# American Institute of Mathematical Sciences

April  2022, 9(2): 253-278. doi: 10.3934/jcd.2022005

## Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach

 1 VU Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands 2 McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada

*Corresponding author: Jean-Philippe Lessard

Received  March 2021 Revised  February 2022 Published  April 2022 Early access  April 2022

In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval $(0,\frac{\pi}{2}]$ with a removable singularity at zero. The singularity is removed by solving the equation with Taylor series on $(0,\delta]$ (with $\delta$ small) while a Chebyshev series expansion is used to solve the problem on $[\delta,\frac{\pi}{2}]$. The two setups are incorporated in a larger zero-finding problem of the form $F(a) = 0$ with $a$ containing the coefficients of the Taylor and Chebyshev series. The problem $F = 0$ is solved rigorously using a Newton-Kantorovich argument.

Citation: Jan Bouwe van den Berg, Gabriel William Duchesne, Jean-Philippe Lessard. Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach. Journal of Computational Dynamics, 2022, 9 (2) : 253-278. doi: 10.3934/jcd.2022005
##### References:
 [1] G. Arioli and H. Koch, Non-radial solutions for some semilinear elliptic equations on the disk, Nonlinear Anal., 179 (2019), 294-308.  doi: 10.1016/j.na.2018.09.001. [2] I. Balázs, J. B. van den Berg, J. Courtois, J. Dudás, J.-P. Lessard, A. Vörös-Kiss, J. F. Williams and X. Y. Yin, Computer-assisted proofs for radially symmetric solutions of PDEs, J. Comput. Dyn., 5 (2018), 61-80.  doi: 10.3934/jcd.2018003. [3] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications Inc., Mineola, NY, second edition, 2001. [4] M. Breden and C. Kuehn, Rigorous validation of stochastic transition paths, J. Math. Pures Appl. (9), 131 (2019), 88-129.  doi: 10.1016/j.matpur.2019.04.012. [5] S. Brendle and F. C. Marques, Recent progress on the Yamabe problem, In Surveys in Geometric Analysis and Relativity, volume 20 of Adv. Lect. Math. (ALM), 29–47. Int. Press, Somerville, MA, 2011. [6] X. Cabré, Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions, Discrete Contin. Dyn. Syst., 20 (2008), 425-457.  doi: 10.3934/dcds.2008.20.425. [7] A. Castro and E. M. Fischer, Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729.  doi: 10.4153/CMB-2015-056-7. [8] L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1 (1963), 149-187. [9] L. Cesari, Functional analysis and Galerkin's method, Michigan Math. J., 11 (1964), 385-414. [10] P. Gonnet, R. Pachón and L. N. Trefethen, Robust rational interpolation and least-squares, Electron. Trans. Numer. Anal., 38 (2011), 146-167. [11] T. C. Hales, A proof of the Kepler conjecture, Ann. Math., 162 (2005), 1065-1185.  doi: 10.4007/annals.2005.162.1065. [12] D. E. Knuth, The Art of Computer Programming. Vol. 2, Addison-Wesley Publishing Co., Reading, Mass., second edition, 1981. Seminumerical algorithms, Addison-Wesley Series in Computer Science and Information Processing. [13] H. Koch, A. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: a case study, SIAM Rev., 38 (1996), 565-604.  doi: 10.1137/S0036144595284180. [14] O. E. Lanford and II I, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X. [15] J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007. [16] J.-P. Lessard and C. Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series, SIAM J. Numer. Anal., 52 (2014), 1-22.  doi: 10.1137/13090883X. [17] P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101. [18] K. Mischaikow and J. D. Mireles James, Encyclopedia of Applied and Computational Mathematics, chapter Computational Proofs in Dynamics, Springer, 2015. [19] M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356.  doi: 10.1081/NFA-100105107. [20] R. Pachón, P. Gonnet and J. van Deun, Fast and stable rational interpolation in roots of unity and Chebyshev points, SIAM J. Numer. Anal., 50 (2012), 1713-1734.  doi: 10.1137/100797291. [21] M. Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl., 324 (2001), 147-187.  doi: 10.1016/S0024-3795(00)00273-1. [22] N. Robertson, D. Sanders, P. Seymour and R. Thomas, The four-colour theorem, J. Combin. Theory Ser. B, 70 (1997), 2-44.  doi: 10.1006/jctb.1997.1750. [23] S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numer., 19 (2010), 287-449.  doi: 10.1017/S096249291000005X. [24] W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.  doi: 10.1007/s002080010018. [25] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. [26] J. B. van den Berg, Introduction to rigorous numerics in dynamics: general functional analytic setup and an example that forces chaos, In Rigorous Numerics in Dynamics, volume 74 of Proc. Sympos. Appl. Math., 1–25. Amer. Math. Soc., Providence, RI, 2018. [27] J. B. van den Berg, G. W. Duchesne and J.-P. Lessard, http://www.math.mcgill.ca/jplessard/ResearchProjects/PatternsSphere/home.html, MATLAB codes to perform the proofs, 2021. [28] J. B. van den Berg, C. M. Groothedde and J. F. Williams, Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem, SIAM J. Appl. Dyn. Syst., 14 (2015), 423-447.  doi: 10.1137/140987973. [29] J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Not. Am. Math. Soc., 62 (2015), 1057-1061.  doi: 10.1090/noti1276. [30] M. J. Ward, Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), R189–R239. doi: 10.1088/1361-6544/aabe4b. [31] N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.

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##### References:
 [1] G. Arioli and H. Koch, Non-radial solutions for some semilinear elliptic equations on the disk, Nonlinear Anal., 179 (2019), 294-308.  doi: 10.1016/j.na.2018.09.001. [2] I. Balázs, J. B. van den Berg, J. Courtois, J. Dudás, J.-P. Lessard, A. Vörös-Kiss, J. F. Williams and X. Y. Yin, Computer-assisted proofs for radially symmetric solutions of PDEs, J. Comput. Dyn., 5 (2018), 61-80.  doi: 10.3934/jcd.2018003. [3] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications Inc., Mineola, NY, second edition, 2001. [4] M. Breden and C. Kuehn, Rigorous validation of stochastic transition paths, J. Math. Pures Appl. (9), 131 (2019), 88-129.  doi: 10.1016/j.matpur.2019.04.012. [5] S. Brendle and F. C. Marques, Recent progress on the Yamabe problem, In Surveys in Geometric Analysis and Relativity, volume 20 of Adv. Lect. Math. (ALM), 29–47. Int. Press, Somerville, MA, 2011. [6] X. Cabré, Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions, Discrete Contin. Dyn. Syst., 20 (2008), 425-457.  doi: 10.3934/dcds.2008.20.425. [7] A. Castro and E. M. Fischer, Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729.  doi: 10.4153/CMB-2015-056-7. [8] L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1 (1963), 149-187. [9] L. Cesari, Functional analysis and Galerkin's method, Michigan Math. J., 11 (1964), 385-414. [10] P. Gonnet, R. Pachón and L. N. Trefethen, Robust rational interpolation and least-squares, Electron. Trans. Numer. Anal., 38 (2011), 146-167. [11] T. C. Hales, A proof of the Kepler conjecture, Ann. Math., 162 (2005), 1065-1185.  doi: 10.4007/annals.2005.162.1065. [12] D. E. Knuth, The Art of Computer Programming. Vol. 2, Addison-Wesley Publishing Co., Reading, Mass., second edition, 1981. Seminumerical algorithms, Addison-Wesley Series in Computer Science and Information Processing. [13] H. Koch, A. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: a case study, SIAM Rev., 38 (1996), 565-604.  doi: 10.1137/S0036144595284180. [14] O. E. Lanford and II I, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X. [15] J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007. [16] J.-P. Lessard and C. Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series, SIAM J. Numer. Anal., 52 (2014), 1-22.  doi: 10.1137/13090883X. [17] P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101. [18] K. Mischaikow and J. D. Mireles James, Encyclopedia of Applied and Computational Mathematics, chapter Computational Proofs in Dynamics, Springer, 2015. [19] M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356.  doi: 10.1081/NFA-100105107. [20] R. Pachón, P. Gonnet and J. van Deun, Fast and stable rational interpolation in roots of unity and Chebyshev points, SIAM J. Numer. Anal., 50 (2012), 1713-1734.  doi: 10.1137/100797291. [21] M. Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl., 324 (2001), 147-187.  doi: 10.1016/S0024-3795(00)00273-1. [22] N. Robertson, D. Sanders, P. Seymour and R. Thomas, The four-colour theorem, J. Combin. Theory Ser. B, 70 (1997), 2-44.  doi: 10.1006/jctb.1997.1750. [23] S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numer., 19 (2010), 287-449.  doi: 10.1017/S096249291000005X. [24] W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.  doi: 10.1007/s002080010018. [25] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. [26] J. B. van den Berg, Introduction to rigorous numerics in dynamics: general functional analytic setup and an example that forces chaos, In Rigorous Numerics in Dynamics, volume 74 of Proc. Sympos. Appl. Math., 1–25. Amer. Math. Soc., Providence, RI, 2018. [27] J. B. van den Berg, G. W. Duchesne and J.-P. Lessard, http://www.math.mcgill.ca/jplessard/ResearchProjects/PatternsSphere/home.html, MATLAB codes to perform the proofs, 2021. [28] J. B. van den Berg, C. M. Groothedde and J. F. Williams, Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem, SIAM J. Appl. Dyn. Syst., 14 (2015), 423-447.  doi: 10.1137/140987973. [29] J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Not. Am. Math. Soc., 62 (2015), 1057-1061.  doi: 10.1090/noti1276. [30] M. J. Ward, Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), R189–R239. doi: 10.1088/1361-6544/aabe4b. [31] N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.
(Left) Two rigorously computed solutions of (1) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding solution of the BVP (3) with Taylor expansion in blue and Chebyshev expansion in orange
Bifurcation diagram of all proven solutions in this paper
Solutions on the branch bifurcating at $\lambda_1 = 6$. (Left) The solution of (1) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding numerical solution of the BVP (3) with Taylor expansion in blue and Chebyshev expansion in orange
Solutions on the branch bifurcating at $\lambda_{16} = 1056$. (Left) The solution of (1) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding numerical solution of the BVP (3) with Taylor expansion in blue and Chebyshev expansion in orange
Parameter values, computational constants and bounds for the solutions depicted in Figures 3 and 4, where the radii polynomial $p(r) < 0$ for $r \in (r_{\min},r_{\max})$
 $n$ 1 16 $\lambda$ 5.6172 451.0710 1055.3153 1057 $\delta$ 0.3 0.3 0.1 0.1 $M$ 80 80 90 90 $N$ 80 110 180 180 $Y_0$ $7.6854\times 10^{-11}$ $5.5146\times 10^{-8}$ $8.1020\times 10^{- 5}$ $3.1482\times 10^{-4}$ $Z_0$ $1.0868\times 10^{-8}$ $1.5306\times 10^{-10}$ $2.1409\times 10^{-4}$ $4.4249\times 10^{-5}$ $Z_1$ 0.1154 0.21536 0.1597 0.1851 $Z_2$ 66.4698 1.1704 165.8340 108.2655 $\alpha$ $[2.2299,2.1083,1.32,10^3]$ $[10^3,9.6736,1.0054,10^3]$ $[10^3,17.376,1.0017,10^3]$ $[10^3,17.3724,1.0017,10^3]$ $r_{\min}$ $8.6883\times 10^{-11}$ $7.0283 10^{-8}$ $9.8353\times 10^{-5}$ $4.0855\times 10^{-4}$ $r_{\max}$ $1.3308\times 10^{-2}$ 0.67040 $4.9674 \times 10^{-3}$ $7.1175\times 10^{-3}$
 $n$ 1 16 $\lambda$ 5.6172 451.0710 1055.3153 1057 $\delta$ 0.3 0.3 0.1 0.1 $M$ 80 80 90 90 $N$ 80 110 180 180 $Y_0$ $7.6854\times 10^{-11}$ $5.5146\times 10^{-8}$ $8.1020\times 10^{- 5}$ $3.1482\times 10^{-4}$ $Z_0$ $1.0868\times 10^{-8}$ $1.5306\times 10^{-10}$ $2.1409\times 10^{-4}$ $4.4249\times 10^{-5}$ $Z_1$ 0.1154 0.21536 0.1597 0.1851 $Z_2$ 66.4698 1.1704 165.8340 108.2655 $\alpha$ $[2.2299,2.1083,1.32,10^3]$ $[10^3,9.6736,1.0054,10^3]$ $[10^3,17.376,1.0017,10^3]$ $[10^3,17.3724,1.0017,10^3]$ $r_{\min}$ $8.6883\times 10^{-11}$ $7.0283 10^{-8}$ $9.8353\times 10^{-5}$ $4.0855\times 10^{-4}$ $r_{\max}$ $1.3308\times 10^{-2}$ 0.67040 $4.9674 \times 10^{-3}$ $7.1175\times 10^{-3}$
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