Computational Results for the Higgs Boson Equation in the de Sitter Spacetime

High performance computations are presented for the Higgs Boson Equation in the de Sitter Space- time using explicit fourth order Runge-Kutta scheme on the temporal discretization and fourth order finite difference discretization in space. In addition to the fully three space dimensional equation its one space dimensional radial solutions are also examined. The numerical code for the three space di- mensional equation has been programmed in CUDA Fortran and was performed on NVIDIA Tesla K40c GPU Accelerator. The radial form of the equation was simulated in MATLAB. The numerical results demonstrate the existing theoretical result that under certain conditions bubbles form in the scalar field. We also demonstrate the known blow-up phenomena for the solutions of the semilinear Klein-Gordon equation with imaginary mass. Our numerical studies suggest several previously not known properties of the solution for which theoretical proofs do not exist yet: 1. smooth solution exists for all time if the initial conditions are compactly supported and smooth; 2. under some conditions no bubbles form; 3. solutions converge to step functions related to unforced, damped Duffing equations.

[1]  K. Yagdjian.  Global existence of the self-interacting scalar field in the de Sitter universe , 2017, Journal of Mathematical Physics.

[2]  Muhammet Yazıcı,et al.  Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime , 2016 .

[3]  Utkarsh Ayachit,et al.  The ParaView Guide: A Parallel Visualization Application , 2015 .

[4]  H. Epstein,et al.  de Sitter Tachyons and Related Topics , 2014, 1403.3319.

[5]  William E. Schiesser,et al.  Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple , 2011 .

[6]  Wilhelm Schlag,et al.  Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein–Gordon equation , 2010, 1011.2015.

[7]  K. Yagdjian.  On the Global Solutions of the Higgs Boson Equation , 2010, 1009.3076.

[8]  J. Macías-Díaz,et al.  Two finite-difference schemes that preserve the dissipation of energy in a system of modified wave equations , 2010, 1112.0594.

[9]  Jalil Rashidinia,et al.  Numerical solution of the nonlinear Klein-Gordon equation , 2010, J. Comput. Appl. Math..

[10]  Mehdi Dehghan,et al.  Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions , 2009 .

[11]  K. Yagdjian.  The semilinear Klein-Gordon equation in de Sitter spacetime , 2009, 0903.0089.

[12]  P. C. Ray,et al.  SOLUTION OF NON-LINEAR KLEIN-GORDON EQUATION WITH A QUADRATIC NON-LINEAR TERM BY ADOMIAN DECOMPOSITION METHOD , 2009 .

[13]  Gilbert Strang,et al.  Computational Science and Engineering , 2007 .

[14]  N. Konyukhova,et al.  On the stability of a self-similar spherical bubble of a scalar Higgs field in de Sitter space , 2005 .

[15]  Salah M. El-Sayed,et al.  A numerical solution of the Klein-Gordon equation and convergence of the decomposition method , 2004, Appl. Math. Comput..

[16]  C. Bogey,et al.  A family of low dispersive and low dissipative explicit schemes for flow and noise computations , 2004 .

[17]  Andrew R. Liddle,et al.  An introduction to modern cosmology , 1999 .

[18]  Patrick Joly,et al.  Construction and Analysis of Fourth-Order Finite Difference Schemes for the Acoustic Wave Equation in Nonhomogeneous Media , 1996 .

[19]  L. Vázquez,et al.  Analysis of Four Numerical Schemes for a Nonlinear Klein-Gordon Equation , 1990 .

[20]  S. Coleman.  Aspects of Symmetry: Selected Erice Lectures , 1988 .

[21]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .

[22]  Victor Pereyra,et al.  Symbolic generation of finite difference formulas , 1978 .

[23]  S. Lipson,et al.  The impossibility of free tachyons , 1971 .