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Superconductivity Within a Slab

Original source of the model: F. Odeh: Existence and bifurcation theorems for the Ginzburg-Landau equations. J. Math. Phys. 8 (1967) 2351-2356.

Solutions are summarized in the branching diagram in Figure 1. A branch of asymmetric solutions branches off a branch of symmetric solutions at a pitchfork bifurcation point. The data of the bifurcation point and four turning points (TP) are listed in the Table.

Investigation of the above boundary-value problem reveals that it supports symmetry. To see this, define

\begin{displaymath}\tilde\Theta(x): = \Theta(d-x), \ \ \tilde\Psi (x): = - \Psi (d-x)\end{displaymath}

and realize that the signs are the same after differentiating twice with respect to x. Hence, $\tilde \Theta$ and $\tilde\Psi$ satisfy the differential equations if $\Theta$ and $\Psi$ do. Analyzing the boundary conditions, as, for example,

\begin{displaymath}\tilde\Theta'(0) = -\Theta'(d) = 0,\end{displaymath}

shows that $\tilde \Theta$ and $\tilde\Psi$ satisfy the boundary conditions. Hence, the equation supports even $\Theta$ and odd $\Psi$. This does not imply that solutions must be even or odd. In fact, the equation has asymmetric solutions. See the solution in Figure 2, which depicts both the solutions $\Theta, \Psi$ and $\tilde\Theta,\tilde\Psi$ for the parameter value $\lambda =0.6647$.

In order to calculate solutions, we rewrite the boundary-value problem as a first-order system. Denoting $y_1 =\Theta$, $y_2 =\Theta'$, $y_3 =\Psi$, and $y_4 =\Psi'$ yields

\begin{displaymath}y'_1 & = y_2 \cr\end{displaymath}


\begin{displaymath}y'_2 & = y_1 (y^2_1 -1+\lambda y^2_3) \cr\end{displaymath}


\begin{displaymath}y'_3 & = y_4 \cr\end{displaymath}


\begin{displaymath}y'_4 & = y^2_1 y_3 \cr\end{displaymath}


\begin{displaymath}y_2 (0) & = y_2 (5) = 0, \ \ y_4 (0) = y_4 (5) = 1\cr\end{displaymath}



The solution profiles of the bifurcation point are shown in Figures 4 and 5; notice the symmetries. This bifurcation point is straddled between the two turning points with asymmetric solutions, which are depicted in Figures 2 and 3. Compare the figures to see how this straddling becomes evident in the solution profiles.

\begin{displaymath}\vbox {\halign {$ ...


Figure 1
Bifurcation diagram $\Psi(0)$ versus $\lambda$. Two branches intersecting in a pitchfork bifurcation.

Figure 2
Two antisymmetric solutions $\Theta(x)$ for $\lambda=0.6647$.

Figure 3
Two antisymmetric solutions $\Psi(x)$ for $\lambda=0.6647$.

Figure 4
Symmetric solution $\Theta(x)$ for $\lambda=0.91196$.

Figure 5
Symmetric solution $\Psi(x)$ for $\lambda=0.91196$.

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Figure 5