Appendix A
Here we show how to derive the normal form (19.3) from (19.4).
Let \(\mathbf{u}(t,x)=(u(x,t),v(x,t))\) be a solution of (19.4). Define the function \(\tilde{\mathbf{u}}(t,x), t>0, x \in (-L,L)\) by
$$ \tilde{\mathbf{u}}(t,x) = \left\{ \begin{array}{ll} \mathbf{u}(t,x) \quad &{} \quad x \in (0,L), \\ \mathbf{u}(t, -x) \quad &{} \quad x \in (-L,0). \end{array} \right. $$
Then \(\tilde{\mathbf{u}}\) is a solution of the system
$$\begin{aligned} \left\{ \begin{array}{l} u_t = D_1 u_{xx} + au + bv + F(u,v) + \dfrac{s}{2L} \displaystyle \int _{-L}^L u(t,x) \, dx, \, x \in (-L,L), \, t>0,\\ v_t = D_2 v_{xx} + cu + dv + G(u,v) , \, x \in (-L,L), \, t>0 \end{array} \right. \end{aligned}$$
(19.29)
with Neumann boundary conditions at the boundary of interval \([-L,L] \subset \mathbb {R}\). Similarly, the solution \(\tilde{\mathbf{u}}\) of (19.4) can be extended to the solution of (19.29) with periodic boundary conditions with period 2L on \(\mathbb {R}\). Conversely, if the function \(\mathbf{u}(t,x)\) satisfying \(\mathbf{u}(t,x) = \mathbf{u}(t,-x)\) is a solution of (19.29) with period 2L, then it satisfies (19.29) and Neumann boundary condition at \(x=0,L\). Therefore, the solution of (19.4) can be identified to the “even” solution of (19.29) with periodic boundary conditions with period 2L. This implies the solution can be expressed by the Fourier series:
$$ u(t,x) = \sum _{n \in \mathbb {Z}} u_m(t) e^{i m \pi x/L}, \quad v(t,x) = \sum _{n \in \mathbb {Z}} v_m(t) e^{i m \pi x/L}, \quad (u_m,v_m) = (u_{-m}, v_{-m}) \in \mathbb {R}^2. $$
Substituting into (19.29), we have
$$\begin{aligned} \frac{d}{dt} \left( \begin{array}{c} u_m \\ v_m \end{array} \right) = M_m \left( \begin{array}{c} u_m \\ v_m \end{array} \right) + \left( \begin{array}{c} f_m \\ g_m \end{array} \right) , m \ge 0, \end{aligned}$$
(19.30)
where
$$\begin{aligned} f_m= & {} \sum _{m_1+m_2+m_3=m \atop m_1, m_2, m_3 \in \mathbb {Z}} ( f_{30} u_{m_1} u_{m_2} u_{m_3} + f_{21} u_{m_1} u_{m_2} v_{m_3} + f_{12} u_{m_1} v_{m_2} v_{m_3} + f_{03} v_{m_1} v_{m_2} v_{m_3} ), \\ g_m= & {} \sum _{m_1+m_2+m_3=m \atop m_1, m_2, m_3 \in \mathbb {Z}} ( g_{30} u_{m_1} u_{m_2} u_{m_3} + g_{21} u_{m_1} u_{m_2} v_{m_3} + g_{12} u_{m_1} v_{m_2} v_{m_3} + g_{03} v_{m_1} v_{m_2} v_{m_3} ),\\ f_{j \ell }= & {} \dfrac{1}{j! \ell !}\dfrac{\partial ^{j \ell } F}{\partial u^j \partial v^\ell }(0,0), \qquad g_{j \ell } = \dfrac{1}{j! \ell !}\dfrac{\partial ^{j \ell } G}{\partial u^j \partial v^\ell }(0,0), \quad j+\ell =3,\,j,\ell \in \mathbb {N}, \end{aligned}$$
$$\begin{aligned} M_m = \left\{ \begin{array}{ll} \left( \begin{array}{cc} a + s &{} b \\ c &{} d\end{array} \right) &{} (m=0), \\ \\ \left( \begin{array}{cc} a - D_1 m^2 k_0^2 &{} b \\ c &{} d- D_2 m^2 k_0^2 \end{array} \right)&(m \not =0 ), \end{array} \right. \end{aligned}$$
(19.31)
and \(k_0 = \pi /L\). We consider the system (19.30) in a phase space
$$\begin{aligned} X_F:= & {} \Big \{ \{ (u_m,v_m) \}_{m \in \mathbb {Z}} ; (u_m,v_m) = (u_{-m}, v_{-m}) \in \mathbb {R}^2, \\&\qquad \Vert \{ (u_m,v_m) \}_{m \in \mathbb {Z}} \Vert ^2_{X_F} = \sum _{m \in \mathbb {Z}} (1+ m^2)^2 |(u_m,v_m)|^2 < \infty \Big \} \end{aligned}$$
Solving \(\det M_0 = \det M_1 = \det M_2 = 0\) for \(s, k_0\) and \(D_2\), a triply degenerate point of 0:1:2-modes is given by the following:
$$\begin{aligned}&k_0 = k_0^{1,2} := \left[ \, \dfrac{1}{8dD_1} \left\{ 5 \varDelta - \sqrt{ 25\varDelta ^2 - 16ad \varDelta } \right\} \, \right] ^{1/2},\\&\\&D_2 = D_2^{1,2}:=\dfrac{\{dD_1 (k_0^{*})^2 - \varDelta \}}{(k_0^{*})^2\{ D_1 (k_0^{*})^2 - a\}},\\&\\&s = s^*:= -\varDelta /d, \end{aligned}$$
where \(\varDelta = ad-bc\). Near this degenerate point, we can apply the center manifold theory (for instance, see [3, 9, 10]). To compute the dynamics on the center manifold of (19.4), we diagonalize the equations in (19.30) for \(m=0,1\) and 2. Set \((k_0,D_2,s) = (k_0^{1,2},D_2^{1,2},s^*)\). Then changing variables \(^t(u_m,v_m) = T_m\, ^t(\tilde{u}_m,\tilde{v}_m), (m=0,1,2)\) by the matrix
$$ T_0 = \left( \begin{array}{cc} -d &{} bc/d \\ c &{} c \end{array} \right) ,\, T_m = \left( \begin{array}{cc} -d + D_2^{1,2} m^2 (k_0^{1,2})^2 &{} a-D_1 m^2 (k_0^{1,2})^2 \\ c &{} c \end{array} \right) , m=1,2, $$
we have
$$ \left( \begin{array}{c} \dot{\tilde{u}}_m \\ \dot{\tilde{v}}_m \end{array} \right) = \left( \begin{array}{cc} 0 &{}0 \\ 0 &{} \mu _m^{-} \end{array} \right) \left( \begin{array}{c} \tilde{u}_m \\ \tilde{v}_m \end{array} \right) + T_m^{-1} \left( \begin{array}{c} \tilde{f}_m \\ \tilde{g}_m \end{array} \right) , m=0,1,2. $$
Here,
$$\begin{aligned}&\mu _0^{-} := d+bc/d, \quad \mu _m^{-} :=(a+d) - m^2 (D_1+D_2^{1,2}) (k_0^{1,2})^2, \\&\tilde{f}_m := f_m|_{^t(u_{m_j},v_{m_j}) = T_m\,^t(\tilde{u}_{m_j},\tilde{v}_{m_j})}, \quad \tilde{g}_m := g_m|_{^t(u_{m_j},v_{m_j}) = T_m\,^t(\tilde{u}_{m_j},\tilde{v}_{m_j})}. \end{aligned}$$
Set
$$ \rho := (k_0^{1,2},D_2^{1,2},s^*)-(k_0,D_2,s), \quad \mu _m^+ := \left\{ \mathrm{tr}~ M_m + \sqrt{(\mathrm{tr} ~ M_m)^2 - 4 \det M_m } \right\} /2. $$
We define a neighborhood \(\mathscr {U}_{\varepsilon }\) of \(X_F \times \mathbb {R}^3\):
$$ \mathscr {U}_\varepsilon := \left\{ ( \{(u_m, v_m)\}_{m \in \mathbb {Z}} , \rho ) \in X_F \times \mathbb {R}^3 ; ||\{(u_m, v_m)\}_{m \in \mathbb {Z}}||_{X_F} + |\rho | < \varepsilon \right\} .$$
Then we have the following theorem.
Theorem A ([13]) For given constants \(a,b,c,d,D_1\)
, there exists a positive constant \(\varepsilon \) such that the local center manifold \(\mathscr {W}_{loc}^c\) of (19.30) is contained in \(\mathscr {U}_{\varepsilon }\). Moreover, the dynamics of (19.30) on the manifold \(\mathscr {W}_{loc}^c\) is governed by the following system:
$$\begin{aligned} \left\{ \begin{array}{l} \dot{z}_0 = (\mu ^+_0 + a_1 z_0^2 + a_2 z_1 ^2 + a_3 z_2^2 ) z_0 + a_4 z_1^2 z_2 + o(3), \\ \dot{z}_1 = (\mu ^+_1 + b_1 z_0^2 + b_2 z_1 ^2 + b_3 z_2^2 ) z_1 + b_4 z_0z_1z_2 + o(3),\\ \dot{z}_2 = (\mu ^+_2 + c_1 z_0^2 + c_2 z_1 ^2 + c_3 z_2^2 ) z_2 + c_4 z_0z_1^2+ o(3). \end{array} \right. \end{aligned}$$
(19.32)
Here, \(z_j(t) \in \mathbb {R}\) denote \(\tilde{u}_j(t)\,(j=0,1,2)\), and o(3) denotes \(o(|(z_0,z_1,z_2)|^3)\). In addition, the coefficients \(\mu _j^{+},a_j,b_j,c_j\) are dependent on the coefficients and parameters appearing in (19.30).
Proof
The first statement of the theorem follows from standard center manifold theory. It also states that for \(m \not =0,1,2 \), there exist functions
$$ h^u_m(\tilde{u}_0,\tilde{u}_1,\tilde{u}_2;\rho ) , m \ge 3, \quad h_m^v (\tilde{u}_0,\tilde{u}_1,\tilde{u}_2;\rho ) ,~ m \ge 0, $$
satisfying
$$\begin{aligned}&\dfrac{\partial h_m^u}{\partial \tilde{u}_j} (0,0,0;0) =\dfrac{\partial h_m^v}{\partial \tilde{u}_j} (0,0,0;0)=0, ~(j=0,1,2) \end{aligned}$$
and
$$\begin{aligned}&\dfrac{\partial h_m^u}{\partial \rho } (0,0,0;0) =\dfrac{\partial h_m^v}{\partial \rho } (0,0,0;0)=0 \end{aligned}$$
such that the local invariant manifold \(\mathscr {W}_{loc}^c\) is expressed by
$$\begin{aligned} \mathscr {W}_{loc}^c= & {} \big \{ \{(\tilde{u}_\ell ,\tilde{v}_\ell ), (u_m,v_m) \}_{ |\ell | \le 2, \, |m| \ge 3 } \in X_F; \tilde{v}_\ell = h^v_\ell (\tilde{u}_0,\tilde{u}_1,\tilde{u}_2;\rho ), \\&\qquad (u_m,v_m) =( h_m^u (\tilde{u}_0,\tilde{u}_1,\tilde{u}_2; \rho ), h_m^v (\tilde{u}_0,\tilde{u}_1,\tilde{u}_2; \rho )), |\ell | \le 2, \, |m| \ge 3 \big \}. \end{aligned}$$
We can check that if \(|(\tilde{u}_0,\tilde{u}_1,\tilde{u}_2; \rho )| < \varepsilon \) then \(h^u_m = h^v_m = o(\varepsilon ^3)\). Then, the cubic truncated equations for \(\tilde{u}_m,\,(m=0,1,2)\) are given by the following:
$$\begin{aligned}&\dot{\tilde{u}}_0 = \mu ^+_0 \tilde{u}_0 - \dfrac{1}{\mu _0^-} \left\{ \tilde{f}_0 - \dfrac{b}{d} \tilde{g}_0 \right\} ,\\&\dot{\tilde{u}}_m = \mu ^+_m \tilde{u}_m - \dfrac{1}{c \mu _m^-} \{ c \tilde{f}_m +(-a + D_1m^2 (k_0^{1,2})^2 ) \tilde{g}_m \}, m=1,2, \end{aligned}$$
where
$$\begin{aligned}&\tilde{f}_m := \sum _{m_1+m_2+m_3=m \atop m_j \in \{ 0, \pm 1, \pm 2\}} ( f_{30} B_{m_1}B_{m_2}B_{m_3} \tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} \\&+\, c f_{21} B_{m_1}B_{m_2} \tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} +c^2 f_{12} B_{m_1}\tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} + c^3 f_{03} \tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} ) \end{aligned}$$
and
$$\begin{aligned}&\tilde{g}_m := \sum _{m_1+m_2+m_3=m \atop m_j \in \{0, \pm 1, \pm 2\}} ( g_{30} B_{m_1}B_{m_2}B_{m_3} \tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} \\&+ \,c g_{21} B_{m_1}B_{m_2} \tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} +c^2 g_{12} B_{m_1}\tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} + c^3 g_{03} \tilde{u}_{m_1} \tilde{u}_{m_2} \tilde{u}_{m_3} ).\\ \end{aligned}$$
Here, \(B_{j} = -d + j^2 D_2^{1,2} (k_0^{1,2})^2\). This gives the cubic truncated system (19.32). We show the explicit form of coefficients in Appendix B.
Appendix B
We show the coefficients of system (19.3) explicitly. We put
$$ \begin{array}{l@{\quad }l} \mu _0^{-} := d+bc/d, &{} \,\mu _m^{-} :=(a+d) - m^2 (D_1+D_2^{1,2}) (k_0^{1,2})^2, \\ A_m := -a + D_1^{1,2} m^2 (k_0^{1,2})^2, \quad &{} B_m = -d + D_2^{1,2} m^2 (k_0^{1,2})^2. \end{array} $$
Then we have the following,
$$ \begin{array}{ll} \mu _m := \left\{ \mathrm{tr\,}M_m + \sqrt{(\mathrm{tr\,}M_m)^2 - 4 \det M_m } \right\} /2, \quad &{} a_j = -\dfrac{1}{\mu _0^+} P_f^{a_j} + \dfrac{b}{d \mu _0^+} P_g^{a_j}, ~j=1 \dots 4, \\ &{} \\ b_j = -\dfrac{1}{\mu _1^-} P_f^{b_j} - \dfrac{A_1}{c \mu _1^-} P_g^{b_j}, ~j=1 \dots 4, \quad &{} c_j = -\dfrac{1}{\mu _2^-} P_f^{c_j} - \dfrac{A_2}{d \mu _2^-} P_g^{c_j}, ~j=1 \dots 4. \end{array} $$
Here,
$$\begin{aligned} \left( \begin{array}{c} P_f^{a_1} \\ P_g^{a_1} \end{array} \right):= & {} B_0^3 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + c B_0^2 \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) + c^2 B_0 \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) , \\ \\ \left( \begin{array}{c} P_f^{a_2} \\ P_g^{a_2} \end{array} \right):= & {} 6 B_1^2 B_0 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 2c B_1(B_1+2B_0) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) +2 c^2 (B_0 + 2B_1) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +6 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{c} P_f^{a_3} \\ P_g^{a_3} \end{array} \right):= & {} 6 B_2^2 B_0 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 2c B_2(B_2+2B_0) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) + 2c^2 (B_0 + 2B_2) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +6 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) ,\\ \\ \left( \begin{array}{c} P_f^{a_4} \\ P_g^{a_4} \end{array} \right):= & {} 6 B_1^2 B_2 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 2c B_1(B_1+2B_2) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) + 2c^2 (B_2 + 2B_1) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +6 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) , \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{c} P_f^{b_1} \\ P_g^{b_1} \end{array} \right):= & {} 3 B_0^2 B_1 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + c B_0(B_0+2B_1) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) + c^2 (B_1+2B_0) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +3c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) ,\\ \\ \left( \begin{array}{c} P_f^{b_2} \\ P_g^{b_2} \end{array} \right):= & {} 3 B_1^3 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 3 c B_1^2 \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) +3 c^2 B_1 \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +3 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) , \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{c} P_f^{b_3} \\ P_g^{b_3} \end{array} \right):= & {} 6 B_2^2 B_1 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 2c B_2(B_2+2B_1) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) + 2c^2 (B_1 + 2B_2) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +6 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) ,\\ \\ \left( \begin{array}{c} P_f^{b_4} \\ P_g^{b_4} \end{array} \right):= & {} 6 B_0 B_1 B_2 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 2c (B_0B_1+B_1B_2+B_2B_0) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) \\&+\, 2c^2 (B_0+B_1+B_2 ) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) + 6 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) , \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{c} P_f^{c_1} \\ P_g^{c_1} \end{array} \right):= & {} 3 B_0^2 B_2 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + c B_0(B_0 + 2 B_2) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) + c^2 (B_2 + 2B_0) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) + 3 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) ,\\ \\ \left( \begin{array}{c} P_f^{c_2} \\ P_g^{c_2} \end{array} \right):= & {} 6 B_1^2 B_2 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 2c B_1(B_1+2B_2) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) +2 c^2 (B_2 + 2B_1) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +6 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) , \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{c} P_f^{c_3} \\ P_g^{c_3} \end{array} \right):= & {} 3 B_2^3 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + 3 c B_2^2 \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) +3 c^2 B_2 \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +3 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) ,\\ \\ \left( \begin{array}{c} P_f^{c_4} \\ P_g^{c_4} \end{array} \right):= & {} 3 B_1^2 B_0 \left( \begin{array}{c} F_{30} \\ G_{30} \end{array} \right) + c B_1(B_1+2B_0) \left( \begin{array}{c} F_{21} \\ G_{21} \end{array} \right) + c^2 (B_0 + 2B_1) \left( \begin{array}{c} F_{12} \\ G_{12} \end{array} \right) +3 c^3 \left( \begin{array}{c} F_{03} \\ G_{03} \end{array} \right) . \end{aligned}$$
