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 +++++++++++++---------
 1 file changed, 56 insertions(+), 35 deletions(-)

(limited to 'source/know/concept/nonlinear-schrodinger-equation')

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.
 
 
 
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