Soliton-like solutions of higher order wave equations of the KdV type

E. Tzirtzilakis, V. Marinakis, C. Apokis and Tassos Bountis

Department of Mathematics and Center for Research and Application of Nonlinear Systems
University of Patras, 26500 Patras, Greece

In this work we study second and third order approximations of water wave equations of the KdV type. First we derive analytical expressions for solitary wave solutions for some special sets of parameters of the equations. Remarkably enough, in all these approximations, the form of the solitary wave and its amplitude-velocity dependence are identical to the sech$^2$-formula of the one-soliton solution of the KdV. Next we carry out a detailed numerical study of these solutions using a Fourier pseudospectral method combined with a finite-difference scheme, in parameter regions where soliton-like behavior is observed. In these regions, we find solitary waves which are stable and behave like solitons in the sense that they remain virtually unchanged under time evolution and mutual interaction. In general, these solutions sustain small oscillations in the form of radiation waves (trailing the solitary wave) and may still be regarded as stable, provided these radiation waves do not exceed a numerical stability threshold. Instability occurs at high enough wave speeds, when these oscillations exceed the stability threshold already at the outset, and manifests itself as a sudden increase of these oscillations followed by a blowup of the wave after relatively short time intervals.

References
A.S. Fokas, ``On a Class of Physically Important Integrable Equations'', Physica 87D (1995), 145-150; see also A.S. Fokas and Q.M. Liu, PRL 77(12) (1996), 2347.
V. Marinakis, T.C. Bountis, ``Special Solutions of a new Class of Water Wave Equations'', Comm. in Appl. Anal. 4(3) (2000), 433-445.
E. Tzirtzilakis, M. Xenos, V. Marinakis, T.C. Bountis, ``Interactions and Stability of Solitary Waves in Shallow Water'', Chaos, Solitons and Fractals, 14 (2000), 87-95.

Localized oscillations in 1-dimensional nonlinear media and homoclinic orbits of invertible maps

Tassos Bountis

Department of Mathematics and Center for Research and Application of Nonlinear Systems
University of Patras, 26500 Patras, Greece

In recent years, a very interesting phenomenon has captured the imagination of many nonlinear scientists: The occurrence of stable, localized oscillations in 1 - and 2 - dimensional lattices. These oscillations have been termed discrete breathers, in analogy to similar solutions found in certain completely integrable continuous systems, like the sine Gordon and the Nonlinear Schrödinger partial differential equations.

In these lectures, we will first review the physical models in which discrete breathers were first discovered and studied. We will then outline the work of Aubry and MacKay who rigorously established the existence of stable discrete breathers in a wide class of infinite chains of linearly coupled anharmonic oscillators. It is interesting that, in many cases, breathers are indeed a discrete phenomenon, as they are not expected to exist in the continuum limit. Furthermore, they have also been recently observed in several experiments, notably some involving arrays of coupled Josephson junctions.

We shall demonstrate that discrete breathers correspond, in fact, to homoclinic orbits at the intersections of invariant manifolds of a saddle point, lying at the origin of a 2N - dimensional map in Fourier space. Exploiting this geometric approach, we will show that discrete breathers can be accurately approximated and even classified using ideas of symbolic dynamics.

In this way, a great variety of such forms (also called multibreathers) can be constructed most of which are found to be linearly unstable. Thus, developing methods for computing homoclinic orbits of invertible maps allows us to obtain accurate representations of discrete breathers and stabilize (or destabilize) them using techniques of continuous feedback control.

References
M. Kollmann, H.W. Capel and T. Bountis, "Breathers and Multi-Breathers in a Damped, Periodically Driven Discretized NLS Equation", Phys. Rev. E 60, 1195 (1998).
T. Bountis, H.W. Capel, M. Kollmann, J.C. Ross, J.M. Bergamin and J.P. van der Weele, "Multibreathers and Homoclinic Chaos in 1-Dimensional Lattices", Phys. Lett. 268A, 50-60(2000).
J. Bergamin, T. Bountis and C. Jung, "A Method for Locating Symmetric Homoclinic Orbits Using Symbolic Dynamics", J. Phys. A: Math. Gen.,33, 8059-8070 (2000).
J. Bergamin, T. Bountis and M. Vrahatis, "Homoclinic Orbits of Invertible Maps", preprint submitted for publication (2002).
T. Bountis, J. Bergamin and V. Basios, "Breather Stabilization Using Continuous Feedback Control", to appear in Phys. Lett. A (2002).
J. M. Bergamin, Sp. Kamvyssis and T. Bountis,"A Numerical Study of the Perturbed Semiclassical Focusing Nonlinear Schrodinger Equation", Phys. Lett. A, to appear (2002).