Multitransition kinks and pulses for fourth order equations with a bistable nonlinearity
The author studies the fourth-order nonlinear differential equation \[ u^{(\text{IV})}-g(u)u''-g'(u)(u')^2/2+f'(u)=0, \] where \(f\) is a symmetric double-well potential with bottoms at \(\pm 1\) and \(g\) is an even function. Such equations determine stationary solutions for some classes of PDEs (for example, for the extended Fisher-Kolmogorov equation). Conditions are given under which there exist kinks (heteroclinic solutions connecting \(-1\) and \(1\)) with arbitrarily many passages from \(-1\) to \(1\). Pulses (homoclinic solutions) are considered, too.
Multitransition heteroclinics and homoclinics
scientific article; zbMATH DE number 2117897