Analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with different kind of noise terms
Imre Ferenc Barna, Gabriella Bognár, Mohammed Guedda, Krisztián Hriczó, László Mátyás
aa r X i v : . [ n li n . PS ] S e p A NALYTIC TR AVELING - WAVE SOLUTIONS OF THE K AR DAR -P ARISI -Z HANG INTER FACE GROWING EQUATION WITHDIFFER ENT KIND OF NOISE TER MS
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Imre Ferenc Barna
Wigner Research CenterHungarian Academy of SciencesH-1525 Budapest, P.O. Box 49, Hungary [email protected]
Gabriella Bognár
Institute of Machine and Product DesignUniversity of MiskolcMiskolc-Egyetemváros 3515, Hungary
Mohammed Guedda
Faculte de Mathematiques et d’InformatiqueUniversité de Picardie Jules Verne Amiens33, rue Saint-Leu 80039 Amiens, France
Krisztián Hriczó
Institute of MathematicsUniversity of MiskolcMiskolc-Egyetemváros 3515, Hungary
László Mátyás
Department of Technical and Natural SciencesSapientia UniversityLibert˘atii sq. 1, 530104 Miercurea Ciuc, RomaniaSeptember 26, 2019 A BSTRACT
The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the traveling-wave Ansatz is analyzed. As a new feature additional analytic terms are added. From the mathemati-cal point of view, these can be considered as various noise distribution functions. Six different caseswere investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutionsare evaluated and analyzed for all cases. All results are expressible with various special functionsMathieu, Bessel, Airy or Whittaker functions showing a very rich mathematical structure with somecommon general characteristics. This study is the continuation of our former work, where the samephysical phenomena was investigated with the self-similar Ansatz. The differences and similaritiesamong the various solutions are enlightened. K eywords traveling-wave solution · KPZ equation · Gaussian noise · Lorentzian noise · Special functions · Heunfunctions
Solidification fronts or crystal growth is a scientific topic which attracts much interest from a long time. Basic physicsof growing crystallines can be found in large number of textbooks (see e.g., [1]). One of the simplest nonlineargeneralization of the ubiquitous diffusion equation is the so called Kardar-Parisi-Zhang (KPZ) model obtained fromLangevin equation ∂u∂t = ν ∇ u + λ ∇ u ) + η ( x , t ) , (1)where u stands for the profile of the local growth [2]. The first term on the right hand side describes relaxation of theinterface by a surface tension preferring a smooth surface. The next term is the lowest-order nonlinear term that can PREPRINT - S
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26, 2019appear in the surface growth equation justified with the Eden model. The origin of this term lies in non-equilibrium.The third term is a Langevin noise which mimics the stochastic nature of any growth process and usually has a Gaussiandistribution. In the last two decades numerous studies came to light about the KPZ equation. Without completenesswe mention some of them. The basic physical background of surface growth can be found in the book of Barabási andStanley [3]. Later, Hwa and Frey [4, 5] investigated the KPZ model with the help of the renormalization group-theoryand the self-coupling method which is a precise and sophisticated method using Green’s functions. Various dynamicalscaling forms of C ( x, t ) = x − ϕ C ( bx, b z t ) were considered for the correlation function (where ϕ, b and z are realconstants). The field theoretical approach by Lässig was to derive and investigate the KPZ equation [6]. Kriecherbauerand Krug wrote a review paper [7], where the KPZ equation was derived from hydrodynamical equations using ageneral current density relation.Several models exist and all lead to similar equations as the KPZ model, one of them is the interface growth ofbacterial colonies [8]. Additional general interface growing models were developed based on the so-called Kuramoto-Sivashinsky (KS) equation which shows similarity to the KPZ model with an extra ∇ u term [9], [10].Kersner and Vicsek investigated the traveling wave dynamics of the singular interface equation [11] which is closelyrelated to the KPZ equation. One may find certain kind of analytic solutions to the problem [12] as already mentionedin [ ? ].Ódor and co-worker intensively examined the two dimensional KPZ equation with dynamical simulations to investi-gate the aging properties of polymers or glasses [13].Beyond these continuous models based on partial differential equations (PDEs), there are large number of purelynumerical methods available to study diverse surface growth effects. As a view we mention the kinetic Monte Carlo[14] model, Lattice-Boltzmann simulations [15], and the etching model [16].In this paper we investigate the solutions to the KPZ equation with the traveling wave Ansatz in one-dimensionapplying various forms of the noise term. The effects of the parameters involved in the problem are examined. In general, non-linear PDEs has no general mathematical theory which could help us to understand general featuresor to derive physically relevant solutions. Basically, there are two different trial functions (or Ansatz) which havewell-founded physical interpretation. The first one is the traveling wave solution, which mimics the wave property ofthe investigated phenomena described by the non-linear PDE of the form u ( x, t ) = f ( x ± ct ) = f ( ω ) , (2)where c means the velocity of the corresponding wave. Gliding and Kersner used the traveling wave Ansatz to inves-tigate study numerous reaction-diffusion equation systems [17]. To describe pattern formation phenomena [18] thetraveling waves Ansatz is a useful tool as well. Saarloos investigated the front propagation into unstable states [19],where traveling waves play a key role.This simple trial function can be generalized in numerous ways, e.g., to e − αt f ( x ± ct ) := e − αt f ( ω ) which describesexponential decay or to g ( t ) · f ( x ± c · t ) := g ( t ) f ( ω ) which can even be a power law function of the time as well.We note, that the application of these Ansatz to the KPZ equation leads to the triviality of e − αt = g ( t ) ≡ . In 2006,He and Wu developed the so-called exp-function method [20] which relying on an Ansatz (a rational combination ofexponential functions), involving many unknown parameters to be specified at the stage of solving the problem. Themethod soon drew the attention of many researchers, who described it as “straightforward”, “reliable”, and “effective”.Later, Aslan and Marinakis [21] summarized various applications of the Ansatz.There is another existing remarkable Ansatz interpolating the traveling-wave and the self-similar Ansatz by Benhami-douche [22].The second one is the self-similar Ansatz [23] of the form u ( x, t ) = t − α f (cid:0) xt β (cid:1) := t − α f ( ω ) . The associated mathe-matical and physical properties were exhaustively discussed in our former publications [24], [25] or in a book chapter[26] in the field of hydrodynamics. All these kind of methods belong to the so-called reduction mechanism, where ap-plying a suitable variable transformation the original PDEs or systems of PDEs are reduced to an ordinary differentialequation (ODE) or systems of ODEs. 2
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Applying the traveling wave Ansatz to the KPZ PDE with η ( x, t ) = 0 , equation (1) leads to the ODE of − νf ′′ ( ω ) + f ′ ( ω ) (cid:20) c − λ f ′ ( ω ) (cid:21) = 0 , (3)From now on we use the Maple 12 mathematical program package to obtain analytic solutions in closed forms. Forequation (3), it can be given as f ( ω ) = 2 λ ln λ (cid:2) c νe cην + c c (cid:3) νc ! ν, (4)where c and c are the constants of integration and c is the speed of the wave.We fix this notation from now on throughout the paper. Note, that this is an equation of a linear function f ( ω ) = aω + b (just given in a complicated form) with any kind of parameter set, except c = 0 which gives a constant solution. Thisphysically means that there is a continuous surface growing till infinity which is quite unphysical. Therefore, someadditional noise is needed to have surface growing phenomena. We remark the general properties of all the forthcomingsolutions. Due to the Hopf-Cole transformation [27, 28] ( h = A ln( y )) convertes the non-linear KPZ equation to theregular heat conduction (or diffusion) equation with an additional stochastic source term eliminating the non-lineargradient-squared term. All the solutions contain a logarithmic function with a complicated argument. In this sense,the solutions have the same structure, the only basic difference is the kind of special function in the argument. If theseargument functions take periodically positive and negative real values then the logarithmic function creates distinctintervals (small islands which describe the surface growing mechanisms, and define the final solution). This statementis generally true for our former study as well [29].Remark that the solution to (1) obtained from the self-similar Ansatz reads f ( ω ) = 2 νλ ln (cid:18) λc √ πν erf [ ω/ (2 √ ν )] + c ν (cid:19) , (5)where erf [ ] means the error function [30]. Figure 1 compares these two solutions. We note the asymptotic conver-gence of the self-similar solution and the divergence of the traveling-wave solution. We have the same conclusion asin our former study [29] (where the self-similar Ansatz was applied) , that without any noise term the KPZ equationcannot be applied to describe surface growth phenomena. The different kind of noise terms define different kind ofextra islands (parts of the solution having compact supports) and these islands show a growth dynamics.To have a better understanding between the two solutions, Fig. 2 shows the projection of both complete solutions u ( x, y = 0 , t ) . The major differences are still present.
As we mentioned in our former study [29] only the additional noise term makes the KPZ solutions interesting. Wesearch the solutions with the traveling-wave Ansatz, therefore is it necessary that the noise term η should be an analyticfunction of ω = x + ct like η ( ω ) = a ( x + ct ) . We will see that for some kind of noise terms it is not possible tofind a closed analytic solution when all the physical parameters are free ( ν, λ, c, a ) , however, if some parameters arefixed it becomes possible to find analytic expressions. It is also clear, that it is impossible to perform a mathematicallyrigorous complete function analysis according to all four physical and two integral parameters c , c . We performednumerous parameter studies and gave the most relevant parameter dependencies of the solutions. n = − As first, case let us consider the brown noise η ( x, t ) = aω . It leads to the following ODE − νf ′′ ( ω ) + f ′ ( ω ) (cid:20) c − λ f ′ ( ω ) (cid:21) − aω = 0 . (6)The solution can be given in the form f ( ω ) = 1 λ cη + ν ln ( λ (cid:2) − c I d (cid:0) cω ν (cid:1) + c K d (cid:0) cω ν (cid:1)(cid:3) c ω (cid:2) K d (cid:0) cω ν (cid:1) I d +1 (cid:0) cω ν (cid:1) + I d (cid:0) cω ν (cid:1) K d +1 (cid:0) cω ν (cid:1)(cid:3) )! (7)3 PREPRINT - S
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26, 2019Figure 1: The two shape functions of the KPZ equation without any kind of noise term. The solid line represents thesolution for traveling-wave and the dashed line is for the self-similar Ansatz. The applied parameter set is c = c = c = 1 , ν = 4 , λ = 3 Figure 2: The two solutions of the KPZ equation without any noise term. The upper lying function represents thetraveling-wave solution. The applied parameter set is the same as used above.4
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26, 2019Figure 3: Three different shape functions for the brown noise n = − . The applied physical parameter set is λ =5 , ν = 3 , a = 2 and c = 2 . The dashed line is for c = 1 , c = 0 , the dotted line is for c = c = 1 and the solid lineis for c = 0 , c = 1 , respectively.where I d ( ω ) and K d ( ω ) are the modified Bessel functions of the first and second kind [30] with the subscript of d = √ ν − aλ ν + 1 . To obtain real solutions for the KPZ equation (which provides the height of the surface) the orderof the Bessel function (notated as the subscript) has to be non-negative and provides the following constrain ν ≥ aλ .This gives us a reasonable relation among the three terms of the right hand side of equation (1). When the magnitudeof the noise term a becomes large enough no surface growth take place. Figure 3 presents solutions with differentcombinations of the integration constants c , c . Having in mind, that the K d () Bessel function of the second kind isregular at infinity, one gets that it has a strong decay at large argument ω . The c = 0 , c = 0 type solutions havephysical relevance. Figure 4 shows the complete solution of the KPZ equation. It can be seen that a sharp and localizedpeak exists for a short time. Therefore, no typical surface growth phenomena is described with this kind of noise andinitial conditions. n = − The noise term η = aω corresponds to the ODE − νf ′′ ( ω ) + f ′ ( ω ) (cid:20) c − λ f ′ ( ω ) (cid:21) − aω = 0 , (8)whose general solution is f ( ω ) = 1 λ + ln (cid:26) − cλ [ c M ( ǫ b ) − c U ( ǫ b )] M ( ǫ d )(2 νcU ( ǫ b ) + aλU ( ǫ b )) + 2 M ( ǫ b ) νcU ( ǫ d ) (cid:27) , (9)where M ( ǫ b ) and U ( ǫ d ) are the Kummer M and Kummer U functions (for more see [30]) with the parameters of ǫ b = ( cν − aλ cν , , cων ) and ǫ d = ( − aλ cν , , cων ) . Figure 5 shows three different shape functions corresponding to the pinknoise. The evaluation of direct parameter dependencies of the solutions are not trivial. In some reasonable parameter5 PREPRINT - S
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26, 2019Figure 4: The solution u ( x, t ) to the KPZ equation for the brown noise n = − with the parameter set of c = c = c = 1 , ν = 4 , λ = 3 . range we found the following trends: for fixed a, c, ν and larger λ values, the solution shows more independent well-defined "bumps" or islands and higher steepness of the line which connects the maxima of the existing peaks of theislands. At fixed parameter values a, c, λ , different values of ν just shift the position of the existing peaks. The role of a and c is not defined. Figure 6 presents a total solution u ( x, t ) to the KPZ equation, the freely traveling three islandsare clearly seen. n = 0 Here, the noise term is η = aω = a which leads to the ODE of − νf ′′ ( ω ) + f ′ ( ω ) (cid:20) c − λ f ′ ( ω ) (cid:21) − a = 0 , (10) f ( ω ) = ωcλ − ω √ c − aλλ − ν ln(2) λ − ν ln c − aλλ " c e ω √ c − aλν − c λ (11)Figure 7 shows two shape functions for two different parameter sets. There exists basically two different functionsdepending on the ratios of the integral constants c and c . The first is a pure linear function with infinite range and itsdomain represents boundless surface growth, which is a physical nonsense. The second solution is a sum of a linearand logarithmic function with a domain bounded from above due to the argument of the ln function. Figure 8 showsthe final solution of the KPZ equation u ( x, t ) . We note that with the substitution ω = x + ct only the first kind ofsolution remains real. For the second parameter set which creates a modified ln function with a cut at well-definedargument becomes complex. 6 PREPRINT - S
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26, 2019Figure 5: Three different shape functions for the pink noise ( n = − . Solid, dashed and dotted lines are for theparameter sets of ( c = c = 1; c = 1 / , ν = 0 . , λ = 3 , a = 2) , ( c = c = 1; c = 1 / , ν = 0 . , λ = 2 . , a = 2) , ( c = c = 1; c = 0 . , ν = 0 . , λ = 5 , a = 2) , respectively.Figure 6: The total solution of the KPZ equation for n = − with the applied parameter set c = c = 1 , c = 1 / , ν =0 . λ = 3 and a = 2 . 7 PREPRINT - S
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26, 2019Figure 7: Two shape functions for the constant or white noise. The solid line is for the parameter set c = 4 , c = − , c = 0 . , ν = 2 , λ = 1 , a = 1 , and the dashed line is for c = c = 1 , c = 4 , ν = 0 . , λ = 1 , a = 0 . ,respectively.Figure 8: The KPZ solution for the constant or white noise. The applied parameter set is c = c = 1 , c = 4 , ν =0 . , λ = 1 , a = 0 . , respectively. 8 PREPRINT - S
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26, 2019Figure 9: The shape function for the blue noise for three parameter sets. The solid, dashed and dotted lines are forthe parameter sets ( c = 1 , c = 0 , c = 3 , a = 0 . , ν = 1 . , λ = 2) , ( c = c = c = 1 , a = 1 , ν = 1 , λ = 3) , and ( c = c = c = 1 , a = 1 , ν = 1 , λ = 3) , respectively. n = 1 The last color noise η = aω leads to the ODE of − νf ′′ ( ω ) + f ′ ( ω ) (cid:20) c − λ f ′ ( ω ) (cid:21) − aω = 0 , (12)with the general solution of f ( ω ) = cωλ − ν ln(2)3 λ + 2 ν λ ln (cid:26) λ [ c Ai (˜ ω ) − c Bi (˜ ω )] νa [ Ai (1 , ˜ ω ) Bi (1 , ˜ ω ) − Bi (1 , ˜ ω ) Ai (˜ ω ))] (cid:27) (13)where Ai (˜ ω ) , Bi ˜ ω ) denote the Airy functions of the first and second kind and Ai (1 , ˜ ω ) and Bi (1 , ˜ ω ) are the firstderivatives of the Airy functions, where we used the following notation: ˜ ω = − (2 aωλ − c )4 ( aλν ) / aλ . Exhaustivedetails of the Airy function can be found in [31]. When the argument ω is positive, Ai ( ω ) is positive, convex, anddecreasing exponentially to zero, while Bi ( ω ) is positive, convex, and increasing exponentially. When ω is negative, Aiω ) and Biω ) oscillate around zero with ever-increasing frequency and ever-decreasing amplitude.Figure 9 represents shape functions with different parameter sets. Our analysis showed that the composite argumentof the ln function is purely real having a decaying oscillatory behavior with alternatively positive and negative values.The ln function creates infinite number of separate "bumps" or islands with compact supports and infinite first spatialderivatives at their boarders. Combining the first two terms of the (13), we get an infinite series of separate islands withincreasing height. The ratio c/λ is the steepness of the line, this automatically defines the steepness of the absoluteheight of the islands. The effects of the various parameters are not quite independent and hard to define, we may saythat in general each parameter ν, λ, a, c alone can change the widths, spacing and absolute height of the peaks. Figure10 shows the total solution of the KPZ equation. The traveling "bumps" are clearly visible.9 PREPRINT - S
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26, 2019Figure 10: The solution u ( x, t ) for the n = 1 or blue noise with the applied parameter set of c = c = c = 1 , ν =2 , λ = 3 , a = 1 As a first non-colour noise let us consider the Lorentzian noise of the form η = a ω . It leads to the ODE of − νf ′′ ( ω ) + f ′ ( ω ) (cid:20) c − λ f ′ ( ω ) (cid:21) − a ω = 0 , (14)We mention, that for the classical exponential and Gaussian noise distributions we could not give solutions in closedanalytic form. Unfortunately, there is no closed analytic expression available if all the parameters ( ν, λ, c, a ) are free.The formal solution contains integrals of the Heun C confluent functions multiplied by some polynomials. However,if the parameters a, λ, ν are fixed, there is analytic solution available for free propagation speed c . The exact solutionfor a = λ = ν = 1 / , and c = 2 is the following f ( ω ) = cω +2 ln (cid:26) c C ( B ) − c ωC ( A )2( ω + ω )[ C ( A ) C ′ ( B ) − C ( B ) C ′ ( A )] + (1 + ω ) C ( A ) C ( B ) (cid:27) , (15)where C ′ () means the first derivative of the Heun C function. For the better transparency we introduce the followingnotations A = 0 , , , c , − c ; − ω and B = 0 , − , , c , − c ; − ω .Figure 11 shows the shape function for given parameter set. There is a broad island close to the origin and numeroustiny ones at larger arguments. The numerical accuracy of Maple 12 was enhanced to reach this resolution. It is well-known that the Heun functions are the most complicated objects among special functions and the evaluations needsmore computer time.Figure 12 presents the total solution of the original KPZ. Due to the substitution ω = x + ct the original local solutionbroke down to several smaller islands which freely propagate in time and space. The last perturbation investigated is a periodic function η = a sin( ω ) and − νf ′′ ( ω ) + f ′ ( ω ) (cid:20) c − λ f ′ ( ω ) (cid:21) − a sin( ω ) = 0 . (16)10 PREPRINT - S
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26, 2019Figure 11: The shape function for the Lorentzian noise. The applied parameters are c = 0 . , c = 2 , c = 1 , ν =1 , a = 1 , λ = 3 , Figure 12: The solution of the KPZ equation for Lorentzian noise, with the parameters mentioned above.11
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26, 2019Figure 13: The shape function for the periodic noise. The applied parameters are c = 0 . , c = 2 , c = 1 , ν = 1 , a =1 , λ = 3 , The general solution can be given as f ( ω ) = 1 λ (cid:18) cω + 2 ln (cid:26) λ [ c C ( ǫ a ) − c S ( ǫ a )] ν [ − C ′ ( ǫ a ) S ( ǫ a ) + C ( ǫ a ) S ′ ( ǫ a )] (cid:27)(cid:19) , (17)where C ( ǫ a ) , S ( ǫ a ) , C ′ ( ǫ a ) and S ′ ( ǫ a ) are the Mathieu S and Mathieu C functions and the first derivatives. For basicproperties we refer to [30]. For a complex study about Mathieu functions see [32, 33, 34]. In (17), we used theabbreviation of ǫ a = − c ν , − aλν , − π + ω .Figure 13 shows a typical shape function for the periodic noise term. Due to the elaborate properties of even thesingle Mathieu C or S functions for some parameter pairs a, q the function is finite with periodic oscillations and forsome neighboring parameters it is divergent for large arguments. No general parameter dependence can be stated. Theparameter space of the set of six real values ( c , c , c, a, ν, λ ) is too large to map. After the evaluation of numerousshape functions we may state, that a typical shape function is presented with two larger islands close to the origin andnumerous smaller intervals. For large argument ω the shape function shows a steep decay.Figure 14 shows the complete solution. Note, that the first two broader islands can be seen as they freely travel. Dueto the finite resolution the smaller islands are represented as irregular noise in the background. In summary, we can say that with an appropriate change of variables applying the self-similar Ansatz one may obtainanalytic solution for the KPZ equation for one spatial dimension with numerous noise terms. We investigated four typeof power-law noise ω n with exponents of − , − , , , called the brown, pink, white and blue noise, respectively. Eachinteger exponent describes completely different dynamics. Additionally, the properties of Gaussian and Lorentziannoises are investigated. Providing completely dissimilar surfaces with growth dynamics. All solutions can be describedwith non-trivial combinations of various special functions, like error, Whittaker, Kummer or Heun. The parameterdependencies of the solutions are investigated and discussed. Future works are planned for the investigations of twodimensional surfaces. 12 PREPRINT - S
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26, 2019Figure 14: The complete traveling wave solution u ( x, t ) for periodic noise with the same parameter set as given above. Acknowledgment
This work was supported by project no. 129257 implemented with the support provided from the National Research,Development and Innovation Fund of Hungary, financed under the K _ funding scheme. The described study wascarried out as part of the EFOP-3.6.1-16-00011 “Younger and Renewing University – Innovative Knowledge City –institutional development of the University of Miskolc aiming at intelligent specialization” project implemented inthe framework of the Szechenyi 2020 program. The realization of this project is supported by the European Union,co-financed by the European Social Fund. References [1] A. Pimpinelli and J. Villain.
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