Expand knowledge base

12 files changed, 632 insertions, 35 deletions

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 a/source/know/concept/optical-soliton/bright-full.png b/source/know/concept/optical-soliton/bright-full.png
new file mode 100644
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+++ b/source/know/concept/optical-soliton/bright-full.png
diff --git a/source/know/concept/optical-soliton/bright-half.avif b/source/know/concept/optical-soliton/bright-half.avif
new file mode 100644
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diff --git a/source/know/concept/optical-soliton/bright-half.jpg b/source/know/concept/optical-soliton/bright-half.jpg
new file mode 100644
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diff --git a/source/know/concept/optical-soliton/bright-half.png b/source/know/concept/optical-soliton/bright-half.png
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diff --git a/source/know/concept/optical-soliton/bright-half.webp b/source/know/concept/optical-soliton/bright-half.webp
new file mode 100644
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diff --git a/source/know/concept/optical-soliton/dark-half.jpg b/source/know/concept/optical-soliton/dark-half.jpg
new file mode 100644
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+++ b/source/know/concept/optical-soliton/dark-half.jpg
diff --git a/source/know/concept/optical-soliton/dark-half.png b/source/know/concept/optical-soliton/dark-half.png
new file mode 100644
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--- /dev/null
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diff --git a/source/know/concept/optical-soliton/dark-half.webp b/source/know/concept/optical-soliton/dark-half.webp
new file mode 100644
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diff --git a/source/know/concept/optical-soliton/index.md b/source/know/concept/optical-soliton/index.md
new file mode 100644
index 0000000..843642f
--- /dev/null
+++ b/source/know/concept/optical-soliton/index.md
@@ -0,0 +1,576 @@
+---
+title: "Optical soliton"
+sort_title: "Optical soliton"
+date: 2024-09-20
+categories:
+- Physics
+- Mathematics
+- Fiber optics
+- Nonlinear optics
+layout: "concept"
+---
+
+In general, a **soliton** is a wave packet
+that maintains its shape as it travels over great distances.
+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.