Nonexistence of monotonic solutions in a model of dendritic growth

Abstract

A simple model for dendritic growth is given by δ 2 θ ′ + θ ′ = cos ⁡ ( θ ) {\delta ^2}\theta ’ + \theta ’ = \cos \left ( \theta \right ) . For δ ≈ 1 \delta \approx 1 we prove that there is no bounded, monotonic solution which satisfies θ ( − ∞ ) = − π / 2 \theta \left ( { - \infty } \right ) = - \pi /2 and θ ( ∞ ) = π / 2 \theta \left ( \infty \right ) = \pi /2 . We also investigate the existence of bounded, monotonic solutions of an equation derived from the Kuramoto-Sivashinsky equation, namely y ′ + y ′ = 1 − y 2 / 2 y’ + y’ = 1 - {y^2}/2 . We prove that there is no monotonic solution which satisfies y ( − ∞ ) = − 2 y\left ( { - \infty } \right ) = - \sqrt 2 and y ( ∞ ) = 2 y\left ( \infty \right ) = \sqrt 2 .