MC 20: Workshop on Nonlinear waves
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Speaker: Christophe Charlier (KTH)
Title: Long-time asymptotics for the good Boussinesq equation via a Riemann-Hilbert approach
Abstract: The good Boussinesq equation models the nonlinear dynamics of waves in a weakly dispersive medium and is also known as the ``nonlinear string equation''. The inverse scattering transform formalism implies that the solution can be expressed in terms of the solution of a 3x3-matrix Riemann-Hilbert problem. I will present long-time asymptotic behavior of the solution by performing a nonlinear steepest descent analysis of this Riemann-Hilbert problem. This is joint work in progress with J. Lenells and D. Wang
Speaker: Evgeniy Lokharu (Linköping University)
Title: Nonexistence of subcritical solitary waves
Abstract: It is well known that solitary waves of elevation are supercritical, that is the Froude number is greater than one. More complicated solitary waves, if they exist, must be of large amplitude and subcritical. The latter means that the corresponding asymptotic depth supports small-amplitude periodic waves. The aim of the talk is to explain why such subcritical solitary waves are not possible both in irrotational and rotational settings. The argument is based on an asymptotic analysis and a maximum principle, which is new even in the irrotational case.
Speaker: Dag Nilsson (NTNU Trondheim)
Title: Solitary-wave solutions of the full dispersion Kadomtsev-Petviashvili equation
Speaker: Long Pei (NTNU Trondheim)
Title: Exponential decay and symmetry of solitary waves to Degasperis-Procesi equation
Abstract: We improve the decay argument by [Bona and Li, J. Math. Pures Appl.,1997] for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady Degasperis-Procesi equation in the nonlocal formulation. In addition, we give a method which confirms the symmetry of solitary waves, including those of the maximum height. Finally, we discover how the symmetric structure is connected to the steady structure of solutions to the Degasperis-Procesi equation, and give a more intuitive proof for symmetric solutions to be traveling waves. The improved argument and new methods above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations.
Speaker: Kristoffer Varholm (Lund University)
Title: On the stability of solitary water waves with a point vortex
Abstract: We establish the conditional orbital stability of small solitary capillary-gravity water waves with an immersed point vortex. This is done through the development of a significant generalization of the seminal stability theory of Grillakis, Shatah, and Strauss; applicable to the (Hamiltonian) water-wave problem with a point vortex. The talk is based on joint work with E. Wahlén (Lund U.) and S. Walsh (U. of Missouri).
Speaker: Erik Wahlén (Lund University)
Title: Large-ampltiude solitary waves of the Whitham equation
Abstract: In the 1960’s G. B. Whitham suggested a non-local version of the KdV equation as a model for water waves. Unlike the KdV equation it is not integrable, but it has certain other advantages. In particular, it has the same dispersion relation as the full water wave problem and it allows for wave breaking. The existence of a highest, cusped periodic wave was recently proved using global bifurcation theory. I will discuss the same problem for solitary waves. This presents several new challenges. The talk is based on joint work with T. Truong (Lund) and M. Wheeler (Bath).