09:00 to 10:00 Tatsuo Iguchi (Keio University)Isobe-Kakinuma model for water waves as a higher order shallow water approximation We justify rigorously an Isobe-Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order $\delta^2$, where $\delta$ is a small nondimensional parameter defined as the ratio of the typical wavelength to the mean depth. The Green-Naghdi equations are known as higher order approximate equations to the water wave equations with an error of order $\delta^4$. In this paper we show that the Isobe-Kakinuma model is a much higher approximation to the water wave equations with an error of order $\delta^6$. INI 1 10:00 to 11:00 Mayumi Shoji (Japan Women's University)Numerical computation of water waves with discontinuous vorticity We consider progressive water waves with a piecewise constant vorticity distribution. Both capillary-gravity waves and gravity waves of finite depth are considered.This is a bifurcation problem of a complicated structure of solutions with many parameters. It is hard to classify the structures of solutions mathematically.We thus resort to a numerical method in order to see their bifurcating phenomena with systematic computations. Another concern of ours is to see whether and when stagnation points appear.The difficulties for numerical computations are that it is a free boundary problem and we need a formulation not to exclude stagnation points.We will show our numerical results with various values of vorticity, depth of fluid and traveling speed. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Yoshimasa Matsuno (Yamaguchi University)Two-component Camassa-Holm system and its reductions My talk is mainly concerned with an integrable two-component Camassa-Holm (CH2) system which describes the propagation of nonlinear shallow water waves.  After a brief review of strongly nonlinear models for shallow water waves including the Green-Naghdi and related systems, I develop a systematic procedure for constructing soliton solutions of the CH2 system.  Specifically, using a direct method combined with a reciprocal transformation, I obtain the parametric representation of the multisoliton solutions, and investigate their properties. Subsequently, I show that the CH2  system reduces to the CH equation and the two-component Hunter-Saxton (HS2) system by means of appropriate limiting procedures. The corresponding expressions of the multisoliton solutions are presented in parametric forms, reproducing the existing results for the reduced equations. Also, I discuss the reduction from the HS2 system to the HS equation. Last, I comment on an interesting issue associated with peaked wave (or peakon) solutions of the CH, Degasperis-Procesi, Novikov and modified CH equations. INI 1 12:30 to 13:30 Lunch @ Wolfson Court 13:30 to 14:30 Vladimir Kozlov (Linköpings Universitet)Small-amplitude steady water waves on flows with counter-currents. The two-dimensional free-boundary problem describing steady  gravity waves with vorticity is considered for water of finite depth. It is known that the whole set of small-amplitude waves on unidirectional rotational flows is exhausted by Stokes and solitary waves. In the case of flows with counter-currents, there occur other patterns of behaviour. Two results of this kind will be discussed. The first one concerns the existence of N-modal steady waves, whereas the second result deals with the following fact.    If the number of roots of the dispersion equation is greater than one, then the major part of the plethora of small-amplitude waves is represented by non-symmetric ones. This is a joint work with E. Lokharu, Lund University. INI 1 14:30 to 15:30 Amin Chabchoub (Aalto University)Rogue and Shock Waves within the Framework of Weakly Nonlinear Evolution Equations - Applicability and Limitations Extreme ocean waves, also referred to as freak or rogue waves (RWs), are known to appear without warning and have a disastrous impact on ships and offshore structures as a consequence of the substantially large wave heights they can reach. Studies on RWs have recently attracted the scientific interest due to the interdisciplinary universal nature of the modulation instability (MI) of weakly nonlinear waves as well as for the sake of accurate modeling and prediction of these mysterious extremes. Indeed, solutions of the nonlinear Schrödinger equation provide advanced backbone models that can be used to describe the dynamics of RWs in time and space, providing therefore prototypes for a deterministic investigation of MI. Experiments as well as related numerical simulations on breathers will be presented. Furthermore, the existence of such localized structures in realistic sea state conditions will be discussed as well.  Moreover, experiments on shock waves in shallow water will be also reported. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 Michio Yamada (Kyoto University)Zonal flows and wave resonance. --- Rossby wave case --- We discuss resonant and nonresonant interaction of waves and time-development of energy distribution in a wave system.   We take, not a water wave system, but a Rossby wave system on a rotating sphere, where spherical harmonic functions correspond to Rossby waves which are labelled by total wave number and longitudinal wavenumber.    In the Rossby wave system, a zonal flow pattern is known to emerge even from random and isotropic initial conditions in the course of time development.  We classify the Rossy waves into 4 groups according to whether it has resonant waves or  not, and also whether  it represents a zonal flow (i.e. with zero longitudinal wavenumber) or not.  Numerical simulation shows that when there is no zonal energy initially, the energy goes into zonal resonant modes from nonzonal and nonresonant modes, which suggests that the nonresonant interaction between the nonzonal nonresonant modes and the zonal resonant modes plays a dominant role in the formation of the zonal flows.  Introducing 'effective energy transfer',  which enables us to talk about the energy transfer from one mode to another mode,  we find that the nonresonant interaction transfers energy from the nonresonant mode mostly to the resonant modes. INI 1 19:30 to 22:00 Formal Dinner at Emmanuel College