One Dimensional Kardar-Parisi-Zhang Equation in Various Initial Condition Amplitudes

Authors

  • Gabriella Bognar University of Miskolc, Institute of Machine and Product Design, Miskolc-Egyetemvaros, 3515, Hungary
  • Okhunjon Sayfidinov University of Miskolc, Institute of Machine and Product Design, Miskolc-Egyetemvaros, 3515, Hungary

DOI:

https://doi.org/10.15377/2409-5761.2020.07.5

Keywords:

KPZ equation, Gaussian noise, white noise, amplitude, initial condition, KPZ universality class

Abstract

The Kardar-Parisi-Zhang (KPZ) equation with different initial conditions has been investigated in this paper. The numerical solutions using fixed data are performed without noise term and with two kinds of noise terms, i.e., Gaussian noise term and white noise term. The solutions to the equation have been simulated with different initial conditions of the form A sin (x/16) Our study introduces the obtained shape of the solutions to the KPZ equation according to noise terms with three different amplitudes A. The effect of the noise and the amplitude of the noises are presented and investigated.

References

Kardar K., Parisi G., Zhang Y.Z. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 1986; 56:889–892. https://doi.org/10.1103/PhysRevLett.56.889

Quastel J. Introduction to KPZ. Current developments in mathematics, 2011; 2011.1 https://doi.org/10.4310/CDM.2011.v2011.n1.a3

Barna I. F., Bognár, G., Guedda, M., Mátyás, L., & Hriczó, K. Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise terms. Mathematical Modelling and Analysis, 2020; 25(2), 241-256. https://doi.org/10.3846/mma.2020.10459

Comets F, Cosco C, Mukherjee C. Space-time fluctuation of the Kardar-Parisi-Zhang equation in $ d¥geq 3$ and the Gaussian free field. 2019; arXiv preprint arXiv:1905.03200 https://doi.org/10.1007/s10955-020-02539-7

Carrasco I.S, Oliveira T.J. Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation. Physical Review E, 2016; 94(5): 050801. https://doi.org/10.1103/PhysRevE.94.050801

Alexandre K., Le Doussal P. Exact short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation with Brownian initial condition. Physical Review E, 2017; 96.2 020102. https://doi.org/10.1103/PhysRevE.96.020102

Vallee O. Functional sensitivity of Burgers and related equations to initial conditions. 2003; arXiv preprint nlin/0304014.

Barabasi A.L., Stanley H.E. Fractal concepts in surface growth. Cambridge University Press, 1995. https://doi.org/10.1017/CBO9780511599798

Hwa T. Frey E. Exact scaling function of interface growth dynamics. Physical Review A, 1991; 44(12): R7873. https://doi.org/10.1103/PhysRevA.44.R7873

Frey E., Täuber U. C., Hwa T. Mode-coupling and renormalization group results for the noisy burgers equation. Physical Review E, 1996; 53(5):4424. https://doi.org/10.1103/PhysRevE.53.4424

Lässig M. On growth, disorder, and field theory. Journal of Physics: Condensed Matter, 1998; 10(44):9905. https://doi.org/10.1088/0953-8984/10/44/003

Kriecherbauer T. Krug J. A pedestrian’s view on interacting particle systems, KPZ universality and random matrices. Journal of Physics A: Mathematical and Theoretical, 2010; 43(40):403001. https://doi.org/10.1088/1751-8113/43/40/403001

Quastel, J., Remenik D. KP governs random growth off a one dimensional substrate. 2019; arXiv preprint arXiv:1908.10353.

Prolhac, S., Spohn H. Height distribution of the Kardar-Parisi-Zhang equation with sharp-wedge initial condition: Numerical evaluations. Physical Review E, 2011; 84(1), 011119. https://doi.org/10.1103/PhysRevE.84.011119

Imamura T., Sasamoto T., Spohn H. On the equal time two-point distribution of the one-dimensional KPZ equation by replica. Journal of Physics A: Mathematical and Theoretical, 2013; 46.35. 355002. https://doi.org/10.1088/1751-8113/46/35/355002

Ferrari, Patrik L., Bálint V. Upper tail decay of KPZ models with Brownian initial conditions. 2020; arXiv preprint arXiv:2007.13496

Calabrese, P., Le Doussal, P. Interaction quench in a Lieb– Liniger model and the KPZ equation with flat initial conditions. Journal of Statistical Mechanics: Theory and Experiment, 2014(5); 005004. https://doi.org/10.1088/1742-5468/2014/05/P05004

Le Doussal P. Crossover between various initial conditions in KPZ growth: flat to stationary. Journal of Statistical Mechanics: Theory and Experiment, 2017(5); 053210. https://doi.org/10.1088/1742-5468/aa6f3e

Krajenbrink A, Doussal PL. Exact short-time height distribution in 1D KPZ equation with Brownian initial condition. 2019; arXiv preprint arXiv:1705.04654.

Quastel J, Remenik D. How flat is flat in random interface growth?. Transactions of the American Mathematical Society. 2019; 371(9), 6047-85. https://doi.org/10.1090/tran/7338

Esipov S.E., Newman T.J. Interface growth and Burgers turbulence: the problem of random initial conditions. Physical Review E. 1993; 48(2), 1046. https://doi.org/10.1103/PhysRevE.48.1046

Kozachenko Y., Orsingher E., Sakhno L., Vasylyk O. Estimates for functionals of solutions to Higher-Order Heat-Type equations with random initial conditions. Journal of Statistical Physics. 2018; 172(6), 1641-62. https://doi.org/10.1007/s10955-018-2111-0

Blömker D., Cannizzaro G., Romito M. Random initial conditions for semi-linear PDEs. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 2020; 150(3) 1533-65. https://doi.org/10.1017/prm.2018.157

Le Doussal, Pierre, Pasquale Calabrese. The KPZ equation with flat initial condition and the directed polymer with one free end. Journal of Statistical Mechanics: Theory and Experiment. 2012; 6001. https://doi.org/10.1088/1742-5468/2012/06/P06001

Gueudré T, Le Doussal P, Rosso A, Henry A, Calabrese P. Short-time growth of a Kardar-Parisi-Zhang interface with flat initial conditions. Physical Review E. 2012; 86(4):041151. https://doi.org/10.1103/PhysRevE.86.041151

Sarkar S, Virág B. Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition. 2020; arXiv preprint arXiv:2002.08496.

Meerson B, Schmidt J. Height distribution tails in the Kardar– Parisi–Zhang equation with Brownian initial conditions. Journal of Statistical Mechanics: Theory and Experiment. 2017(10);103207. https://doi.org/10.1088/1742-5468/aa8c12

Bognár G. Roughening in Nonlinear Surface Growth Model. Applied Sciences, 2020; 10 (4), 1422 https://doi.org/10.3390/app10041422

da Silva R.G., Lyra M.L., da Silva C.R., Viswanathan GM. Roughness scaling and sensitivity to initial conditions in a symmetric restricted ballistic deposition model. The European Physical Journal B-Condensed Matter and Complex Systems. 2000; 17(4), 693-7. https://doi.org/10.1007/s100510070110

Barna I. F., Bognár, G., Guedda, M., Mátyás, L., Hriczó, K. Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise terms. Mathematical Modelling and Analysis, 2020; 25(2), 241-256. https://doi.org/10.3846/mma.2020.10459

Barna I. F., Bognár, G., Guedda, M., Mátyás, L., Hriczó, K. Analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with different kind of noise terms, (2019). Differential and Difference Equations with Applications, Springer Proceedings in Mathematics & Statistics, 2019; 333, pp 239-253 https://doi.org/10.1007/978-3-030-56323-3_19

Inc M. The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method. Journal of Mathematical Analysis and Applications. 2008; 345(1) 476-84. https://doi.org/10.1016/j.jmaa.2008.04.007

Bukharev I.A, Kosterlitz J.M. Influence of initial conditions on KPZ growth. InAPS March Meeting Abstracts 1996; H33-04

Fukai Y.T, Takeuchi K.A. Kardar-Parisi-Zhang interfaces with curved initial shapes and variational formula. Physical review letters. 2020; 124(6), 060601. https://doi.org/10.1103/PhysRevLett.124.060601

Meerson B, Sasorov P.V, Vilenkin A. Nonequilibrium steady state of a weakly-driven Kardar–Parisi–Zhang equation. Journal of Statistical Mechanics: Theory and Experiment. 2018; 2018(5), 053201. https://doi.org/10.1088/1742-5468/aabbcc

Sayfidinov O., Bognár G. Numerical Solutions of the Kardar-Parisi-Zhang Interface Growing Equation with Different Noise Terms. In: Jármai K., Voith K. (eds) Vehicle and Automotive Engineering 3. VAE 2020. Lecture Notes in Mechanical Engineering. Springer, Singapore. 2020; 302-311. https://doi.org/10.1007/978-981-15-9529-5_27

Corwin I, Ghosal P. KPZ equation tails for general initial data. Electronic Journal of Probability, 2020; 25. https://doi.org/10.1214/20-EJP467

Corwin I., Ghosal P. Lower tail of the KPZ equation. Duke Mathematical Journal. 2020; 169(7): 1329-95. https://doi.org/10.1215/00127094-2019-0079

Korutcheva E., Cuerno, R. (editors): Advances in Condensed Matter and Statistical Physics, Nova Scientific Publishing. Inc., 2004; 237-259.

Quastel J., Spohn H. The one-dimensional KPZ equation and its universality class. Journal of Statistical Physics, 2015; 160.4. 965-984. https://doi.org/10.1007/s10955-015-1250-9

Takeuchi K. A., Sano, M. Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Physical Review Letters, 2010; 104(23), 230601. https://doi.org/10.1103/PhysRevLett.104.230601

Halpin-Healy T., Zhang Y. C. Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Physics Reports, 1995; 254(4-6), 215-414. https://doi.org/10.1016/0370-1573(94)00087-J

Hoshino, M., KPZ equation with fractional derivatives of white noise. Stochastics and Partial Differential Equations: Analysis and Computations, 2016; 4(4), 827-890. https://doi.org/10.1007/s40072-016-0078-x

Squizzato D, Canet L. Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach. Physical Review E. 2019 Dec 30;100(6):062143. https://doi.org/10.1103/PhysRevE.100.062143

Kardar M. Statistical physics of fields. Cambridge University Press, 2007. https://doi.org/10.1017/CBO9780511815881

Fogedby H. C. Kardar-Parisi-Zhang equation in the weak noise limit: Pattern formation and upper critical dimension. Physical Review E, 2006; 73(3), 031104. https://doi.org/10.1103/PhysRevE.73.031104

Prolhac S., Spohn H. Height distribution of the Kardar-Parisi-Zhang equation with sharp-wedge initial condition: Numerical evaluations. Physical Review E, 2011; 84(1), 011119. https://doi.org/10.1103/PhysRevE.84.011119

Sasamoto T., Spohn H. Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Physics B, 2010; 834(3), 523-542. https://doi.org/10.1016/j.nuclphysb.2010.03.026

Cosco C., Nakajima, S., Nakashima M. Law of large numbers and fluctuations in the sub-critical and $ L^ 2$ regions for SHE and KPZ equation in dimension $ d¥geq 3$. 2020; arXiv preprint arXiv:2005.12689. https://doi.org/10.1214/17-AAP1338

Chhita S, Ferrari P.L, Spohn H. Limit distributions for KPZ growth models with spatially homogeneous random initial conditions. Annals of Applied Probability, 2018; 28(3):1573-603.

Sasamoto, Tomohiro. Spatial correlations of the 1D KPZ surface on a flat substrate. Journal of Physics A: Mathematical and General, 2005; 38-33, L549. https://doi.org/10.1088/0305-4470/38/33/L01

Baik J, Liu Z. Periodic TASEP with general initial conditions. Probability Theory and Related Fields. 2020; 1-98. https://doi.org/10.1007/s00440-020-01004-6

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Published

2020-11-20

How to Cite

Bognar, G. ., & Sayfidinov, O. . (2020). One Dimensional Kardar-Parisi-Zhang Equation in Various Initial Condition Amplitudes. Journal of Advances in Applied &Amp; Computational Mathematics, 7(1), 32–37. https://doi.org/10.15377/2409-5761.2020.07.5

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