This investigation is devoted to the study of the Fermi-Pasta-Ulam (FPU) lattice dynamics.
Approximations of the FPU lattice dynamics have been an old subject, it is believed that
the stability of the FPU traveling waves depends on the stability of the KDV solitary waves.
The key question is: Are the traveling waves of the FPU lattice stable if the traveling waves
of KDV type equation are stable?.
We consider the FPU lattice with the nonlinear potential which leads to the generalized
Korteweg-de Vries (gKDV) equation, which is known to have orbitally stable traveling waves
in a subcritical case and orbitally unstable traveling waves in critical and supercritical cases.
In order to pursue the question asked above, we use the energy method.
We establish that the H^s(R) norm of the solution of the gKDV equation is bounded by
a time-independent constant in the subcritical case, whereas the H^s(R) norm grows at most
exponentially in the critical and supercritical cases. With the help of these results,
we extend the time scale for the approximation of the traveling waves of the FPU lattice
by the traveling waves of the gKDV equation logarithmically in the subcritical case. In the
critical and supercritical cases, we extend the time scale by a double-logarithmic factor.
Our results show that the traveling waves of the FPU lattice are stable if the solitary
waves of the gKDV equation are stable in the subcritical case. On the other hand, in the
critical and supercritical cases, our results are restricted to small-norm initial data, which
exclude solitary waves. / Thesis / Master of Science (MSc)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/17274 |
| Date | 06 1900 |
| Creators | Khan, Amjad |
| Contributors | Pelinovsky, Dmitry, Mathematics and Statistics |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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