Location: Mathematics Institute, Cardiff University, Wales.

Room: All talks will be in room M/0.40 (ground floor)

Date: 25th of September 2017

Programme:

**11.00-12.00 Registration and welcome coffee, UCAS room (1st floor)**

12.05-12.45 Nearly singular solutions for plasmas, Jonathan Ben-Artzi (Cardiff)

**12.45-13.45 Lunch, UCAS room (1st floor)**

13.45-14.25 Nonlinear stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains, Karsten Matthies (Bath)

14.25-15.05 Travelling waves in anisotropic smectic C* liquid crystals, Elaine Crooks (Swansea)

**15.05-15.15 Coffee break**

15.15-15.55 Fractional Pearson Diffusions, Nikolai Leonenko (Cardiff)

15.55-16.35 On non propagating waves in a one- dimensional dislocation problem, Hartmut Schwetlick (Bath)

Abstracts:

1-Nearly singular solutions for plasmas (Jonathan Ben-Artzi Cardiff)

Since the late 1980s it is well-known that the Vlasov-Poisson system has global classical solutions. Since then, there has been a significant effort to study the long-time behavior of solutions, e.g. the phenomenon known as “Landau damping” (the dissipation of the electric field). In this talk, I will present a new result for the intermediate regime, showing that solutions can get arbitrarily close to being singular. This has recently been done by Rein & Taegert in the gravitational case, but the plasma case is entirely different due to the different nature of particle trajectories. This is joint work with Simone Calogero and Stephen Pankavich.

2-Travelling waves in anisotropic smectic C* liquid crystals (Elaine Crooks, Swansea).

We consider minimality conditions for the speed of monotone travelling waves in a model of a sample of smectic C* liquid crystal subject to a constant electric field, dealing with both isotropic and anisotropic cases. Such conditions are important in understanding switching properties of a liquid crystal, and our focus is on understanding how the presence of anisotropy can affect the speed and nature of switching. Through a study of travelling-wave solutions of a quasilinear parabolic equation, we obtain an estimate of the influence of anisotropy on the minimal speed, sufficient conditions for linear and non-linear minimal speed selection mechanisms to hold in different parameter regimes, and a characterisation of the boundary separating the linear and non-linear regimes in parameter space. This is joint work with Michael Grinfeld and Geoff Mackay (Strathclyde).

3- Fractional Pearson Diffusions (Nikolai N. Leonenko, Cardiff)

Pearson diffusions have stationary distributions of Pearson type. They includes Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, and several others processes. Their stationary distributions solve the Pearson equation, developed by Pearson in 1914 to unify some important classes of distributions (e.g.,normal, gamma, beta, reciprocal gamma, Student, Fisher-Snedecor). Their eigenfunction expansions involve the traditional classes of orthogonal polynomials (e.g., Hermite, Laguerre, Jacobi), and the finite classes of orthogonal polynomials (Bessel, Routh-Romanovski, Fisher-Snedecor). The self-adjointness of the semigroup generator of one dimensional diffusions implies a spectral representation which has found many useful applications, for example for the prediction of second order stationary sequences and in mathematical finance. However, on non-compact state spaces the spectrum of the generator will typically include both a discrete and a continuous part, with the latter starting at a spectral cutoff point related to the nonexistence of stationary moments. The significance of this fact for statistical estimation is not yet fully understood. We develop fractional Pearson diffusions, constructing by a non-Markovian inverse stable time change. Their transition densities are shown to solve a time-fractional analogue to the diffusion equation with polynomial coefficients. Because this process is not Markovian, the stochastic solution provides additional information about the movement of particles that diffuse under this model. Anomalous diffusions have proven useful in applications to physics, geophysics, chemistry, and finance.

This talk is based on joint works with Mark M. Meerschaert , Alla Sikorskii (Michigan State University) and Nenad Suvak and Ivan Papic (Osijek University, Croatia).

4- Nonlinear stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains (Karsten Matthies, Bath)

The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a longstanding open problem and has been solved so far only in a certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in which the waves propagate with near sonic speed, have large wave length, and carry low energy. Here we derive a similar result in a complementary asymptotic regime related to fast and strongly localized waves with high energy. In particular, we show that the spectrum of the linearized FPUT operator contains asymptotically no unstable eigenvalues except for the neutral ones that stem from the shift symmetry and the spatial discreteness. This ensures that high-energy waves are linearly stable in some orbital sense, and the corresponding nonlinear stability is granted by the general, non-asymptotic part of the seminal Friesecke-Pego result. Our analytical work splits into two principal parts. First we refine two-scale techniques that relate high-energy wave to a nonlinear asymptotic shape ODE and provide accurate approximation formulas. In this way we establish the existence, local uniqueness, smooth parameter dependence, and exponential localization of fast lattice waves for a wide class of interaction potentials with algebraic singularity. Afterwards we study the crucial eigenvalue problem in exponentially weighted spaces, so that there is not unstable essential spectrum. Our key argument is that all proper eigenfunctions can asymptotically be linked to the unique bounded and normalized solution of the linearized shape ODE, and this finally enables us to disprove the existence of unstable eigenfunctions in the symplectic complement of the neutral ones. Joint work with Michael Herrmann.

References:

[1] Avram, F., Leonenko, N.N and Suvak, N. (2013) On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions, Markov Processes and Related Fields, Volume 19, N 2 , 249-298.

[2] Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for the Fisher-Snendecor diffusion, Bernoulli, Volume 19, No. 5B, 2294-2329.

[3] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional Pearson diffusion, Journal of Mathematical Analysis and Applications, Volume 403, 532-546.

[4] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation Structure of Fractional Pearson diffusion, Computers and Mathematics with Applications, 66, 737-745.

[5] Leonenko,N.N., Meerschaert,M.M., Schilling, R.L. and Sikorskii, A. (2014) Correlation Structure of Time-Changed Lévy Processes, Communications in Applied and Industrial Mathematics, Vol. 6 , No. 1, p. e-483 (22 pp).

[6] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, in press.

[7] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017) Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, in press.

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