1. 27 October 2005 Konstantin Blyuss, University of Exeter 1 Konstantin Blyuss, University of Exeter 1 Spatio-temporal dynamics in a phase- field model with phase-dependent heat absorption Konstantin Blyuss (joint work with Peter Ashwin, David Wright and Andrew Bassom) University of Exeter, UK 27 October 2005

2. Konstantin Blyuss, University of Exeter 2 Konstantin Blyuss, University of Exeter 2 Problem formulation

3. 27 October 2005 Konstantin Blyuss, University of Exeter 3 Konstantin Blyuss, University of Exeter 3 Phase-field model The dynamics is characterized by two fields: the temperature T(x,t) and the phase  x,t). The convention is  is melt and  is solid. Phase-field equation has the form where Here, p is the interface thickness, is the strength of coupling between the phase field and the temperature field.

4. 27 October 2005 Konstantin Blyuss, University of Exeter 4 Konstantin Blyuss, University of Exeter 4 Temperature evolution is determined by with being the radiative absorption coefficients, b is the thermal emission coefficient, T a is the ambient temperature and d is a thermal diffusivity. Basic features: energy throughput of the system is much larger than the latent heat two phases have different rates of heat absorption

5. 27 October 2005 Konstantin Blyuss, University of Exeter 5 Konstantin Blyuss, University of Exeter 5 Steady states Uniform equilibria of the model solve the following system of equations Two of the roots are where Provided we have This also necessarily requires a -1 >a 1.

6. 27 October 2005 Konstantin Blyuss, University of Exeter 6 Konstantin Blyuss, University of Exeter 6 Assuming one has a cubic equation for the equilibrium phase where Depending on the relation between A, B and there are one or more roots with These roots correspond to the mushy layers. For our system the steady states with intermediate values of the phase and temperature can be interpreted as the states of mushy layer, where a transition from melt to solid takes place.

7. 27 October 2005 Konstantin Blyuss, University of Exeter 7 Konstantin Blyuss, University of Exeter 7 Dispersion relation for stability of the steady states is From this relation it follows that the linearly stable steady states are further stabilised by diffusion. The possibility of linearly unstable steady states is not excluded, but the Turing instability cannot occur since we have an inhibitor-inhibitor system.

8. 27 October 2005 Konstantin Blyuss, University of Exeter 8 Konstantin Blyuss, University of Exeter 8 Travelling waves These solutions have the form Substituting this into the system gives Two equilibria are and a travelling wave is a heteroclinic connection between them. Spectrum of the linearization near the steady states has two positive and two negative real eigenvalues, and so heteroclinic connections can exist only for isolated values of velocity c.

9. 27 October 2005 Konstantin Blyuss, University of Exeter 9 Konstantin Blyuss, University of Exeter 9 a)Sinusoidal initial profile. b)Initial profile in the form of tanh

10. 27 October 2005 Konstantin Blyuss, University of Exeter 10 Konstantin Blyuss, University of Exeter 10 Stability of travelling waves Linearizing the system near the travelling wave solution and looking for solutions of the form one arrives at the following eigenvalue problem for the growth rate  : with the auxiliary function This system can be recast as a first-order system

11. 27 October 2005 Konstantin Blyuss, University of Exeter 11 Konstantin Blyuss, University of Exeter 11 Stability of travelling waves In the limit the matrix A reduces to a constant matrix with the eigenvalues Let U + (z,  ) be the two-dimensional subspace of solutions that decay exponentially as and U – (z,  ) be a two- dimensional subspace decaying exponentially as The non-trivial intersection of these two subspaces indicates the presence of unstable eigenmodes.

12. 27 October 2005 Konstantin Blyuss, University of Exeter 12 Konstantin Blyuss, University of Exeter 12 One can define the Evans function as Zeros of this function correspond to the eigenvalues of the linearized stability problem. For actual evaluation it is convenient to integrated the induced systems which describe the dynamics of the corresponding subspaces. Also, in this way one can preserve analyticity in spectral parameter.

13. 27 October 2005 Konstantin Blyuss, University of Exeter 13 Konstantin Blyuss, University of Exeter 13 satisfies the following induced system Here acts on a decomposable two-form as The limiting matrix has an eigenvalue with the corresponding eigenvector Similarly, for the adjoint system The Evans function can now be written as

14. 27 October 2005 Konstantin Blyuss, University of Exeter 14 Konstantin Blyuss, University of Exeter 14 Evans function results

15. 27 October 2005 Konstantin Blyuss, University of Exeter 15 Konstantin Blyuss, University of Exeter 15 Transverse stability Consider now the stability of travelling waves in the direction orthogonal to the basic direction of propagation. To model this, we replace

16. 27 October 2005 Konstantin Blyuss, University of Exeter 16 Konstantin Blyuss, University of Exeter 16 Looking for solutions of the linearized problem in the form one arrives at the eigenvalue problem with transverse wavenumber as a parameter: Evans function can be defined in the same way as before.

17. 27 October 2005 Konstantin Blyuss, University of Exeter 17 Konstantin Blyuss, University of Exeter 17 Conclusions Phase-dependent absorption is sufficient to provide a bi-stability in the system Travelling fronts are stable with respect to both longitudinal and transversal perturbations The model can be extended to study explosive crystallisation

18. 27 October 2005 Konstantin Blyuss, University of Exeter 18 Konstantin Blyuss, University of Exeter 18 Thanks for your attention