Klaus Kirchgässner

Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation

Publisher Summary This chapter examines the nonlinearly resonant surface waves and homoclinic bifurcation. The behavior of steady nonlinear water waves on the surface of an inviscid heavy fluid layer has received much attention, both from the mathematical and from the physical side. The basic equations for the interaction of traveling nonlinear surface waves with in-phase external pressure waves are derived. An inviscid fluid layer of mean depth is considered under gravity. On its free upper surface, where capillary forces may also act, it supports nonlinear surface waves of permanent form, traveling from right to left with constant speed. The upper bound could be estimated explicitly by quantitatively showing the validity of subsequently described, almost identical transformations and estimates on noncritical eigenvalues of the operator. The linear dispersion relation for cnoidal waves corresponds to the imaginary eigenvalues. It is found that the reduced phase space is four-dimensional and the existence of a homoclinic orbit is equivalent to the intersection of two curves. The existence of solitary waves on a curve in the parameter space is also elaborated.

NONLINEAR WAVE MOTION AND HOMOCLINIC BIFURCATION

Internal- and surface solitary waves can be understood as homoclinic orbits in infinite-dimensional space. The influence of localized and periodic external forces is studied for different physical models. The mathematical method used leads to a low order ordinary differential equation describing all steady flows of small but finite amplitude.

Three-Dimensional Steady Capillary-Gravity Waves

Three-dimensional steady capillary-gravity water-waves are studied in this paper. Potential flow of an ideal fluid in a layer with finite depth and upper free surface is considered. The existence of these waves is derived through bifurcation processes from the state of rest. The waves are assumed to be periodic in the direction of propagation and just bounded in the transverse direction (modulated periodic travelling waves — MPTW). Restricting the analysis to small amplitude waves, one can reduce the problem to a finite-dimensional reversible and reflectionally symmetric dynamical system. Existence and full information about the geometry of the shape of possible crests then follows via normal form analysis and persistence.

Permanent Fronts in Two-Phase Flows in a Porous Medium

A family of exact solutions for a model of a one-dimensional horizontal flow of two immiscible, incompressible fluids in a porous medium, including the effects of capillary pressure, is obtained analytically by solving the governing singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is shown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting–outflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analytically and numerically in detail in the case when the permeabilities and the capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks–Corey model are also presented. In both models three global bifurcations occur. By using a geometric dynamical system approach, the nonlinear stability of the permanent fronts is established analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is given as follows. The instantaneous upstream (downstream) velocity of an arbitrary non-quasistationary front is equal to the velocity of a permanent front whose shape coincides up to two leading orders with the instantaneous shape of the non-quasistationary front at the upstream (respectively, downstream) location. The upstream and downstream locations of the front undergo instantaneous translations governed by modified nonsingular hyperbolic equations. The portion of the front in between these locations undergoes a diffusive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic–hyperbolic domain decomposition for computing non-quasistationary fronts.

PERIODIC AND NONPERIODIC SOLUTIONS OF REVERSIBLE SYSTEMS

Publisher Summary This chapter describes the periodic and nonperiodic solutions of reversible systems. It discusses the recent results about the existence of bounded solutions for finite systems of ordinary differential equations, which are reversible in the sense of G.D. Birkhoff . These solutions appear under various conditions when, for some value of an external parameter, an equilibrium point exhibits bifurcation. For the case n = 2, the local and global existence of periodic solutions has been proved by Wolkowisky, the global aspects depending strongly on the validity of nodal properties and the restriction to two dimensions. The extension of the local bifurcation result to arbitrary dimensions is straightforward if one follows the general lines of theorem. However, the global result is by no means trivial. Using a new uniqueness result, global existence of periodic solutions for arbitrary dimensions could be established. A particularly deep existence problem arises when A has k pairs of single eigenvalues on the imaginary axis. As related questions for Hamiltonian systems indicate, one external parameter generally does not suffice to make the bifurcation of the expected quasiperiodic solutions.

Theoretical and Experimental Studies of an S-Catamaran

Using a nonlinear shallow-water solitary-wave theory it was demonstrated that for a ship moving at supercritical speed along the centerline of a rectangular channel, if the hull sectional-area curve is of a special form determined by the solution of an oblique double-soliton interaction and the channel width is chosen to ensure complete wave cancelation through sidewall reflection, the ship waves can be made to form a purely localized pattern around the ship so that its wave resistance, which results only from far-field free waves, theoretically vanishes. To get rid of the crucial dependence on impractical sidewall reflection, this mechanism was developed further to obtain a novel catamaran comprising twin hulls with curved centerlines, yaw and skegs; it has theoretically zero wave-resistance at a chosen supercritical design speed in laterally unrestricted shallow water. Despite certain deviations from the ideal form for practical reasons, the wave-resistance of the new curved-yawed-hull catamaran with and without skeg was numerically found to be less than that of an equivalent straight-unyawed-hull catamaran by 50 and 30%, respectively. Now, the new design, albeit without skeg, has been validated by model experiment and comparison with a state-of-the-art reference catamaran of equal main dimensions that was developed and tested earlier in the VBD. Up to 28% wave-resistance reduction was achieved in the experiment, although not in the originally designed configuration but at a reduced yaw angle found by trial and error.

Periodic Solutions of Semilinear Elliptic Equations in A Strip

A parameter-dependent semilinear elliptic boundary value problem is considered in a strip. It is shown for some parameter interval that, if the nonlinearity satisfies certain symmetry conditions, all “small” solutions are periodic in the unbounded variable. The method described is generalisable to higher order elliptic equations.