Fig. 1

Effect of Dirichlet boundary condition \(u_{BC} = 0\) on the bifurcation diagram of a bump solution. (A): bump solutions for Eq. (1) with \(\varOmega= [-10\pi, 10 \pi]\). (B): bumps solutions for Eqs. (3)–(4) posed on \(\varOmega= [-10\pi, 10 \pi]\) with Dirichlet boundary condition \(u_{BC} = 0\). Stable (unstable) solutions are indicated with solid (dashed) lines. The insets show the shapes of the lower (stable) bumps at \(\kappa= 0.7\) for \(q(x)\) (blue) and \(z(x)\) (red). Parameters are \(a_{1} =14\), \(a_{2} = 13\), \(b_{1} = 24\), \(b_{2} = 150\), \(c=5\)