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 pointlike 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. Finding such critical parameter values 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 show the existence of a critical behavior with certain boundary conditions. Then 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 * Corresponding author the critical value, respectively. Due to the nonlinearity of the equation, 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. The results are also compared with the results using the Monte-Carlo method.

An efficient determination of critical parameters of nonlinear Schrödinger equation with a point-like potential using generalized polynomial chaos methods

“…This means that the spectral reconstruction of u(x, t, V ) for any V ∈ [V a , V b ] using ûl (x, t), l = 0, • • • , Q may fail to converge to the right solution due to the discontinuity at V = V c . This was also addressed in our previous work for the critical behavior of the soliton solution for the sine-Gordon equation [5]. Here note that the proposed method in this paper uses only the first moment û0 (x, t) to estimate the critical velocity V c but not the reconstruction of u(x, t, V ).…”

“…7 for both small and large values of ǫ by the gPC collocation method. We use the gPC method for the solution of the NLSE using the Wiener-Askey scheme [19,21], in which Hermite, Legendre, Laguerre, Jacobi and generalized Laguerre orthogonal polynomials are used for modeling the effect of continuous random variables described by the normal, uniform, exponential, beta and gamma probability distribution functions (PDFs), respectively [5,20]. These orthogonal polynomials are optimal for those PDFs since the weight function in the inner product and its support range correspond to the PDFs for those continuous distributions.…”

Accelerated and Improved Stabilization for High Order Moments of Racah Polynomials

“…For instance, Usman et al [8] employed Gegenbauer polynomials to solve nonlinear physical models, while Chakraborty and Jung [9] utilized Hermite, Legendre, Laguerre, Jacobi, and generalized Laguerre polynomials to model the impact of continuous random variables described by normal, uniform, exponential, beta, and gamma probability distributions, respectively. Feinberg and Langtangen [10] discussed the application of orthogonal polynomials for uncertainty quantification.…”

A basic class of symmetric orthogonal polynomials of a discrete variable

“…Also there exists a number of numerical and symbolic methods for solving hypergeometric equations of type (1) or (2), which are of interest in applications, particularly for cases containing symmetric solutions, such as resolution of the Gibbs phenomenon [7,8], Fourier-Kravchuk transform used in Optics [9], approximation of harmonic oscillator wave functions [10], tissue segmentation of human brain MRI through preprocessing [11], reconstructions for electromagnetic waves in the presence of a metal nanoparticle [12], efficient determination of the critical parameters and the statistical quantities for Klein-Gordon and sine-Gordon equations with a singular potential [13], image representation [14,15] and quantitative theory for the lateral momentum distribution after strong-field ionization [16].…”