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1 files changed, 30 insertions, 24 deletions

diff --git a/content/know/concept/optical-wave-breaking/index.pdc b/content/know/concept/optical-wave-breaking/index.pdc
index 757a633..2ab3ff1 100644
--- a/content/know/concept/optical-wave-breaking/index.pdc
+++ b/content/know/concept/optical-wave-breaking/index.pdc
@@ -75,11 +75,10 @@ the instantaneous frequencies for these separate effects:
 $$\begin{aligned}
     \omega_i(z,t)
     &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
-    \\
 %    &= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
 %    + \frac{2\gamma P_0 z}{T_0^2} t \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
 %    \\
-    &= \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
+    = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
     + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
 \end{aligned}$$
 
@@ -97,14 +96,14 @@ and $N_\mathrm{sol}$ is the **soliton number**,
 which is defined as:
 
 $$\begin{aligned}
-    N_\mathrm{sol}^2 = \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|}
+    N_\mathrm{sol}^2 \equiv \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|}
 \end{aligned}$$
 
 This quantity is very important in anomalous dispersion,
 but even in normal dispesion, it is still a useful measure of the relative strengths of GVD and SPM.
 As was illustrated earlier, $\omega_i$ overtakes itself at the edges,
-so OWB only occurs when $\omega_i$ is not monotonic,
-which is when its $t$-derivative,
+so OWB occurs when $\omega_i$ oscillates there,
+which starts when its $t$-derivative,
 the **instantaneous chirpyness** $\xi_i$,
 has *two* real roots for $t^2$:
 
@@ -122,11 +121,17 @@ leading to the following exact minimum value $N_\mathrm{min}^2$ for $N_\mathrm{s
 such that OWB can only occur when $N_\mathrm{sol}^2 > N_\mathrm{min}^2$:
 
 $$\begin{aligned}
-    N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12
+    \boxed{
+        N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12
+    }
 \end{aligned}$$
 
-Now, consider two times $t_1$ and $t_2$ in the pulse, separated by
-a small initial interval $(t_2 - t_1)$.
+If this condition $N_\mathrm{sol}^2 > N_\mathrm{min}^2$ is not satisfied,
+$\xi_i$ cannot have two roots for $t^2$, meaning $\omega_i$ cannot overtake itself.
+GVD is unable to keep up with SPM, so OWB will not occur.
+
+Next, consider two points at $t_1$ and $t_2$ in the pulse,
+separated by a small initial interval $(t_2 - t_1)$.
 The frequency difference between these points due to $\omega_i$
 will cause them to displace relative to each other
 after a short distance $z$ by some amount $\Delta t$,
@@ -136,21 +141,21 @@ $$\begin{aligned}
     \Delta t
     &\approx z \Delta\beta_1
     \qquad
-    &&\Delta\beta_1 = \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
+    &&\Delta\beta_1 \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
     \\
     &\approx z \beta_2 \Delta\omega_i
     \qquad
-    &&\Delta\omega_i = \omega_i(z,t_2) - \omega_i(z,t_1)
+    &&\Delta\omega_i \equiv \omega_i(z,t_2) - \omega_i(z,t_1)
     \\
     &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
     \qquad \quad
-    &&\Delta\xi_i = \xi_i(z,t_2) - \xi_i(z,t_1)
+    &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1)
 \end{aligned}$$
 
 Where $\beta_1(\omega)$ is the inverse of the group velocity.
 OWB takes place when $t_2$ and $t_1$ catch up to each other,
 which is when $-\Delta t = (t_2 - t_1)$.
-The distance where this happens, $z = L_\mathrm{WB}$,
+The distance where this happens first, $z = L_\mathrm{WB}$,
 must therefore satisfy the following condition
 for a particular value of $t$:
 
@@ -161,7 +166,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The time $t$ of OWB must be where $\omega_i(t)$ has its steepest slope,
-which is at the minimum value of $\xi_i(t)$, and, by extension $f(x)$.
+which is at the minimum value of $\xi_i(t)$, and by extension $f(x)$.
 This turns out to be $f(3/2)$:
 
 $$\begin{aligned}
@@ -170,16 +175,17 @@ $$\begin{aligned}
     = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2
 \end{aligned}$$
 
-Clearly, $f_\mathrm{min} \ge 0$ when
-$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$, which, when inserted in the
-condition above, confirms that OWB cannot occur in that case. Otherwise,
-if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then:
+Clearly, $f_\mathrm{min} \ge 0$ when $N_\mathrm{sol}^2 \le N_\mathrm{min}^2$,
+which, when inserted above, leads to an imaginary $L_\mathrm{WB}$,
+confirming that OWB cannot occur in that case.
+Otherwise, if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then:
 
 $$\begin{aligned}
-    L_\mathrm{WB}
-    = - \frac{T_0^2}{\beta_2 \, \sqrt{f_\mathrm{min}}}
-    = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
-    = \frac{L_D}{\sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}}
+    \boxed{
+        L_\mathrm{WB}
+        = \frac{T_0^2}{\beta_2 \, \sqrt{- f_\mathrm{min}}}
+        = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
+    }
 \end{aligned}$$
 
 This prediction for $L_\mathrm{WB}$ appears to agree well
@@ -196,7 +202,7 @@ Filling $L_\mathrm{WB}$ in into $\omega_\mathrm{SPM}$ gives:
 
 $$\begin{aligned}
     \omega_{\mathrm{SPM}}(L_\mathrm{WB},t)
-    = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
+    = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
 \end{aligned}$$
 
 Assuming that $N_\mathrm{sol}^2$ is large in the denominator, this can
@@ -205,8 +211,8 @@ be approximately reduced to:
 $$\begin{aligned}
     \omega_\mathrm{SPM}(L_\mathrm{WB}, t)
 %   = \frac{2 \gamma P_0 t \exp(-t^2 / T_0^2)}{\beta_2 \sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
-    \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
-    = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
+    \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
+    = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
 \end{aligned}$$
 
 The expression $x \exp(-x^2)$ has its global extrema