Asymmetric bound states of differential equations in nonlinear optics
Consider the boundary value problem \[ -u''(x)+\beta^2 u(x)=n^2 \bigl(x,u^2(x)\bigr) u(x),x\in \mathbb{R},\lim_{| x|\to+\infty} u(x)= \lim_{ | x|\to +\infty}u'(x)=0. \tag{1} \] It is known that at certain value \(\beta_0\) of \(\beta\) a family of asymmetric solutions bifurcates from the branch of symmetric ones. The authors investigate the same phenomenon for a class of equations (1) which cannot be integrated directly. The nonlinearity \(n^2\) is taken of the form \[ n^2(x,s)=q^2+ c^2h(x/ \varepsilon)+s-\alpha(x/ \varepsilon,s) \] where \(\varepsilon\) is a small positive parameter; \(q,c \in \mathbb{R}\), \(h\) and \(\alpha\) are given functions. The authors use a variational method related to the Poincaré-Melnikov theory of homoclinics.
scientific article; zbMATH DE number 1279137