Evolution of pattern formation under ion bombardment of substrate

Kudryashov, Nikolay A.; Ryabov, Pavel N.; Fedyanin, Timofey E.; Кутуков, А. А.

doi:10.1016/j.physleta.2013.01.007

Cited by 25 publications

(11 citation statements)

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“…Step 3: Substituting Equation (10) into Equation ( 7) along with Equation (11), we arrive at a polynomial in Q i−j , (i, j = 0, 1, 2, . .…”

Section: Description Of the Generalized Kudryashov Methodsmentioning

confidence: 99%

“…There are many robust, stable, and effective methods that have been developed for constructing exact, approximate analytical and numerical solutions for NPDEs. Particularly, the methods have been extensively used to find exact solutions for NPDEs, such as the (G /G, 1/G)-expansion method [1], the modified G /G 2 -expansion method [2], the Jacobi elliptic equation method [3], the sine-Gordon expansion method [4,5], the extended direct algebraic method [6,7], the modified Exp-function method [8,9], and the Kudryashov method [10,11]. However, the focus of this work is to search for exact solutions of certain nonlinear partial integro-differential equations (PIDEs) converted into NPDEs in some ways.…”

Section: Introductionmentioning

confidence: 99%

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Application of the Exp-Function and Generalized Kudryashov Methods for Obtaining New Exact Solutions of Certain Nonlinear Conformable Time Partial Integro-Differential Equations

2021

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The key objective of this paper is to construct exact traveling wave solutions of the conformable time second integro-differential Kadomtsev–Petviashvili (KP) hierarchy equation using the Exp-function method and the (2 + 1)-dimensional conformable time partial integro-differential Jaulent–Miodek (JM) evolution equation utilizing the generalized Kudryashov method. These two problems involve the conformable partial derivative with respect to time. Initially, the conformable time partial integro-differential equations can be converted into nonlinear ordinary differential equations via a fractional complex transformation. The resulting equations are then analytically solved via the corresponding methods. As a result, the explicit exact solutions for these two equations can be expressed in terms of exponential functions. Setting some specific parameter values and varying values of the fractional order in the equations, their 3D, 2D, and contour solutions are graphically shown and physically characterized as, for instance, a bell-shaped solitary wave solution, a kink-type solution, and a singular multiple-soliton solution. To the best of the authors’ knowledge, the results of the equations obtained using the proposed methods are novel and reported here for the first time. The methods are simple, very powerful, and reliable for solving other nonlinear conformable time partial integro-differential equations arising in many applications.

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“…Step 3: Substituting Equation (10) into Equation ( 7) along with Equation (11), we arrive at a polynomial in Q i−j , (i, j = 0, 1, 2, . .…”

Section: Description Of the Generalized Kudryashov Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of the Exp-Function and Generalized Kudryashov Methods for Obtaining New Exact Solutions of Certain Nonlinear Conformable Time Partial Integro-Differential Equations

2021

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“…The partial integro-differential Equations ( 1)-( 2) is hard to analyze. In the works (Makeev et al, 2002;Park et al, 1999;Kahng et al, 2001;Feix et al, 2005;Kudryashov et al, 2013;Kudryashov and Ryabov, 2014) the authors reduce this equation to the stochastic nonlinear partial differential equation of the fourth or sixth order. At that almost all theories are based on the Sigmund's (1981) sputtering theory, according to which the ion depositing energy is approximated by the Gaussian distribution.…”

Section: Introductionmentioning

confidence: 99%

Statistical simulation of pattern formation by ion-beam sputtering

Kudryashov

Skachkov

2015

Multidiscipline Modeling in Materials and Structures

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Users who downloaded this article also downloaded: (2015),"Effect of phase-lags on Rayleigh wave propagation in thermoelastic medium with mass diffusion", Multidiscipline Modeling in Materials and Structures, Vol. 11 Iss 4 pp. 474-493 http:// dx.(2015),"The effect of negative capacitance in varistor structure on the basis of its models with voltage drop on the intergranular interlayer", Multidiscipline Modeling in Materials and Structures, Vol. 11 Iss 4 pp. 598-615 http://dx.If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services.Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. AbstractPurpose -The purpose of this paper is to investigate the influence of ion-flow parameters on surface topography and making numerical simulation at the times when the process of surface erosion becomes strongly nonlinear. Design/methodology/approach -The base of the mathematical model of target ion-sputtering is nonlinear evolutionary equation in which the erosion velocity dependence on ion flux is evaluated by means of a Monte Carlo method. The difference between this equation and the one of continuum theory is that the ion flux is not smooth function. Instead, it is a set of separate incident ions. Findings -Some simulations with using independent random points of arrival for the incident ions leads to results uncorrelated with the continuum model at early times. The ripples are not quite developed or observed. This phenomenon is explained by random fluctuations of the target sputtering depth. Sufficiently big values of the random fluctuations destroy the ripple structure on target surface. The simulation with using equally distributed sequence (Holton sequence) of points of arrival for the incident ions leads to results well correlated with the continuum model. Originality/value -The discrete model which goes into the equation of continuum theory within the appropriate asymptotic limit has been proposed. The discretization parameters influence on surface morphology formation has been studied. This paper may be interesting to researchers making the theoretical and numerical analysis of pattern formation on plane target surfaces undergoing ion-beam sputtering.

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“…In the literature, many of these methods among which we find are: the Hirotas bilinear method [19], the Backlund transformation method [20], the Darboux transformation method [21], the Painleve singularity structure analysis method [22], the Riccati expansion with constant coefficients [23], the variational iteration method [24], the exp-function method [25], the algebraic method [26], the collocation method [27], the Kudryashov method [28][29][30][31][32], the (G /G)-expansion method [33][34][35][36][37][38], the simplest equation method [39][40][41][42][43], and so on. However, some of these analytical methods are not easy to handle and are often subject to tedious mathematical developments.…”

mentioning

confidence: 99%