p d e 2 p a th is a free and easy to use M atlab continuation/bifurcation pack age for elliptic systems of PDEs with arbitrary many components, on general two dimensional domains, and with rather general boundary conditions. The package is based on the FEM of the M atlab p d e to o lb o x , and is explained by a number of examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard convection, and von Karman plate equations. These serve as templates to study new problems, for which the user has to provide, via M atlab function files, a description of the geometry, the boundary conditions, the coefficients of the PDE, and a rough initial guess of a solution. The basic algorithm is a one parame ter arclength-continuation with optional bifurcation detection and branch-switching. Stability calculations, error control and mesh-handling, and some elementary timeintegration for the associated parabolic problem are also supported. The con tinuation, branch-switching, plotting etc are performed via M atlab command-line function calls guided by the AUTO style. The software can be downloaded from w w w .s ta f f .u n i-o ld e n b u r g .d e /h a n n e s .u e c k e r/p d e 2 p a th , where also an on line documentation of the software is provided such that in this paper we focus more on the mathematics and the example systems.

The dynamics of the envelopes of spatially and temporarily oscillating wave packets advancing in spatially periodic media can approximately be described by solutions of a Nonlinear Schrödinger equation. Here we prove estimates for the error made by this formal approximation using Bloch wave analysis, normal form transformations, and Gronwall's inequality.

Gap solitons near a band edge of a spatially periodic nonlinear PDE can be formally approximated by solutions of Coupled Mode Equations (CMEs). Here we study this approximation for the case of the 2D Periodic Nonlinear Schrödinger / Gross-Pitaevskii Equation with a non-separable potential of finite contrast. We show that unlike in the case of separable potentials [T. Dohnal, D. Pelinovsky, and G. Schneider, J. Nonlin. Sci. 19, 95-131 (2009)] the CME derivation has to be carried out in Bloch rather than physical coordinates. Using the Lyapunov-Schmidt reduction we then give a rigorous justification of the CMEs as an asymptotic model for reversible non-degenerate gap solitons and even potentials and provide H s estimates for this approximation. The results are confirmed by numerical examples including some new families of CMEs and gap solitons absent for separable potentials.

We consider reaction-diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u0(kx − ωt; k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u0(kx + φ±; k) as x → ±∞ with different phases φ− = φ+ at infinity for solutions that initially converge to these states as x → ±∞. The proof is based on Bloch wave analysis, renormalization theory, and a rigorous decomposition of the perturbations of these wave solutions into a phase mode, which shows diffusive behavior, and an exponentially damped remainder. Depending on the dispersion relation, the asymptotic states mix linearly with a Gaussian profile at lowest order or with a nonsymmetric non-Gaussian profile given by Burgers equation, which is the amplitude equation of the diffusive modes in the case of a nontrivial dispersion relation.

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution of a single scalar field like a density or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn-and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dipcoating. Through the different examples we illustrate how to employ the numerical tools provided by the packages AUTO07P and PDE2PATH to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues.Published as: Engelnkemper, S., Gurevich, S.V.,

For a Selkov-Schnakenberg model as a prototype reaction-diffusion system on two dimensional domains we use the continuation and bifurcation software pde2path to numerically calculate branches of patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with a planar interface between the two patterns. We use the Ginzburg-Landau reduction to approximate the locations of these branches by Maxwell points for the associated Ginzburg-Landau system. For our basic model, some but not all of these branches show a snaking behaviour in parameter space, over the given computational domains. The (numerical) non-snaking behaviour appears to be related to too narrow bistable ranges with rather small Ginzburg-Landau energy differences. This claim is illustrated by a suitable generalized model. Besides the localized patterns with planar interfaces we also give a number of examples of fully localized patterns over patterns, for instance hexagon patches embedded in radial stripes, and fully localized hexagon patches over straight stripes.

We consider the sine-Gordon equation on metric graphs with simple topologies and derive vertex boundary conditions from the fundamental conservation laws together with successive spacederivatives of sine-Gordon equation. We analytically obtain traveling wave solutions in the form of standard sine-Gordon solitons such as kinks and antikinks for star and tree graphs. We show that for this case the sine-Gordon equation becomes completely integrable just as in case of a simple 1D chain. This simple analysis provides a cornerstone for the numerical solution of the general case, including a quantification of the vertex scattering. Applications of the obtained results to Josephson junction networks and DNA double helix are discussed.

We explain the setup for using the pde2path libraries for Hopf bifurcation and continuation of branches of periodic orbits and give implementation details of the associated demo directories. See [Uec19a] for a description of the basic algorithms and the mathematical background of the examples. Additionally we explain the treatment of Hopf bifurcations in systems with continuous symmetries, including the continuation of traveling waves and rotating waves in O(2) equivariant systems as relative equilibria, the continuation of Hopf bifurcation points via extended systems, and some simple setups for the bifurcation from periodic orbits associated to critical Floquet multipliers going through ±1. MSC: 35J47, 35B22, 37M20