Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise term
Imre Ferenc Barna, Gabriella Bognár, Mohammed Guedda, Krisztián Hriczó, László Mátyás
aa r X i v : . [ n li n . PS ] A p r A NALYTIC SELF - SIMILAR SOLUTIONS OF THE K AR DAR -P ARISI -Z HANG INTER FACE GROWING EQUATION WITHVAR IOUS 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, RomaniaApril 4, 2019 A BSTRACT
The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similarAnsatz is analyzed. As a new feature additional analytic terms are added. From the mathematicalpoint 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 solutionswere evaluated and analyzed for all cases. All results are expressible with various special functionslike Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure withsome common general characteristics. K eywords self-similar solution · KPZ equation · Gaussian noise · Lorentzian noise · Special functions · Heunfunctions
Growth patterns in clusters and sodification fronts are challenging problems from a long time. Basic knowledge of theroughness of growing crystalline facets has obvious technical applications [1]. The simplest nonlinear generalization ofthe ubiquitous diffusion equation is the so called Kardar-Parisi-Zhang (KPZ) model obtained from Langevin 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, which prefers a smooth surface. The second term is the lowest-order nonlinear termthat can appear in the surface growth equation justified with the Eden model and originates from the tendency of thesurface to locally grow normal to itself and has a non-equilibrium in origin. The last term is a Langevin noise to PREPRINT - A
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4, 2019mimic the stochastic nature of any growth process and has a Gaussian distribution usually. There are numerous studiesavailable about the KPZ equation in the literature in the last two decades. Without completeness we mention some ofthem. The foundation of the physics of surface growth can be found in the book of Barabási and Stanley [3]. Hwaand Frey [4, 5] investigated the KPZ model with the help of the self-mode-coupling method and with renormalizationgroup-theory, which is an exhaustive and sophisticated method using Green’s functions. They considered variousdynamical scaling form of C ( x, t ) = x − ϕ C ( bx, b z t ) for the correlation function (where ϕ, b and z are real constants).Lässig showed how the KPZ model can be derived and investigated with field theoretical approach [6]. In a topicalreview paper Kriecherbauer and Krug [7] derived the KPZ model from hydrodynamical conservation equations with ageneral current density relation. Later, Einax et al. [8] published a review study on cluster growth on surfaces.Numerous models exist, which may lead to similar equations as the KPZ model, i.e., the interface growth of bacterialcolonies [9]. More general interface growing models were developed based on the so-called Kuramoto- Sivashinsky(KS) equation which is similar to the KPZ model with and extra −∇ u term on the right hand side of (1) (see [10],[11]).Guedda has already investigated the generalized deterministic KPZ equation, when the gradient term is on an arbitraryexponent, with the self-similar Ansatz [12]. Kersner and Vicsek investigated the traveling wave dynamics of the singu-lar interface equation [13], which is closely related to he KPZ equation. Ódor and co-worker intensively examined thetwo dimensional KPZ equation with extended dynamical simulations to study the physical aging properties of differentsystems like glasses or polymers [14].Beyond these continuous models based on partial differential equations (PDEs) there are numerous purely numericalmethods available to study diverse surface growth phenomena. Without completeness, we mention the kinetic MonteCarlo [15], Lattice-Boltzmann simulations [16] and the etching model [17].By present work, one may find certain kind of solutions to the problem [18, 19]. It is already mentioned in [20] theseare for droplet initial conditions.The first term on the right hand side of the equation can be also related to diffusion [21], and it can be found in thedescription of such processes [22, 23].In this paper we analyze the solutions of the KPZ equation with the self-similar Ansatz in one-dimension applyingvarious forms of the noise term. Numerical results are provided both for similarity solutions with similarity variablesand for the solutions with the original variables as well. The effect of the parameters involved in the problem isexamined.The similarity method was used for the investigation of analytic solution of the two dimensional Navier-Stokes equa-tion with a non-Newtonian type of viscosity [24]. Non-linear PDEs has no general mathematical theory, which could help us to derive physically relevant solutions.Basically, there are two different trial functions (or Ansatz) having well-founded physical interpretation. The firstone is the traveling wave solution, which mimics the wave property of the investigated phenomena described by thenon-linear PDE. The second one is the self-similar Ansatz of the form u ( x, t ) = t − α f (cid:16) xt β (cid:17) := t − α f ( ω ) , (2)where u ( x, t ) can be an arbitrary variable of a PDE and t denotes time and x means spatial dependence. The similarityexponents α and β are of primary physical importance since α represents the rate of decay (or sharpening process if α < ) of the magnitude u ( x, t ) , while β is the rate of spread (or contraction if β < ) of the space distribution for t > . The most powerful result of this Ansatz is the fundamental or Gaussian solution of the Fourier heat conductionequation (or for Fick’s diffusion equation) with α = β = 1 / . These solutions are exhibited on Fig. 1 for fixed times t < t . We can generally state, that this Ansatz mimics the diffusive properties (the similarities to normal diffusion)of the investigated PDE. This is the key point why this Ansatz is used. We note, that in some cases [25] the travelingwave and self-similar solutions are intertwinted and can be transformed into one another.This transformation is based on the assumption that a self-similar solution exists, i.e., every physical parameter pre-serves its shape during the expansion. Self-similar solutions usually describe the asymptotic behavior of an unboundedor a far-field problem; the time t and the spacial coordinate x appear only in the combination of f ( x/t β ) . It means thatthe existence of self-similar variables implies the lack of characteristic lengths and times. These solutions are usuallynot unique and do not take into account the initial stage of the physical expansion process. It is also transparent from(2) that to avoid singularity at t = 0 the following transformation ˜ t = t + t is valid.2 PREPRINT - A
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4, 2019Figure 1: A self-similar solution of Eq. (2) for t < t . The presented curves are Gaussians for regular heat conduction.We should note that with the application of the Hopf-Cole transformation h = A ln ( y ) the non-linear KPZ equationcan be converted to the regular heat conduction (or diffusion equation) with a stochastic term.There are numerous reasonable generalization of (2) available, one of them is u ( x, t ) = h ( t ) · f [ x/g ( t )] , where h ( t ) and g ( t ) are continuous functions. The choice of h ( t ) = g ( t ) = √ t − t is a special kind, called the blow-up solution.It means that the solution becomes infinity after a well-defined finite time duration.These kind of solutions describe the intermediate asymptotics of a problem. They hold when the precise initial con-ditions are no longer important, but before the system has reached its final steady state. For some systems it can beshown that the self-similar solution fulfills the source type (Dirac delta) initial condition. They are much simpler thanthe full solutions and so easier to understand and study in different regions of parameter space. A final reason forstudying them is that they are solutions of a system of (ordinary differential equations (ODEs) and they do not sufferthe extra inherent numerical problems related to PDEs. In some cases self-similar solutions helps us to understandglobal physical properties of the solutions like finite oscillations, diffusion-like properties, discontinuous solutions orthe existence of compact supports. Such kind of general information is hard to find from purely numerical calculations.Applicability of this Ansatz is quite wide and comes up in various mechanical systems [26], in transport phenomenalike heat conduction [25], in Euler equation [27] or even in various two or three dimensional Navier-Stokes equations[28, 29]. We start our investigation with the KPZ equation (1) in one spatial dimension neglecting the noise term ( η ( x, t ) = 0 ).Calculating the time and spacial derivatives of (2) and substituting to (1) one gets the following constrains for theexponents: α = 0 and β = 1 / . In regular heat conduction (or diffusion) process both exponents are equal to / , which means that the decay (perpendicular dynamics to the surface) and the spreading (parallel dynamics to thesurface) of the solution have the same strength in time. For the KPZ equation the general features are different. Now, α vanishes, which means that we cannot identify any kind of decaying dynamics of the solution perpendicular to thesurface. The non-zero value of β can be understood as a kind of spreading parallel to the surface. These are generaland relevant statements of the surface growth process described by our solution. The remaining non-linear ODE reads νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) = 0 . (3)The general solution can be given with the logarithm of the error function f ( ω ) = 2 νλ ln (cid:18) λc √ πν erf [ ω/ (2 √ ν )] + c ν (cid:19) , (4)3 PREPRINT - A
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4, 2019Figure 2: The self-similar solution of the KPZ equation without any noise term for c = c = 1 . Solid line is for λ = ν = 1 , dashed line is for ν = 1 and λ = 2 and the dotted line represents ν = 2 and λ = 1 .Figure 3: The self-similar solution of the original KPZ equation without the noise term for the parameter set c = c = λ = ν = 1 .where erf is the error function [30] and c and c are integration constants. The role of c is just a shift of the solution.For physical reasons the surface tension ν should be larger than zero. Analyzing the solution to Eq. (4), the value of λ should be positive as well. Figure 2 presents three different shape function solutions of the ODE with c = c = 1 andfor three different combinations of λ and ν . Note, that all solution has the same simple qualitative behavior, a quickramp-up and a converged plateau. Figure 3 shows the complete solutions of the original PDE showing the spatialand time dependence for c = c = λ = ν = 1 . The function has a similar structure like Fig. 2 a quick ramp-upand a slow convergent plateau. We may say that different numerical values of η and ν do not drastically change thequalitative structure of the solution surface. A closer look of the solution shows, that at t = 0 the height of the surfacehas a constant value, later at small times a thin valley is formed, which becomes wider and wider as time goes on.The physical parameters λ, ν and the integration constants (only shifts the solutions) set the shape and the depth ofthe valley. Even at large times at large spatial distances, the height of the surface, i.e., the asymptotic solution remainsconstant. The growth of the valley (or void) can be understood as a kind of front propagation as well, and thereforecan be explained with the non-zero β exponent.At this point is comes clear to us, that any kind of surface growth mechanism can be only described and investigatedwith the additional noise term. The direct application of the self-similar solution to the KPZ equation without anyadditional noise term η ( x, t ) cannot describes any kind of growth process.4 PREPRINT - A
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It is obvious that every physical process is perturbed with some kind of perturbations. Perturbations which carry noinformation are called noise. The KPZ equation, very correctly, include an additive noise term. Our similarity Ansatzof the from (2) satisfy the general ODE with the form of νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) + tη ( ω ) = 0 . (5)If we want to apply the self-similar Ansatz (2) to the noisy KPZ equation than the noise term tη ( x, t ) = l ( ω ) = l ( x/t β ) should be some kind of analytic function of the original variables of x, t , on the other side, we want to handle noisein a statistically correct manner, η ( x, t ) should be a density function of a probability distribution as well. This secondcondition dictates that the density function should be positive and should have an existing finite integral on a finite orinfinite support. Note, the extra time dependence of the last term in (5) is dictated from a dimensional analysis reason.First, we investigate noises with various power-law dependencies l ( ω ) = aω n . Noises with different integer powervalues of n are named after different colors n = − , − , , which are brown, pink, white and blue, respectively.Two additional cases, the Gaussian and Lorenzian noises, are investigated as well. To avoid further misunderstandingwe must state that in our calculations a Gaussian noise means that the noise term explicitly depends on the scaledspatial coordinate x/t / and not on the Fourier spectra as usually considered. The argument ω of the shape functionis the time-scaled spatial coordinate and not the angular frequency. Of course, in principle it is possible to evaluatethe Fourier spectra of our noise terms and interpret them in the frequency domain but that is not the aim of the presentstudy. n = − Our first case leads to the ODE of νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) + aω = 0 . (6)Using the mathematical program package Maple 12, the solution can be obtained in a closed form f ( ω ) = − ω λ + 1 λ (cid:20) ln (( ω λ { c M − , d ( r ) − c W − , d ( r ) } ) / ( M , d ( r ) · ν · W − , d ( r ) + M , d ( r ) · ν · d · W − , d ( r ) +4 M − , d ( r ) · ν · W , d ( r )) ) ν (cid:21) , (7)where M and W are the Whittaker M and Whittaker W functions [30]. For the better transparency we used thefollowing notations d = √ ν − λa/ν for the second parameter and r = ω / (4 ν ) for the argument of the Whittakerfunctions. Both parameters of the Whittaker functions must be real numbers, which means that ν − λa ≥ thereforefor any kind of fixed and positive ν and λ , there is an upper limit for a , which is the strength of the noise term. So, ifthe magnitude of the noise reaches a definite level, the Whittaker function and the solution of the problem becomesundefined and meaningless. This is consistent with our physical picture about noisy processes.Due to the Whittaker function, the solution is undefined for negative arguments η for any kind of parameter set. Figure4 presents the solution of Eq. (7) for two different parameter sets. The numerical value of ν defines the position of thesingularity in a non trivial way, larger ν shifts the position to larger arguments. At fixed physical parameters a , ν and λ , the first integration constant c is equal with the asymptotic value of the solution for large arguments. The secondintegration constant c directly defines the function in the origin in a non-trivial way, the larger the value the larger thefunction as well.Figure 5 shows the solution profile u ( x, t ) of the KPZ equation as the function of time and spatial coordinate. Thesharp cusp is clearly seeing. It is also evident that the position of the cusp moves to larger spatial coordinates as timegoes on, compared to the free KPZ solution of Eq. (3.1), which means that the additional noise term puts a small islandinto the origin which is growing and pushing the cusp before. Another interesting feature is that the cusps surviveseven at large times and is not filled up as would we expect from our physical intuition.The sharp but finite cusp that arises in this solutions is the so-called Van Hove singularity, which was first seen incrystals in the function of elastic frequency distribution [31]. We note, that the finite number of peaks on Fig. 5 underthe cusp are just an artifact of the finite resolution of the Maple software.5 PREPRINT - A
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4, 2019Figure 4: The shape functions of the Brownian noise Eq. (7) for a = λ = 1 , ν = 2 physical parameters . The red lineis for c = 3 , c = 1 and and the green line is for the integration constants c = 1 , c = 3 .Figure 5: The solution of the KPZ eq. with Brownian noise, for the parameter set of λ = ν = c = c = 1 and a = 1 / . n = − The corresponding ODE reads νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) + aω = 0 . (8)In the most general case when all three parameters are undefined ( λ, ν, a ), there is no closed formula available forthe solution. There is an existing expression containing the integral of the HeunB functions [30] together with otherfunctions. For given values ( λ = ν = a = 1 ), the formula becomes a bit more transparent f ( η ) = − η + 2 ln (cid:20) c η H B (cid:18) , , − , , − η (cid:19) − c η H B (cid:18) , , − , , − η (cid:19) × (Z e − η η H B (cid:0) , , − , , − η (cid:1) ) . (9)Unfortunately, if the strength of the noise a is different, then the term containing the integral of the function H C cannotbe separated from the pure H C function and the final from cannot be evaluated numerically.Figure 6 shows the shape function where the physical parameters are set to unity. At first sight, the solution looks thesame as the solution without noise, however there is a small positive island, which is created next to the valley. Inother words the solution has a local maximum at finite ω , which means a finite time and space coordinate x/t . As6 PREPRINT - A
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4, 2019Figure 6: The shape function of the pink noise for a = λ = ν = 1 and for c = 1 and c = 0 . Figure 7: The complete solution of the KPZ equation for the pink noise with the parameters given above.time goes on the valley becomes wider and wider pushing this tiny island to the right with a continuous smearing. Wecan say that the surface growing phenomena breaks down in the dynamics of this tiny island or rather "reef". So, thereis no general surface growth phenomena along the whole axis.Figure 7 shows the complete solution of the KPZ equation with the pink noise, note the tiny positive bump at smalltime and space coordinates. n = 0 The associated ODE is now νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) + c = 0 . (10)There is no general closed formula available for a general real constant c . However, if the constant noise term is writtenin the form of η = nλ , which can be identified as a kind of external mechanism due to the work of [32], then otheranalytical solutions become available which can be expressed via Kummer M and Kummer U functions [30] f ( ω ) = 2 νλ ln (cid:18)(cid:26) λ (cid:20) − c M (cid:18) − nλ , , ω nλ (cid:19) − c U (cid:18) − nλ , , ω nλ (cid:19)(cid:21)(cid:27) / (cid:26) ν (cid:20) M (cid:18) − nλ , , ω nλ (cid:19) U (cid:18) − nλ , , ω nλ (cid:19) + PREPRINT - A
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4, 2019Figure 8: The shape function for the white noise for parameter set a = λ = ν = 1 = c = c = 1 .Figure 9: The complete solution for the white noise for a = 1 / , λ = 2 , ν = 1 / and for c = c = 1 .nλM (cid:18) − nλ , , ω nλ (cid:19) U (cid:18) − nλ , , ω nλ (cid:19) +2 M (cid:18) − nλ , , ω nλ (cid:19) U (cid:18) − nλ , , ω nλ (cid:19)(cid:21)(cid:27)(cid:19) − ν ln(2) λ . (11)Figure 8 shows the shape function for the white noise. The new feature is that the solution fell apart to numerousdistinct intervals with compact supports. The function has large but finite negative values at the supports with infinitelylarge derivatives, which can be called cusps as well. Note, that there are finite intervals where the solution is not defined.Figure 9 shows the solution u ( x, t ) of the KPZ equation. We mention that the separate islands continuously grow astime goes on, but, they cannot touch each other even at large times. The finite number of peaks under the cusp areagain an artifact of the finite resolution of the Maple software. Such kind of surface growth, where separate "barrys"are created, can be noticed on coral reefs or on dripstones. n = 1 The last power law noise case is the following νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) + aω = 0 . (12)8 PREPRINT - A
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4, 2019Figure 10: The shape function of the blue noise for the parameter set a = λ = ν = 1 and initial conditions c = 1 and c = 1 , respectively.Figure 11: The complete solution u ( x, t ) for the blue noise with the parameters given above.The structure of the solution shows some similarity to the brown and white noise and can be expressed with the helpof the Kummer U and M functions f ( ω ) = − ω + 2 νλ ln ( − λ (4 λ − ω ) (cid:2) c U ( ǫ − , , σ ) − c M ( ǫ − , , σ ) (cid:3) (cid:2) M ( ǫ + , , σ ) U ( ǫ − , , σ ) λ + M ( ǫ , , σ ) U ( ǫ + , , σ ) ν (cid:3) ) . (13)For the better transparency, we use the following notations ǫ − = λ − νν , ǫ + = λ + νν and σ = (4 λ − ω ) ν . Figure 10presents the shape function for the blue noise. It shows some similar features to the former white noise. The solutionon the positive axis can be interpreted only on two separate finite intervals. The function has finite values, however,the first derivatives at the right hand side of the intervals become infinite, which can be interpreted as a kind of "semi-cusp". With some vertical shifts parallel to the axis f ( ω ) the solution can be physically interpreted as two distinctislands growing in time. As an additional fineness, we note that at left side of the right island there is a gap.Figure 11 shows the solution function u ( x, t ) of the original PDE. The spatial range of the two distinct intervals iscontinuously growing in time, however, it remains separate even at infinite times. With this kind of noise and Ansatz,there is no way to grow a constant surface above the whole positive semi-axis.9 PREPRINT - A
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The first non-power law noise gives us the ODE of νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) + ae − ω n = 0 . (14)There is no general formula available for arbitrary parameters λ, η, a and n . Fortunately, if two parameters are fixede.g. ν = 1 / and n = 1 , than there is a closed expression available for the solution f ( ω ) = − λ ln " (cid:26) √ λaπ · erf ( r ω c (cid:27) + c , (15)where erf means the error function [30]. Figure 12 presents various solutions for different values of λ . The larger thevalue λ an the smaller the parameter a the more the number of initial islands are, which is a remarkable new feature.The solution itself is a continuous function on the whole ω axis. The Van Hove singularities at finite ω are still present.Note, that at larger value of λ , the depth of the singularity valleys become shallower.Figure 13. presents the final solution of the PDE. The general features are very similar to the formerly investigatednoise a/ω but now three independent islands increases as time goes on. The islands never grow together, the valleysstay present even at large time.At this point we mention, that for the exponential distribution η = e − ω/a as noise term, there is no analytic solutionavailable at all. As last system we have to investigate the ODE of νf ′′ ( ω ) + f ′ ( ω ) (cid:20) ω λ f ′ ( ω ) (cid:21) + a ω = 0 . (16)Unfortunately, the general solution cannot again be given in a closed form. In the formal solution some integrals ofthe Heun functions remain. For positive and given parameters ν, λ , the solution becomes well-defined. As an examplefor a = 1 / , λ = 2 and ν = 1 / , the shape function reads f ( ω ) = − ω (cid:26)(cid:20) − c ωH C (cid:18) − , , , , , − ω (cid:19) + c H C (cid:18) − , − , , , , − ω (cid:19)(cid:21) / (cid:20) − ω H C (cid:18) − , − , , , , − ω (cid:19) H CP rime (cid:18) − , , , , , − ω (cid:19) +2 ω H CP rime (cid:18) − , − , , , , − ω (cid:19) H C (cid:18) − , , , , , − ω (cid:19) + ω H C (cid:18) − , − , , , , − ω (cid:19) H C (cid:18) − , , , , , − ω (cid:19) +2 ω H CP rime (cid:18) − , − , , , , − ω (cid:19) H C (cid:18) − , , , , , − ω (cid:19) − ω H C (cid:18) − , − , , , , − ω (cid:19) H CP rime (cid:18) − , , , , , − ω (cid:19) + H C (cid:18) − , − , , , , − ω (cid:19) H C (cid:18) − , , , , , − ω (cid:19) , (cid:21)(cid:27) (17)where H C and H CP rime means the Heun functions and the derivative of the Heun C function, respectively [30].Figure 14 shows the shape function for the Lorenzian noise term. The new feature compared to the former Gaussiannoise term is that the domain of the solution is just a finite interval. Just a single island is born at the beginning of thesurface growth process. The solution blows up (or blows down) on a finite one is the compact support of the solution.The last figure (Fig. 15) presents the final solution of the KPZ PDE. The solution has a compact support as well. Itmeans that the small island which was positioned at the origin just grows for large times, but cannot diffuse onto thewhole surface. 10 PREPRINT - A
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4, 2019Figure 12: Various shape functions of Eq.(15) for the parameter set ν = 1 / , n = 1 , c = 1 and c = 0 for different λ values. Black, red and blue lines are for a = 1 , λ = 25 , a = 0 . , λ = 25 , and a = 0 . , λ = 55 , respectively.Figure 13: The complete solution for the Gaussian noise for a = 0 . , λ = 55 value. Other parameters are unchanged. Conclusions
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 fourpower-law-type noise ω n with exponents of − , − , , , called the brown, pink, white and blue noise, respectively.Each integer exponent describes completely different dynamics. Additionally, we investigated the properties of Gaus-sian and Lorenzian noises. Providing completely dissimilar surfaces with growth dynamics. All solutions can bedescribed with non-trivial combinations of various special functions, like error, Whittaker, Kummer or Heun. The pa-rameter dependencies of the solutions are investigated and discussed. Future works are planned for the investigationsof the two dimensional surfaces.We also remark that applying transformations k = e λ ν u , k = t α m ( z ) and z = xt − β to equation (1), one gets thelinear ordinary differential equation m ′′ + zm ′ + ( λ ν tη − α ) m = 0 for any arbitrary value of α and β = 1 / . 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.11 PREPRINT - A
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4, 2019Figure 14: The shape function for the Lorenz noise for a = 1 / , λ = 2 , ν = 1 / and for c = c = 1 . Figure 15: The complete solution u ( x, t ) for the Lorenz noise with the parameters given above. References [1] A. Pimpinelli and J. Villain.
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