Convert differential equation with substitutions?

Questions
,
1

Hi
How do we get from equation 11 to equation 13 with mathematica by converting equations 12?

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2

I have only:

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3

Thank you
My version of Mathematica is different and does not give that answer

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4

It dosen’t work because, DSolveChangeVariables command was introduced in 2022 in version 13.1.
You need update to newer version of Mathematica.

Reagards M.I.

5

This should work in version 12:

eq0 = D[u[x, t], {t, 2}] - (v - 2)*
    u[x, t] - (v + v1/\[Mu])*(-u[x, t]^3 + u[x, t]^5);
eq0 /. u -> Function[{x, t}, U[Sqrt[k] (x - c t)]]
% /. Sqrt[k] (x - c t) -> \[Xi]
% /. {U -> Function[\[Xi], Sqrt[W[\[Xi]]]]} // Simplify
Collect[%*4 \[Mu] W[\[Xi]]^(3/2)/(4 c^2 \[Mu] W[\[Xi]]),
 {W'[\[Xi]], W''[\[Xi]]}]
6

@gianlucagorni

I’m so grateful.
It has only one extra C^2.

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7

Sorry, I had started with the wrong derivative:

eq0 = D[u[x, t], {x, 2}] - (v - 2)*
    u[x, t] - (v + v1/\[Mu])*(-u[x, t]^3 + u[x, t]^5);
eq0 /. u -> Function[{x, t}, U[Sqrt[k] (x - c t)]]
% /. Sqrt[k] (x - c t) -> \[Xi]
% /. {U -> Function[\[Xi], Sqrt[W[\[Xi]]]]} // Simplify
eq = Collect[%*4 \[Mu] W[\[Xi]]^(3/2)/(4 \[Mu] W[\[Xi]]),
  {W[\[Xi]], W'[\[Xi]], W''[\[Xi]]}, Factor]
8

Much obliged.

9

Hello
Where did I go wrong for this conversion?

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10

Here is my interpretation:

eq0 = I*D[
    Subscript[\[Phi], 
     n][\[Tau]], \[Tau]] + (1 + \[Mu]*
      Abs[Subscript[\[Phi], n][\[Tau]]]^2)*
   (Subscript[\[Phi], n + 1][\[Tau]]*Exp[I*k] + 
     Subscript[\[Phi], n - 1][\[Tau]]*Exp[-I*k]) -
  \[Nu]*(Abs[Subscript[\[Phi], n][\[Tau]]]^2)*
   Subscript[\[Phi], n][\[Tau]]
% /. Subscript[\[Phi], n_] -> 
  Function[\[Tau], Subscript[\[Chi], n] Exp[-I*\[Theta]*\[Tau]]]
FullSimplify[% /. Subscript[\[Chi], n_] :> Subscript[u, n]/Sqrt[\[Mu]],
 Element[\[Theta] | \[Tau] | Subscript[u, n_], Reals] && \[Mu] > 0]
Collect[%*\[Mu]^(
   1/2)/(E^(-I (k + \[Theta] \[Tau])) (1 + (Subscript[u, n])^2)),
 {Subscript[u, n - 1], Subscript[u, n + 1]}]

I don’t know how to get rid of the exponentials.

11

Thank you.
What is this part for?

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12

This eliminates some common factors and brings the expression into a form more similar to equation (3).

13

Thanks for your hard work on this But the answers are different.

14

Your equation (1) depends on k, while equation (3) does not. How is it possible? Are you tacitly assuming that k is an even integer?

15

Hi,
It is like this and we get from equation 6 to equation 10.

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16

It was resolved ,Thank you. I couldn’t have done it without you.