We evaluate a method to describe subcritical transitions in systems with a finitewavelength instability (pattern forming systems) by means of a direct expansion in the amplitude of the linearly least stable mode. We apply the method to two model equations, a subcritical generalization of the Swift-Hohenberg equation and an extension of the Kuramoto-Sivashinsky equation. We assess the reliabilit...

We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strateg...

We analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified Kuramoto–Sivashinsky equations. Time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization. The general case analyzed covers several physical applications arising in multi-phase hydrodynamics and t...

Steady premixed flames subjected to space-periodic steady forcing are studied via inhomogeneous Michelson-Sivashinsky (MS) and then Burgers equations. For both, the flame slope is posited to comprise contributions from complex poles to locate, and from a base-slope profile chosen in three classes (pairs of cotangents, single-sine functions or sums thereof). Base-slope-dependent equations for th...

The dynamics of patterns in large two-dimensional domains remains a challenge in nonequilibrium phenomena. Often it is addressed through mild extensions of one-dimensional equations. We show that full two-dimensional generalizations of the latter can lead to unexpected dynamic behavior. As an example we consider the anisotropic Kuramoto-Sivashinsky equation, which is a generic model of anisotro...

An algorithm is presented to integrate nonlinear partial differential equations, which is particularly useful when accurate estimation of spatial derivatives is required. It is based on an analytic approximation method, referred to as distributed approximating functionals (DAF's), which can be used to estimate a function and a finite number of derivatives with a specified accuracy. As an applic...

In this work we investigate a technique for accelerating convergence of adjoint–based optimization of PDE systems based on a nonlinear change of variables in the control space. This change of variables is accomplished in the differentiate–then–discretize approach by constructing the descent directions in a control space not equipped with the Hilbert structure. We show how such descent direction...

Most evolution equations are partially integrable and, in order to explicitly integrate all possible cases, there exist several methods of complex analysis, but none is optimal. The theory of Nevanlinna and Wiman-Valiron on the growth of the meromorphic solutions gives predictions and bounds, but it is not constructive and restricted to meromorphic solutions. The Painlevé approach via the a pri...

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (pdes) [1]. However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretisation of pdes such as Burgers’ equation [2] and the Kuramoto-Sivashinsky equation [3]. In this paper the g...

We process snapshots of trajectories of evolution equations with intrinsic symmetries, and demonstrate the use of recently developed eigenvector-based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise. Our illustrative examples include a one-dimensional evolutionary partial differential (the Kuramoto-Sivashinsky) equation with...