Numerical Analysis of the Geometric Model for Dendritic Growth of Crystals

Abstract

This paper investigates the numerical properties of solutions θ = θ(t) of the third-order equation

$$\varepsilon {\theta _{3}} + {\theta _{1}} = \frac{{\cos \;\theta }}{{1 + \alpha \cos 4\theta }},\;\theta \left( {\pm \infty } \right) = \pm \pi /2$$

((1.1))

Suffices to θ in (1.1) denote differentiation with respect to t; that is to say θ _n = dⁿ θ/dtⁿ; the boundary conditions θ(±∞) = ±π/2 are abbreviations for θ(t) → π±/2 as t → ±∞ respectively; and ε and α are prescribed parameters satisfying ε > 0 and 0 ≤ α < 1. It is convenient to write ε = 2^k, and to tabulate results as functions of α and k = log₂ ε. Kruskal and Segur [1] give references to the appearance of (1.1) as a model for the dendritic growth of crystals in a supercooled liquid. [Warning: there is some variation of notation in the literature; and, in particular, Kruskal and Segur write ε ² for the coefficient of θ ₃, thus entailing k = 21og₂ ε for their use of ε.] A strictly monotonic solution of (1.1) is called a needle crystal solution: and interest centres upon the question of the existence or non-existence of needle solutions. Our earlier paper [2] proved that needle solutions could not exist for α = 0. Kruskal and Segur [1] concluded that, for sufficiently small 6, needle solutions would exist for certain discrete values of α = α(k) > 0. Thus we have an eigenvalue problem with a discrete spectrum. Our analysis of this problem is incomplete, and several interesting questions remain unresolved.