Load the package.

In[1]:=

<< ReplaceVariables`

Define the original equation.

u[x_] := -a Tanh[a x/Sqrt[2]] eqorig = v '''[x] + v '[x] - 3 u[x]^2 v '[x] + Sqrt[2] (1 - a^2) u[x] v[x] == 0

Out[3]=

-2^(1/2) a (1 - a^2) Tanh[(a x)/2^(1/2)] v[x] + v^'[x] - 3 a^2 Tanh[(a x)/2^(1/2)]^2 v^'[x] + v^(3)[x] == 0

Make the first replacement:    x->z 2^(1/2)/a and a -> A^(1/2).

In[8]:=

eq = ReplaceVariables[eqorig, x -> Sqrt[2] z/a, v[x] -> v[z], v[z]] /. a^2 -> A

Out[8]=

4 Tanh[z] v[z] - 4 A Tanh[z] v[z] - 2 v^'[z] + 6 A Tanh[z]^2 v^'[z] - A v^(3)[z] == 0

The second replacement reads  v(z) -> w(z)/ cosh z  and  A -> 2/(2 + α).

In[11]:=

eq1 = Simplify[ReplaceVariables[eq, z -> z, v[z] -> 1/Cosh[z] w[z], w[z]] /. A -> 2/(α + 2)]

Out[11]=

1/(2 + α) (2 (3 (-1 + α) Tanh[z] w[z] - (-1 + α) w^'[z] + 3 Tanh[z] w^''[z] - w^(3)[z])) == 0

The third replacement is   z -> arctan ξ, w(z) -> ww(ξ) cosh z.

In[13]:=

EQ1 = ReplaceVariables[eq1, z -> ArcTanh[ξ], w[z] -> ww[ξ] Cosh[z], ww[ξ]]

Out[13]=

2 α ξ ww[ξ] - α ww^'[ξ] + α ξ^2 ww^'[ξ] + 6 ξ ww^''[ξ] - 12 ξ^3 ww^''[ξ] + 6 ξ^5 ww^''[ξ] - ww^(3)[ξ] + 3 ξ^2 ww^(3)[ξ] - 3 ξ^4 ww^(3)[ξ] + ξ^6 ww^(3)[ξ] == 0

The last replacement reads   ξ ->( 1 - ζ)^(1/2),  ww(ξ)=y(ζ)/ζ .

In[18]:=

EQ1a = ReplaceVariables[EQ1, ξ -> Sqrt[1 - ζ], ww[ξ] -> y[ζ]/ζ, y[ζ]]

Out[18]=

6 y[ζ] - α y^'[ζ] - 6 ζ y^'[ζ] + 6 ζ^2 y^''[ζ] - 4 ζ^2 y^(3)[ζ] + 4 ζ^3 y^(3)[ζ] == 0

The resulting equation is of hypergeometric type and is solved by the built-in function DSolve.

In[23]:=

sol = DSolve[EQ1a, y[ζ], ζ] /. α -> 2 (1/a^2 - 1) // FullSimplify

Out[23]=

{{y[ζ] -> -((1 + a^2 (-1 + 3 ζ)) C[1])/(-1 + a^2) - i i^(-(3 - 2/a^2)^(1/2)) 2^(-(3 - 2/a^2)^(1/2)) (1 + (1 - ζ)^(1/2))^(3 - 2/a^2)^(1/2) ζ^(3/2 - (-2 + 3 a^2)^(1/2)/(2 a^2^(1/2))) C[2] - i (2 i)^(3 - 2/a^2)^(1/2) (1 + (1 - ζ)^(1/2))^(-(3 - 2/a^2)^(1/2)) ζ^(1/2 (3 + (3 - 2/a^2)^(1/2))) C[3]}}

Converted by Mathematica  (July 21, 2004)