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Efficient determination of the critical parameters and the statistical quantities for Klein-Gordon and sine-Gordon equations with a singular potential using generalized polynomial chaos methods
We consider the Klein-Gordon and sine-Gordon type equations with a point-like
potential, which describes the wave phenomenon in disordered media with a
defect. The singular potential term yields a critical phenomenon--that is, the
solution behavior around the critical parameter value bifurcates into two
extreme cases. Pinpointing the critical value with arbitrary accuracy is even
more challenging. In this work, we adopt the generalized polynomial chaos (gPC)
method to determine the critical values and the mean solutions around such
values. First, we consider the critical value associated with the strength of
the singular potential for the Klein-Gordon equation. We expand the solution in
the random variable associated with the parameter. The obtained partial
differential equations are solved using the Chebyshev collocation method. Due
to the existence of the singularity, the Gibbs phenomenon appears in the
solution, yielding a slow convergence of the numerically computed critical
value. To deal with the singularity, we adopt the consistent spectral
collocation method. The gPC method, along with the consistent Chebyshev method,
determines the critical value and the mean solution highly efficiently. We then
consider the sine-Gordon equation, for which the critical value is associated
with the initial velocity of the kink soliton solution. The critical behavior
in this case is that the solution passes through (particle-pass), is trapped by
(particle-capture), or is reflected by (particle-reflection) the singular
potential if the initial velocity of the soliton solution is greater than,
equal to, or less than the critical value, respectively. We use the gPC mean
value rather than reconstructing the solution to find the critical parameter.
Numerical results show that the critical value can be determined efficiently
and accurately by using the proposed method.Comment: 39 pages, 27 figures. Submitted to Journal of Computational Physics.
Assignment Manuscript Number JCOMP-D-11-0081