Heriot-Watt Mathematics Report Series
HWM99-49, 18 Jan 2000
Numerical Computation of Solitary Waves on Infinite Cylinders
G Lord, D Peterhof, B Sandstede and A Scheel
Abstract
The numerical computation of solitary waves to semilinear elliptic
equations in infinite cylindrical domains is investigated. Rather than
solving on the infinite cylinder, the equation is approximated by a
boundary-value problem on a finite cylinder. Convergence and stability
results for this approach are given. It is also shown that Galerkin
approximations can be used to compute solitary waves of the elliptic
problem on the finite cylinder. In addition, it is demonstrated that
the aforementioned procedures simplify in cases where the elliptic
equation happens to admit an additional reversibility structure.
Finally, the theoretical predictions are compared with numerical
computations. In particular, post buckling of an infinitely long
cylindrical shell under axial compression is considered; it is shown
numerically that, for a fixed spatial truncation, the error in the
truncation on the length of the cylinder scales in accordance with the
theoretical predictions.
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