Differential and Integral Equations

A domain-wall between single-mode and bimodal states

G. J. B. van den Berg and R. C. A. M. van der Vorst

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We examine a model equation describing spatial patterns in a class of physical systems where instabilities to travelling waves occur. The spatial patterns are modelled by a system of two second order ordinary differential equations, in which the cross-coupling coefficient is spatially dependent. The system has two clearly distinct types of stationary states, of which the stability depends on the cross-coupling coefficient. Under mild assumptions on the cross-coupling coefficient, we apply a variational method to prove the existence of a heteroclinic orbit between both types of states, corresponding to a domain-wall in the physical picture. This solution is found as a minimizer of a Lyapunov functional and the variational structure is exploited to obtain detailed information about the shape of the solution. In the case of a constant cross-coupling coefficient we find heteroclinic solutions connecting stationary states of the same type.

Article information

Differential Integral Equations, Volume 13, Number 1-3 (2000), 369-400.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B40: Boundary value problems on infinite intervals
Secondary: 34B15: Nonlinear boundary value problems 35K55: Nonlinear parabolic equations 49J10: Free problems in two or more independent variables


van den Berg, G. J. B.; van der Vorst, R. C. A. M. A domain-wall between single-mode and bimodal states. Differential Integral Equations 13 (2000), no. 1-3, 369--400. https://projecteuclid.org/euclid.die/1356124304

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