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Suppression of unbounded gradients in a SDE associated with the Burgers equation  [PDF]
Sergio Albeverio,Olga Rozanova
Mathematics , 2008,
Abstract: We consider the Langevin equation describing a stochastically perturbed by uniform noise non-viscous Burgers fluid and introduce a deterministic function that corresponds to the mean of the velocity when we keep the value of position fixed. We study interrelations between this function and the solution of the non-perturbed Burgers equation. Especially we are interested in the property of the solution of the latter equation to develop unbounded gradients within a finite time. We study the question how the initial distribution of particles for the Langevin equation influences this blowup phenomenon. It is shown that for a wide class of initial data and initial distributions of particles the unbounded gradients are eliminated. The case of a linear initial velocity is particular. We show that if the initial distribution of particles is uniform, then the mean of the velocity for a given position coincides with the solution of the Burgers equation and in particular does not depend on the constant variance of the stochastic perturbation. Further, for a one space space variable we get the following result: if the decay rate of the power-behaved initial particles distribution at infinity is greater or equal $|x|^{-2},$ then the blowup is suppressed, otherwise, the blowup takes place at the same moment of time as in the case of the non-perturbed Burgers equation.
Linearizability of the Perturbed Burgers Equation  [PDF]
R. A. Kraenkel,J. G. Pereira,E. C. de Rey Neto
Physics , 1997, DOI: 10.1103/PhysRevE.58.2526
Abstract: We show in this letter that the perturbed Burgers equation $u_t = 2uu_x + u_{xx} + \epsilon ( 3 \alpha_1 u^2 u_x + 3\alpha_2 uu_{xx} + 3\alpha_3 u_x^2 + \alpha_4 u_{xxx} )$ is equivalent, through a near-identity transformation and up to order \epsilon, to a linearizable equation if the condition $3\alpha_1 - 3\alpha_3 - 3/2 \alpha_2 + 3/2 \alpha_4 = 0$ is satisfied. In the case this condition is not fulfilled, a normal form for the equation under consideration is given. Then, to illustrate our results, we make a linearizability analysis of the equations governing the dynamics of a one-dimensional gas.
Spontaneously generated waves and obstacles to integrability in perturbed evolution equations  [PDF]
Alex Vekser,Yair Zarmi
Physics , 2005,
Abstract: The perturbed Burgers and KdV equations are considered. Often, the perturbation excites waves that are different from the solution one is seeking. In the case of the Burgers equation, the spontaneously generated wave is also a solution of the equation. In contrast, in the case of the KdV equation, this wave is constructed from new (non-KdV) solitons that undergo an elastic collision around the origin. Their amplitudes have opposite signs, which they exchange upon collision. The perturbation then contains terms, which represent coupling between the solution and these spontaneously generated waves. Whereas the unperturbed equations describe gradient systems, these coupling terms may be non-gradient. In that case, they turn out to be obstacles to asymptotic integrability, encountered in the analysis of the solutions of the perturbed evolution equations.
CONDITIONAL STABILITY OF THE GENERAL KdV SOLITON SOLUTION AND KdV-BURGERS TRAVELING WAVE SOLUTION
广义KdV孤子解和广义KdV-Burgers行波解的条件稳定性

LI CHAO-HONG,
李朝红

物理学报 , 1998,
Abstract: For a general KdV soliton solution and a general KdV-Burgers traveling wave solution,the direct perturbation method is used to construct their general perturbed corrections.Theoretical analysis reveals that the solution possesses conditional stability,namely,their stability depends sensitively on the physical parameters and initial conditions of the corresponding system.The results have extended and corrected some assertions of instability in the recent articles.
Wave interactions and the analysis of the perturbed Burgers equation  [PDF]
Alex Veksler,Yair Zarmi
Physics , 2005, DOI: 10.1016/j.physd.2005.08.001
Abstract: In multiple-front solutions of the Burgers equation, all the fronts, except for two, are generated through the inelastic interaction of exponential wave solutions of the Lax pair associated with the equation. The inelastically generated fronts are the source of two difficulties encountered in the standard Normal Form expansion of the approximate solution of the perturbed Burgers equation, when the zero-order term is a multiple-front solution: (i) The higher-order terms in the expansion are not bounded; (ii) The Normal Form (equation obeyed by the zero-order approximation) is not asymptotically integrable; its solutions lose the simple wave structure of the solutions of the un-perturbed equation. The freedom inherent in the Normal Form method allows a simple modification of the expansion procedure, making it possible to overcome both problems in more than one way. The loss of asymptotic integrability is shifted from the Normal Form to the higher-order terms (part of which has to be computed numerically) in the expansion of the solution. The front-velocity update is different from the one obtained in the standard analysis.
EXPLICIT SOLUTION OF BURGERS’ AND GENERALIZED BURGERS’ EQUATION USING HOMOTOPY PERTURBATION METHOD  [PDF]
Miral Bhagat,P. H. Bhathawala,Kaushal Patel
International Journal of Advances in Engineering and Technology , 2013,
Abstract: In recent years, many more of the numerical methods were used to solve a wide range of mathematical, physical and engineering problems, linear and nonlinear. In this article, Homotopy Perturbation Method (HPM) is employed to approximate the solution of the Burgers’ equation which is a one-dimensional non-linear differential equation in fluid dynamics. The explicit solution of the Burgers’ equation was obtained and compared with the exact solutions. We take the cases where the exact solution was not available for viscosity smaller than 0.01, we apply the HPM structure for obtaining the explicit solution. The results reveal that the HPM is very effective, convenient and quite accurate to partial differential equation.
A DIRECT PERTURBATION THEORY OF THE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

CHEN SHI-RONG,CHEN ZHI-DE,YUAN XIAN-ZHANG,HUANG NIAN-NING,

中国物理 B , 1999,
Abstract: An exact direct perturbation theory of nonlinear Schrodinger equation with corrections is developed under the condition that the initial value of the perturbed solution is equal to the value of an exact multisoliton solution at a particular time. After showing the squared Jost functions are the eigenfunctions of the linearized operator with a vanishing eigenvalue,suitable definitions of adjoint functions and inner product are introduced. Orthogonal relations are derived and the expansion of the unity in terms of the squared Jost functions is naturally implied. The completeness of the squared Jost functions is shown by the generalized Marchenko equation. As an example,the evolution of a Raman loss compensated soliton in an optical fiber is treated.
On the Exact Solution of Burgers-Huxley Equation Using the Homotopy Perturbation Method  [PDF]
S. Salman Nourazar, Mohsen Soori, Akbar Nazari-Golshan
Journal of Applied Mathematics and Physics (JAMP) , 2015, DOI: 10.4236/jamp.2015.33042
Abstract: The Homotopy Perturbation Method (HPM) is used to solve the Burgers-Huxley non-linear differential equations. Three case study problems of Burgers-Huxley are solved using the HPM and the exact solutions are obtained. The rapid convergence towards the exact solutions of HPM is numerically shown. Results show that the HPM is efficient method with acceptable accuracy to solve the Burgers-Huxley equation. Also, the results prove that the method is an efficient and powerful algorithm to construct the exact solution of non-linear differential equations.
Assessment of He`s Homotopy Perturbation Method in Burgers and Coupled Burgers` Equations  [PDF]
Abdoul R. Ghotbi,A. Avaei,A. Barari,M.A. Mohammadzade
Journal of Applied Sciences , 2008,
Abstract: Various methods for finding explicit solution to nonlinear evolution equations have been proposed in this letter Homotopy Perturbation Method (HPM) is employed for solving Korteweg-de Vries-Burgeres (KdVB) equation and coupled Burgers` equations which both of them are very applicable in mathematics, physics and engineering. The final results obtained by means of HPM are compared with those results obtained from the exact solution and the Adomian Decomposition Method (ADM). The comparison shows a precise agreement between the results and introduces this new method as an applicable one which it needs less computations and is much easier and more convenient than others, so it can be widely used in engineering too.
Fronts in randomly advected and heterogeneous media and nonuniversality of Burgers turbulence: Theory and numerics  [PDF]
Jackson R. Mayo,Alan R. Kerstein
Physics , 2007, DOI: 10.1103/PhysRevE.78.056307
Abstract: A recently established mathematical equivalence--between weakly perturbed Huygens fronts (e.g., flames in weak turbulence or geometrical-optics wave fronts in slightly nonuniform media) and the inviscid limit of white-noise-driven Burgers turbulence--motivates theoretical and numerical estimates of Burgers-turbulence properties for specific types of white-in-time forcing. Existing mathematical relations between Burgers turbulence and the statistical mechanics of directed polymers, allowing use of the replica method, are exploited to obtain systematic upper bounds on the Burgers energy density, corresponding to the ground-state binding energy of the directed polymer and the speedup of the Huygens front. The results are complementary to previous studies of both Burgers turbulence and directed polymers, which have focused on universal scaling properties instead of forcing-dependent parameters. The upper-bound formula can be heuristically understood in terms of renormalization of a different kind from that previously used in combustion models, and also shows that the burning velocity of an idealized turbulent flame does not diverge with increasing Reynolds number at fixed turbulence intensity, a conclusion that applies even to strong turbulence. Numerical simulations of the one-dimensional inviscid Burgers equation using a Lagrangian finite-element method confirm that the theoretical upper bounds are sharp within about 15% for various forcing spectra (corresponding to various two-dimensional random media). These computations provide a new quantitative test of the replica method. The inferred nonuniversality (spectrum dependence) of the front speedup is of direct importance for combustion modeling.
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