Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise terms
Abstract
The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions are evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common general characteristics.
Keyword : self-similar solution, KPZ equation, Gaussian noise, Lorentzian noise, special functions, Heun functions
How to Cite
Barna, I. F., Bognár, G., Guedda, M., Mátyás, L., & Hriczó, K. (2020). Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise terms. Mathematical Modelling and Analysis, 25(2), 241-256. https://doi.org/10.3846/mma.2020.10459
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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W.J.F. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark. NIST handbook of mathematical functions. Cambridge University Press, 2010.
A. Pimpinelli and J. Villain. Physics of Crystal Growth. Cambridge University Press, 1998. https://doi.org/10.1017/CBO9780511622526
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L.I. Sedov. Similarity and dimensional methods in mechanics. CRC press, 1993.
D. Sergi, A. Camarano, J.M. Molina, A. Ortona and J. Narciso. Surface growth for molten silicon infiltration into carbon millimetersized channels: Lattice–Boltzmann simulations, experiments and models. International Journal of Modern Physics C, 27(06):1650062, 2016. https://doi.org/10.1142/S0129183116500625
G.I. Sivashinsky. Large cells in nonlinear Marangoni convection. Physica D: Nonlinear Phenomena, 4(2):227–235, 1982. https://doi.org/10.1016/01672789(82)90063-X
I.F. Barna. Self-similar solutions of three-dimensional Navier-Stokes equation. Communications in Theoretical Physics, 56(4):745, 2011.
I.F. Barna. Self-similar analysis of various Navier-Stokes equations in two or three dimensions. In D. Campos(Ed.), Handbook on Navier-Stokes Equations, Theory and Applied Analysis, New York, 2017. Nova Publishers. https://doi.org/10.1088/0253-6102/56/4/25
I.F. Barna, G. Bognár and K. Hriczó. Self-similar analytic solution of the two-dimensional Navier-Stokes equation with a non-Newtonian type of viscosity. Mathematical Modelling and Analysis, 21(1):83–94, 2016. https://doi.org/10.3846/13926292.2016.1136901
I.F. Barna and R. Kersner. Heat conduction: a telegraph-type model with selfsimilar behavior of solutions. Journal of Physics A: Mathematical and Theoretical, 43(37):375210, 2010. https://doi.org/10.1088/1751-8113/43/37/375210
I.F. Barna and L. Mátyás. Analytic solutions for the one-dimensional compressible Euler equation with heat conduction and with different kind of equations of state. Miskolc Mathematical Notes, 14(3):785–799, 2013. https://doi.org/10.18514/MMN.2013.694
P. Calabrese and P.L. Doussal. Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions. Physical review letters, 106(25):250603, 2011. https://doi.org/10.1103/PhysRevLett.106.250603
P. Calabrese, P.L. Doussal and A. Rosso. Free-energy distribution of the directed polymer at high temperature. EPL (Europhysics Letters), 90(2):20002, 2010. https://doi.org/10.1209/0295-5075/90/20002
Z. Csahók, K. Honda, E. Somfai, M. Vicsek and T. Vicsek. Dynamics of surface roughening in disordered media. Physica A: Statistical Mechanics and its Applications, 200(1–4):136–154, 1993. https://doi.org/10.1016/0378-4371(93)905123
P.L. Doussal and T. Thiery. Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation. Physical Review E, 96(1):010102, 2017. https://doi.org/10.1103/PhysRevE.96.010102
M. Einax, W. Dieterich and P. Maass. Colloquium: Cluster growth on surfaces: Densities, size distributions, and morphologies. Reviews of modern physics, 85(3):921, 2013. https://doi.org/10.1103/RevModPhys.85.921
E. Frey, U.C. Täuber and T. Hwa. Mode-coupling and renormalization group results for the noisy Burgers equation. Physical Review E, 53(5):4424, 1996. https://doi.org/10.1103/PhysRevE.53.4424
A. Gladkov. Self-similar blow-up solutions of the KPZ equation. International Journal of Differential Equations, 2015, 2015. https://doi.org/10.1155/2015/572841
A. Gladkov, M. Guedda and R. Kersner. A KPZ growth model with possibly unbounded data: correctness and blow-up. Nonlinear Analysis: Theory, Methods & Applications, 68(7):2079–2091, 2008. https://doi.org/10.1016/j.na.2007.01.033
M. Guedda and R. Kersner. Self-similar solutions to the generalized deterministic KPZ equation. Nonlinear Differential Equations and Applications NoDEA, 10(1):1–13, 2003. https://doi.org/10.1007/s00030-003-1036-z
L. Van Hove. The occurrence of singularities in the elastic frequency distribution of a crystal. Physical Review, 89(6):1189, 1953. https://doi.org/10.1103/PhysRev.89.1189
T. Hwa and E. Frey. Exact scaling function of interface growth dynamics. Physical Review A, 44(12):R7873, 1991. https://doi.org/10.1103/PhysRevA.44.R7873
J. Kelling, G. Ódor and S. Gemming. Suppressing correlations in massively parallel simulations of lattice models. Computer Physics Communications, 220:205– 211, 2017. https://doi.org/10.1016/j.cpc.2017.07.010
R. Kersner and M. Vicsek. Travelling waves and dynamic scaling in a singular interface equation: analytic results. Journal of Physics A: Mathematical and General, 30(7):2457–2465, 1997. https://doi.org/10.1088/0305-4470/30/7/024
T. Kriecherbauer and J. Krug. A pedestrian’s view on interacting particle systems, KPZ universality and random matrices. Journal of Physics A: Mathematical and Theoretical, 43(40):403001, 2010. https://doi.org/10.1088/17518113/43/40/403001
Y. Kuramoto and T. Tsuzuki. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progress of theoretical physics, 55(2):356–369, 1976. https://doi.org/10.1143/PTP.55.356
M. Lässig. On growth, disorder, and field theory. Journal of physics: Condensed matter, 10(44):9905, 1998. https://doi.org/10.1088/0953-8984/10/44/003
G. Parisi M. Kardar and Yi-C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56:889, 1986. https://doi.org/10.1103/PhysRevLett.56.889
T. Martynec and S.H.L. Klapp. Impact of anisotropic interactions on nonequilibrium cluster growth at surfaces. Phys. Rev. E, 98:042801, Oct 2018. https://doi.org/10.1103/PhysRevE.98.042801
M. Matsushita, J. Wakita, H. Itoh, I. Rafols, T. Matsuyama, H. Sakaguchi and M. Mimura. Interface growth and pattern formation in bacterial colonies. Physica A: Statistical Mechanics and its Applications, 249(1–4):517–524, 1998. https://doi.org/10.1016/S0378-4371(97)00511-6
L. Mátyás and P. Gaspard. Entropy production in diffusion-reaction systems: The reactive random Lorentz gas. Physical Review E, 71(3):036147, 2005. https://doi.org/10.1103/PhysRevE.71.036147
L. Mátyás T. Tél and J. Vollmer. Multibaker map for shear flow and viscous heating. Physical Review E, 64(5):056106, 2001. https://doi.org/10.1103/PhysRevE.64.056106
B.A. Mello. A random rule model of surface growth. Physica A: Statistical Mechanics and its Applications, 419:762–767, 2015. https://doi.org/10.1016/j.physa.2014.10.064
A.B. Muravnik. On absence of global positive solutions of elliptic inequalities with KPZ-nonlinearities. Complex Variables and Elliptic Equations, 64(5):736– 740, 2019. https://doi.org/10.1080/17476933.2018.1501037
P.J. Olver. Applications of Lie groups to differential equations, volume 107. Springer Science & Business Media, 2012.
W.J.F. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark. NIST handbook of mathematical functions. Cambridge University Press, 2010.
A. Pimpinelli and J. Villain. Physics of Crystal Growth. Cambridge University Press, 1998. https://doi.org/10.1017/CBO9780511622526
T. Sasamoto and H. Spohn. One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Physical review letters, 104(23):230602, 2010. https://doi.org/10.1103/PhysRevLett.104.230602
L.I. Sedov. Similarity and dimensional methods in mechanics. CRC press, 1993.
D. Sergi, A. Camarano, J.M. Molina, A. Ortona and J. Narciso. Surface growth for molten silicon infiltration into carbon millimetersized channels: Lattice–Boltzmann simulations, experiments and models. International Journal of Modern Physics C, 27(06):1650062, 2016. https://doi.org/10.1142/S0129183116500625
G.I. Sivashinsky. Large cells in nonlinear Marangoni convection. Physica D: Nonlinear Phenomena, 4(2):227–235, 1982. https://doi.org/10.1016/01672789(82)90063-X