yields: + +$$\begin{aligned} + \tanh^{-1}(F) + &= \pm \sqrt{-\frac{B}{2}} (x - t_0) + \qquad\qquad + t_0 + \equiv x_0 \mp \sqrt{-\frac{2}{B}} \tanh^{-1}(F_0) +\end{aligned}$$ + +Next, we take the $$\tanh$$ of both sides. +It is an odd function, so the $$\pm$$ can be moved outside, +where it can be ignored entirely thanks to the NLS equation's Gauge symmetry. +Using $$F = \phi / \sqrt{-B}$$: + +$$\begin{aligned} + \phi(x) + &= \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} (x - t_0) \Bigg) +\end{aligned}$$ + +Combining this with our expression for $$\theta$$, +we arrive at the full **dark soliton** solution for $$u$$: + +$$\begin{aligned} + \boxed{ + u(z, t) + = \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} (t - v z - t_0) \Bigg) + \exp\!\bigg( i \frac{v}{2} t - i \Big( \frac{v^2}{4} - B \Big) z \bigg) + } +\end{aligned}$$ + +There are two free parameters here: $$B < 0$$ and $$v$$. +Once again, we can always transform to a moving coordinate system such that $$v = 0$$, +resulting in a stationary soliton: + +$$\begin{aligned} + \boxed{ + u(z, t) + = \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} (t - t_0) \Bigg) + \exp(i B z) + } +\end{aligned}$$ + +Like we did for the bright solitons, +let us specialize this result to fiber optics. +Making a similar transformation $$u \to U/U_c$$, +$$t \to T/T_c$$ and $$z \to Z/Z_c$$ yields: + +$$\begin{aligned} + \frac{U(Z, T)}{U_c} + = \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} \frac{T - T_0}{T_c} \Bigg) + \exp\!\bigg( i B \frac{Z}{Z_c} \bigg) +\end{aligned}$$ + +Where we again choose $$U_c$$ manually, +and then find $$T_c$$ and $$Z_c$$ using these relations +(note the opposite signs because $$r = -1$$ in this case): + +$$\begin{aligned} + Z_c + = \frac{-1}{\gamma_0 U_c^2} + \qquad\qquad + T_c + = \sqrt{\frac{\beta_2}{2 \gamma_0 U_c^2}} +\end{aligned}$$ + +Recall that $$r = -1$$ implies $$\beta_2 > 0$$ assuming $$\gamma_0 > 0$$, +meaning dark solitons can only exist in the normal dispersion regime. +Inserting this into the expression +and defining the background power $$P_0 \equiv -B U_c^2$$ +such that $$|U|^2 \to P_0$$ for $$t \to \pm \infty$$, +we arrive at: + +$$\begin{aligned} + U(Z, T) + = \sqrt{P_0} \tanh\!\Bigg( \sqrt{\frac{\gamma_0 P_0}{\beta_2}} (T - T_0) \Bigg) \exp(i \gamma_0 P_0 Z) +\end{aligned}$$ + +Which, as for bright solitons, can be rewritten +with a pulse width $$T_\mathrm{w}$$ satisfying $$N_\mathrm{sol} = 1$$: + +$$\begin{aligned} + \boxed{ + U(Z, T) + = \sqrt{P_0} \tanh\!\bigg( \frac{T - T_0}{T_\mathrm{w}} \bigg) \exp(i \gamma_0 P_0 Z) + } +\end{aligned}$$ + + + +## References + +1. A. Scott, + *Nonlinear science: emergence and dynamics of coherent structures*, + 2nd edition, Oxford. +2. O. Bang, + *Nonlinear mathematical physics: lecture notes*, + 2020, unpublished. -- cgit v1.2.3