# Nonlinear Dynamics and Asymptotics

Dynamical systems, nonlinear waves, asymptotic analysis

### Transseries analysis and Stokes phenomena

The space of (formal) power series is often too small to completely solve problems. This space can be extended to transseries and that space is big enough to solve most analytic problems. This will involve a deep understanding of the Stokes phenomenon, but also the recently discovered higher order Stokes phenomenon. The Stokes phenomenon is the switching on of small exponentials when certain curves are crossed. This phenomenon is well understood. Recently it has been discovered is that there are also curves where the Stokes multiplier itself can change its value. This is the higher order Stokes phenomenon, and a deep understanding is still missing. In this project you could at the start first focus on transseries analysis and try to obtain all solutions for, for example,  the first Painlevé equation from their transseries representations. The general local behaviour for solutions of  the first Painlevé equation is elliptic (double periodic). What is the link with these local ‘periods’ and the two free constants in the transseries? One other starting point can also be the higher order Stokes phenomenon. There are many interesting identities that have been obtained formally, but rigorous proofs are still missing. The ultimate goal for both directions could be rigorous exponential asymptotics for PDEs with a small parameter.

 Chapman, S. J. and Howls, C. J. and King, J. R. and Olde Daalhuis, A. B., Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation, Nonlinearity, 20, 2007, 2425–2452.

 Chapman, S. Jonathan and Mortimer, David B., Exponential asymptotics and Stokes lines in a partial differential equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 2005, 2385–2421.

 Howls, C. J. and Langman, P. J. and Olde Daalhuis, A. B., On the higher-order Stokes phenomenon, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460, 2004, 2285–2303.

 Olde Daalhuis, A. B., Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 2005, 3005–3021.

### Exponentially-improved asymptotics for solutions of q-difference equations

It is well understood how to obtain formal solutions of differential equations. Typically the coefficients in the infinite series representations grow like factorials, and the so called Borel transform (inverse Laplace transform) is needed to make sense of these solutions, and to deal with exponentially small phenomena. Recently q-difference equations start playing an important role in many applications, especially combinatorics. For example discrete Painlevé equations show in many places in physics. Again many of the formal solutions will be divergent, but one major change is that the divergence is much stronger than factorial. The q-Borel transform has been created, but so far it has hardly been used to deal with exponentially small phenomena. Hence, this is a new promising field and many tools (mainly complex analysis) still have to be created.

 G. Bonelli, A. Grassi, and A. Tanzini, Quantum curves and q-deformed Painlevé equations, Lett. Math. Phys., 109 (2019), pp. 1961–2001.

 T. Dreyfus and C. Zhang, On the product of two 1-q-summable series, 2023.

 N. Joshi, Quicksilver solutions of a q-difference first Painlevé equation, Stud. Appl. Math.,

 A. B. Olde Daalhuis, Asymptotic expansions for q-gamma, q-exponential, and q-Bessel functions, J. Math. Anal. Appl., 186 (1994), pp. 896–913.  H. Tahara, q-analogues of Laplace and Borel transforms by means of q-exponentials, Ann. Inst. Fourier (Grenoble), 67 (2017), pp. 1865–1903.

### Voigt waves in bianisotropic materials

Voigt-wave propagation  represents an unusual form of electro-magnetic plane-wave propagation that is supported by certain anisotropic dielectric materials. It may be distinguished from the usual form of plane-wave propagation, as encountered in standard textbooks on electromagnetics and optics, on the basis of rate of amplitude decay. That is, the decay of Voigt waves is governed by the product of a linear function and an exponential function of propagation distance whereas in the usual case of plane-wave propagation there is only exponential decay with propagation distance. Bianisotropic materials over much greater scope for Voigt-wave propagation than do anisotropic materials, because of the intrinsic coupling between electric and magnetic fields and the much larger constitutive parameter space that is associated with bianisotropic materials. Conditions for Voigt-wave propagation will be derived for certain types of physically-realizable bianisotropic materials. These bianisotropic materials will take the form of engineered materials that arise from the homogenization of relatively simple component materials that may not themselves support Voigt-wave propagation. Furthermore, the prospects of controlling the directions in which Voigt waves propagate by means of an applied DC electric  field will be investigated for certain bianisotropic materials arising from electro-optic component materials . The prospects of harnessing Voigt waves in bianisotropic materials for optical sensing application will also be investigated . In addition, the prospects of realizing Voigt waves which exhibit a linear gain in amplitude with propagation distance will be investigated for certain bianisotropic materials arising from active component materials .  Informal enquiries can be made to Tom Mackay (T.Mackay@ed.ac.uk).

 Electromagnetic Anisotropy and Bianisotropy, 2nd edition. T.G. Mackay & A. Lakhtakia, World Scientific (2019)

 T.G. Mackay, Controlling Voigt waves by the Pockels e ect, Journal of Nanophotonics 9 093599 (2015)

### Mixed-mode dynamics

Mixed-mode dynamics is a type of complex oscillatory behaviour that is characterised by an alternation of oscillations of small and large amplitudes. Mixed-mode oscillations (MMOs) frequently occur in fast-slow ('multi-scale') systems of ordinary differential equations; they are, for instance, found in models from mathematical neuroscience, where they correspond to complicated firing patterns seen in experimental recordings of neural activity.Various mechanisms have been proposed to explain mixed-mode dynamics; however, the relationship between them has not been investigated systematically yet. Moreover, relevant neurological models are typically too high-dimensional to be amenable to mathematical analysis, and have to be reduced efficiently to lower-dimensional normal forms which still capture the essential model dynamics. Informal enquiries can be made to Nikola Popovic (Nikola.Popovic@ed.ac.uk).

### Cut-off reaction-diffusion systems

Reaction-diffusion equations, such as the Fisher or the Nagumo equations, have found widespread use as 'minimal' models in the sciences. Solutions that retain a fixed profile in time and space, known as propagating fronts, are frequently relevant as asymptotic ('limiting') states to which general solutions converge. While reaction-diffusion models have been widely applied in the continuum limit of discrete (many-particle) systems, their utility is often limited due to stochastic effects which need to be considered for finite particle numbers. These effects can be approximated by introducing a 'cut-off' function which deactivates the reaction terms whenever the particle concentration lies below some threshold.The impact of such a cut-off approximation has been studied in detail in a number of propagation regimes; however, an in-depth understanding of the stability and convergence of the resulting front solutions is lacking, as is the investigation of cut-off systems in more than one spatial dimension. Informal enquiries can be made to Nikola Popovic (Nikola.Popovic@ed.ac.uk).