i
u
t
+v
u
xx
+σu
2
|
u0 (1)
​is one of the fundamental equations for wave propagation (from the oceans to the optical fibers). The function
u(x,t)
is a complex function of real variables
x∈R,t>0
. We denote by
i
the complex unit. The parameters
ν,σ∈R
and we consider being positive. One of its most important solutions is the so-called soliton solution (soliton)
u(x,t)
2
Ω-
2
κ
ν
σ
sech
Ω-
2
κ
ν
ν
(x-2κνt)exp[i(κx-Ωt)](2)
.where
κ,Ω∈R
.​For
Ω2
,
κνσ1
, draw snapshots of the evolution for the times
t0,2.5,3.5
when
x∈[-50,50]
, of both t the solution
u
and the real part of
Re(u)
(or the imaginary part
Im(u)
) . What do you observe? Considering both the form of the solution, in conjunction with the graphs, give a brief explanation of the behavior of the solution
u
.
In[]:=
Clear["Global`*"]​​(*defineparameters*)​​v=1;​​k=1;​​σ=1;​​Ω=2;​​(*definesolitonsolution*)​​u[x_,t_]=Sqrt[2]*Sqrt[Abs[(Ω-k^2*v)/σ]]*Sech[Sqrt[Abs[(Ω-k^2*v)/v]]*(x-2*k*v*t)]*Exp[I(k*x-Ω*t)]
Out[]=
2
(-2t+x)

Sech[2t-x]

For
t0

In[]:=
fig1=Plot[Evaluate[Im[u[x,0]]],{x,-50,50},PlotRange->All,PlotLabel->"Imu"]
Out[]=
In[]:=
fig2=Plot[Evaluate[Re[u[x,0]]],{x,-50,50},PlotRange->All,PlotLabel->"Re[u]"]
Out[]=
In[]:=
fig3=Plot[Evaluate[Abs[u[x,0]]],{x,-50,50},PlotRange->All,PlotStyle->{Thick,Dashed,Blue},PlotLabel->"Abs[u]"]
Out[]=

For
t2.5

In[]:=
fig4=Plot[Evaluate[Im[u[x,2.5]]],{x,-50,50},PlotRange->All,PlotLabel->"Im[u]"]
Out[]=
In[]:=
fig5=Plot[Evaluate[Re[u[x,2.5]]],{x,-50,50},PlotRange->All,PlotLabel->"Re[u]"]​​
Out[]=
In[]:=
fig6=Plot[Evaluate[Abs[u[x,2.5]]],{x,-50,50},PlotRange->All,PlotStyle->{Thick,Dashed,Red},PlotLabel->"Abs[u]"]
Out[]=

For
t3.5

In[]:=
fig7=Plot[Evaluate[Im[u[x,3.5]]],{x,-50,50},PlotRange->All,PlotLabel->"Im[u]"]
Out[]=
In[]:=
fig8=Plot[Evaluate[Re[u[x,3.5]]],{x,-50,50},PlotRange->All,PlotLabel->"Re[u]"]
Out[]=
In[]:=
fig9=Plot[Evaluate[Abs[u[x,3.5]]],{x,-50,50},PlotRange->All,PlotStyle->{Thick,Dashed,Purple},PlotLabel->"Abs[u]"]

Numerical Solution

Problem 1

3-D visualization

Problem 2

CITE THIS NOTEBOOK

Nonlinear Schrödinger equation: solution and visualization​
by Athanasios Paraskevopoulos​
Wolfram Community, STAFF PICKS, April 24 2023
​https://community.wolfram.com/groups/-/m/t/2906358