Rewrite "Lagrange multiplier", various improvements

1 files changed, 10 insertions, 5 deletions

diff --git a/source/know/concept/self-steepening/index.md b/source/know/concept/self-steepening/index.md
index 9666167..e06b0b5 100644
--- a/source/know/concept/self-steepening/index.md
+++ b/source/know/concept/self-steepening/index.md
@@ -48,15 +48,16 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The phase $$\phi$$ is not so interesting, so we focus on the latter equation for $$P$$.
-As it turns out, it has a general solution of the form below, which shows that
-more intense parts of the pulse will tend to lag behind compared to the rest:
+As it turns out, it has a general solution of the form below (you can verify this yourself),
+which shows that more intense parts of the pulse
+will lag behind compared to the rest:
 
 $$\begin{aligned}
     P(z,t) = f(t - 3 \varepsilon z P)
 \end{aligned}$$
 
 Where $$f$$ is the initial power profile: $$f(t) = P(0,t)$$.
-The derivatives $$P_t$$ and $$P_z$$ are then given by:
+The derivatives $$P_t$$ and $$P_z$$ are given by:
 
 $$\begin{aligned}
     P_t
@@ -76,12 +77,15 @@ These derivatives both go to infinity when their denominator is zero,
 which, since $$\varepsilon$$ is positive, will happen earliest where $$f'$$
 has its most negative value, called $$f_\mathrm{min}'$$,
 which is located on the trailing edge of the pulse.
-At the propagation distance where this occurs, $$L_\mathrm{shock}$$,
+At the propagation distance $$z$$ where this occurs, $$L_\mathrm{shock}$$,
 the pulse will "tip over", creating a discontinuous shock:
 
 $$\begin{aligned}
+    0
+    = 1 + 3 \varepsilon z f_\mathrm{min}'
+    \qquad \implies \qquad
     \boxed{
-        L_\mathrm{shock} = -\frac{1}{3 \varepsilon f_\mathrm{min}'}
+        L_\mathrm{shock} \equiv -\frac{1}{3 \varepsilon f_\mathrm{min}'}
     }
 \end{aligned}$$
 
@@ -135,5 +139,6 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## References
 1. B.R. Suydam, [Self-steepening of optical pulses](https://doi.org/10.1007/0-387-25097-2_6), 2006, Springer.