Complex singularities and PDEs

In this paper we give a review on the computational methods used to characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the singularity tracking method based on the analysis of the Fourier spectrum. We then introduce other methods generally used to detect the hidden singularities. In particular we show some applications of the Pad\'e approximation, of the Kida method, and of Borel-Polya method. We apply these techniques to the study of the singularity formation of some nonlinear dispersive and dissipative one dimensional PDE of the 2D Prandtl equation, of the 2D KP equation, and to Navier-Stokes equation for high Reynolds number incompressible flows in the case of interaction with rigid boundaries

Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for ``depth ratios'' across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figure

Geometry and Analysis of some Euler-Arnold Equations

In 1966, Arnold showed that the Euler equation for an ideal fluid can arise as the geodesic flow on the group of volume preserving diffeomorphisms with respect to the right invariant kinetic energy metric. This geometric interpretation was rigorously established by Ebin and Marsden in 1970 using infinite dimensional Riemannian geometry and Sobolev space techniques. Many other nonlinear evolution PDEs in mathematical physics turned out to fit in this universal approach, and this opened a vast research on the geometry and analysis of the Euler-Arnold equations, i.e., geodesic equations on a Lie group endowed with one-sided invariant metrics. In this thesis, we investigate two Euler-Arnold equations; the Camassa-Holm equation from the shallow water equation theory and quasi-geostrophic equation from geophysical fluid dynamics. First, we will prove the local-wellposedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms, satisfying certain asymptotic conditions at infinity. Motivated by the work of Misio{\l}ek, we will re-express the equation in Lagrangian variables, by which the PDE becomes an ODE on a Banach manifold with a locally Lipschitz right-side. Consequently, we obtain the existence and uniquenss of the solution, and the topological group property of the diffeomorphism group ensures the continuous dependence on the initial data. Second, we will construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We will use a simple Lagrangian change of variables, which removes the wave breaking singularity of the original equation and allows the weak continuation. Furthermore, we obtain the global spatial smoothness of the Lagrangian trajectories via this construction. This work was motivated by Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation. Lastly, we will study some geometric aspects for the quasi-geostrophic equation, which is the geodesic on the quantomorphism group, a subgroup of the contactomorphism group. We will derive an explicit formula for the sectional curvature and discuss the nonpositive curvature criterion, which extends the work of Preston on two dimensional incompressible fluid flows

Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores in the fully nonlinear dissipationless shallow- water dynamics described by the Green-Naghdi system. We use the Whitham modula- tion theory to obtain an asymptotic analytical description for a full range of the depth ratio across the bore. The positions of the edges of the undular bore and the amplitude of the leading solitary wave are found as functions of this depth ratio. The formation of a partial undular bore with a rapidly-varying finite-amplitude rear wave front is predicted for depth ratios across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with the full numerical solution for the development of an undular bore in the Green-Naghdi system

Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure