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Rigorous continuation of bifurcation points in the diblock copolymer equation

  • * Corresponding author: T. Wanner

    * Corresponding author: T. Wanner
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  • We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through ${\mathbb{Z}}_2$ symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.

    Mathematics Subject Classification: Primary: 35B40, 65G20, 65P30; Secondary: 35B41, 35K55, 65G30, 65N35, 74N99.


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  • Figure 1.  Equilibrium bifurcation diagrams of the Cahn-Hilliard model (left image) and the diblock copolymer model for $\sigma = 6$ (right image). In both figures, $\mu = 0$. Most of the shown branches actually correspond to two or more solution branches, since the $L^2$-norm of the associated solutions is used as the vertical diagram axis. The colors correspond to the indices of the solutions. Along the horizontal trivial solution branch they increase from zero (black) to five (cyan)

    Figure 2.  Two degenerate symmetry-breaking bifurcations as described in Example 2.14

    Figure 3.  Equilibrium solutions $u_0$ (in blue) of the diblock copolymer equation for $\sigma_0 = 6$, together with their associated kernel functions $\varphi _0$ (in orange). These two distinct stationary solutions are both saddle-node bifurcation points at the same parameter value $\lambda_0 \approx 262.9$ and the same $L^2$-norm close to $0.562$. In fact, the entire non-trivial portion of the bifurcation diagram is multiply covered. These equilibria are rigorously proved in Theorem 3.7

    Figure 4.  Equilibrium solutions $u_0$ (in blue) and associated kernel functions $\varphi _0$ (in orange) of the diblock copolymer equation for $\sigma_0 = 6$, with $\lambda_0 \approx 142.1, 53.6,203.1$ for top left, top right, and bottom right, respectively. All four solutions are pitchfork bifurcation points. They are the first bifurcation points on the first four branches bifurcating from the trivial solution in the right image sof Figure 1. Note that as in Figure 3, the bifurcation diagram is a double cover; corresponding to each of these four solutions, there is another solution at the same point in the bifurcation diagram. See also Theorems 3.13, 3.14, and 3.15

    Figure 5.  Piecewise linear curve approximation (in black) constructed using parameter continuation and existence of a global solution curve $\mathcal C$ of $f=0$ (in blue) nearby the approximations

    Figure 6.  Equilibrium solutions $u_0$ (in blue) of the diblock copolymer equation for $\sigma_0 = 6$, together with their associated kernel functions (in red). On left $\lambda_0 \approx 681.4$, on right $\lambda_0 \approx 1343.3$. These two distinct stationary solutions are both saddle-node bifurcation points. The equilibrium on the left (respectively right) is rigorously proved in Theorem 3.8 (respectively Theorem 3.9)

    Figure 7.  Three global $C^\infty$ branches of saddle-node bifurcation points of the diblock copolymer equation. The red (respectively green, blue) branch is proven in Theorem 3.10 (respectively Theorem 3.11, Theorem 3.12)

    Figure 8.  (Left) Global $C^\infty$ branches of pitchfork bifurcations points of the nonlinear diblock-copolymer equation (1). The red (respectively green, blue) branch is proven in Theorem 3.16 (respectively Theorem 3.17, Theorem 3.18). (Right) Zoom-in of the branches

    Figure 9.  The cosine Fourier coefficients of the saddle-node bifurcation point from Theorem 3.7. We show ${\bar a }_k$ for $k \ge 1$. Note that all even coefficients are $0$

    Figure 10.  The cosine Fourier coefficients of the saddle-node bifurcation point from Theorem 3.8. We show ${\bar a }_k$ for $k \ge 1$. Note that all even coefficients are $0$

    Figure 11.  The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.13. We show ${\bar a }_k$ for $k \ge 1$. Note that all even coefficients are $0$

    Figure 12.  The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.14. We show ${\bar a }_k$ for $k \ge 2$. Note that all other coefficients are $0$

    Figure 13.  The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.15. We show ${\bar a }_k$ for $k \ge 2$. Note that all other coefficients are $0$

    Table 1.  Some of the partial derivatives of the bifurcation function $b(\lambda,\nu)$ at the point $(\lambda_0,0)$ up to order three, together with the required partial derivatives of $W$

    $\begin{align*}D_{\lambda} b(\lambda_0,0)&= \psi_0^* D_{\lambda}F(\lambda_0,u_0) \; , \\[1ex] D_{\lambda\alpha} b(\lambda_0,0)&= \psi_0^* D_{\lambda u}F(\lambda_0,u_0)[\varphi _0] \\[1ex] & ~~~~ + \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,D_{\lambda}W(\lambda_0,v_0)] \; , \\[1ex] D_{\alpha\alpha} b(\lambda_0,0)&= \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,\varphi _0] \; , \\[1ex] D_{\alpha\alpha\alpha} b(\lambda_0,0)&= \psi_0^* D_{uuu}F(\lambda_0,u_0)[\varphi _0,\varphi _0,\varphi _0] \\[1ex] & ~~~~ + 3 \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,D_{vv}W(\lambda_0,v_0)[\varphi _0,\varphi _0]] \; , \\[1ex] L D_{\lambda} W(\lambda_0,v_0)&= -(I-P) D_{\lambda}F(\lambda_0,u_0) \; , \\[1ex] L D_{vv} W(\lambda_0,v_0)[\varphi _0,\varphi _0]&= -(I-P) D_{uu}F(\lambda_0,u_0)[\varphi _0,\varphi _0] \; . \end{align*} $
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