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

Abstract

We consider the Klein–Gordon and sine-Gordon type equations with a point-like potential, which describe the wave phenomenon 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. Finding such critical parameter value and the associated statistical quantities demands a large number of individual simulations with different parameter values. 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 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 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. Due to the nonlinearity of the equation, we use the gPC mean value instead of reconstructing the solution to find the critical parameter. Numerical results show that the critical value can be determined efficiently and accurately by the proposed method.