From 270adb174e9f536f408296ab0141478666dd1690 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 6 Oct 2024 10:43:21 +0200
Subject: Improve knowledge base

---
 source/know/concept/self-steepening/index.md | 22 ++++------------------
 1 file changed, 4 insertions(+), 18 deletions(-)

(limited to 'source/know/concept/self-steepening')

diff --git a/source/know/concept/self-steepening/index.md b/source/know/concept/self-steepening/index.md
index 80d9fcb..409f6c9 100644
--- a/source/know/concept/self-steepening/index.md
+++ b/source/know/concept/self-steepening/index.md
@@ -74,10 +74,10 @@ but in fact most authors make a small additional approximation.
 Let us write out the derivative of $$\gamma(\omega)$$:
 
 $$\begin{aligned}
-    \pdv{\gamma}{\omega}
+    \dv{\gamma}{\omega}
     = \frac{n_2}{c A_\mathrm{eff}}
-    + \frac{\omega}{c A_\mathrm{eff}} \pdv{n_2}{\omega}
-    - \frac{\omega n_2}{c A_\mathrm{eff}^2} \pdv{A_\mathrm{eff}}{\omega}
+    + \frac{\omega}{c A_\mathrm{eff}} \dv{n_2}{\omega}
+    - \frac{\omega n_2}{c A_\mathrm{eff}^2} \dv{A_\mathrm{eff}}{\omega}
 \end{aligned}$$
 
 In practice, the $$\omega$$-dependence of $$n_2$$ and $$A_\mathrm{eff}$$
@@ -102,11 +102,10 @@ is still conserved, defined as:
 $$\begin{aligned}
     \boxed{
         N(z)
-        \equiv \int_{-\infty}^\infty \frac{|\tilde{A}(z, \Omega)|^2}{\Omega} \dd{\Omega}
+        \equiv \int_0^\infty \frac{|A(z, \omega)|^2}{\omega} \dd{\omega}
     }
 \end{aligned}$$
 
-
 A pulse's intensity is highest at its peak,
 so the nonlinear index shift is strongest there,
 meaning that the peak travels slightly slower than the rest of the pulse,
@@ -244,19 +243,6 @@ Nevertheless, the trend is nicely visible:
 the trailing slope becomes extremely steep, and the spectrum
 broadens so much that dispersion can no longer be neglected.
 
-{% comment %}
-When self-steepening is added to the nonlinear Schrödinger equation,
-it no longer conserves the total pulse energy $$\int |A|^2 \dd{t}$$.
-Fortunately, the photon number $$N_\mathrm{ph}$$ is still
-conserved, which for the physical envelope $$A(z,t)$$ is defined as:
-
-$$\begin{aligned}
-    \boxed{
-        N_\mathrm{ph}(z) = \int_0^\infty \frac{|\tilde{A}(z,\omega)|^2}{\omega} \dd{\omega}
-    }
-\end{aligned}$$
-{% endcomment %}
-
 
 
 ## References
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