Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced which consequently give rise to explicit solutions. Potential symmetries, which are realized as local symmetries of a related auxiliary system, are obtained. For some special nonlinearities new symmetry reductions and exact solutions are derived by using the nonclassical method.
It is shown that the novel Lie group of transformations method is a competent and prominent tool in solving nonlinear partial differential equations(PDEs) in mathematical physics. Lie group analysis is used to carry out the similarity reduction and exact solutions of the (3 + 1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. This research deals with the similarity solutions of CBS equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reduction for the CBS equation. For the different Lie algebra, Lie symmetry method r…
We present a complete description of the similarity solutions $u_α(x,t)=t^{-α/2}f(\Vert x \Vert/\sqrt{t};α)$ for the following nonlinear diffusion equation $$ u_{t}+γ\vert u_{t} \vert =Δu\qquad(-1<γ<1) $$ The behaviors of these solutions are obtained through the explicit representation of $f(η;α)$, in terms of Kummer and Tricomi functions. Considering results about confluent hypergeometric functions, new methods to describe asymptotic and oscillatory behaviors of the similarity solutions are obtained. We prove that there exists an increasing and unbounded sequence of positive similarity expone…
We investigate the large-time asymptotics of nonlinear diffusion equations $u_t = Δu^p$ in dimension $n \ge 1$, in the exponent interval $p > n/(n+2)$, when the initial datum $u_0$ is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative Rényi entropy are demonstrated for finite-mass solutions defined in the whole space when they are re-normalized at each time $t> 0$ with respect to their own second moment. The analysis shows that the relative Rényi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston-…
This paper is concerned with the transient dynamics described by the solutions of the reaction-diffusion equations in which the reaction term consists of a combination of a superlinear power-law absorption and a time-independent point source. In one space dimension, solutions of these problems with zero initial data are known to approach the stationary solution in an asymptotically self-similar manner. Here we show that this conclusion remains true in two space dimensions, while in three and higher dimensions the same conclusion holds true for all powers of the nonlinearity not exceeding the S…
Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers' equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self-similar variables. Furthermore, by a diffusion approximation, the long time convergence to the self-similar solution is proved in the sense of distribution.
In this article we obtain exact solutions of (2+1)-dimensional Boiti-Leon-Pempinelli system of nonlinear partial differential equations which describes the evolution of horizontal velocity component of water waves propagating in two directions. We perform the Lie symmetry analysis to the given system and construct one-dimensional optimal subalgebra which involves some arbitrary functions of spatial variables. Several new exact solutions are obtained by symmetry reduction using each of the optimal subalgebra. We then study the physical behavior of some exact solutions by numerical simulations a…
We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite limit along any ray as $|x| \to \infty$. Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solut…
In this paper, we developed the group analysis to the time Fractional Porous Medium Equation (FPME) and the time Fractional Dual Porous Medium Equation (FDPME). The symmetry groups and the corresponding optimal systems of these two equations are obtained. Based on the above results, similarity reductions are performed and some explicit group invariant solutions are constructed.
The Michaelis-Menten mechanism is probably the best known model for an enzyme-catalyzed reaction. For spatially homogeneous concentrations, QSS reductions are well known, but this is not the case when chemical species are allowed to diffuse. We will discuss QSS reductions for both the irreversible and reversible Michaelis-Menten reaction in the latter case, given small initial enzyme concentration and slow diffusion. Our work is based on a heuristic method to obtain an ordinary differential equation which admits reduction by Tikhonov-Fenichel theory. We will not give convergence proofs but we …
We present a class of one-dimensional systems of nonlinear parabolic equations for which long-time phase dynamics can be described by an ODE with a Lipschitz vector field in R^n. In the considered case of the Dirichlet boundary value problem sufficient conditions for a finite-dimensional reduction turn out to be much wider than the known conditions of this kind for a periodic situation.
Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers' equation driven by some stochastic forced is studied. Under…
We consider general models of coupled reaction-diffusion systems for interacting variants of the same species. When the total population becomes large with intensive competition, we prove that the frequencies (i.e. proportions) of the variants can be approached by the solution of a simpler reaction-diffusion system, through a singular limit method and a relative compactness argument. As an example of application, we retrieve the classical bistable equation for Wolbachia's spread into an arthropod population from a system modeling interaction between infected and uninfected individuals.
For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x,y,z. In this paper, the Clarkson-Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.
We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa-Holm (mCH) equation and Novikov's equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov's equation to a negative mKdV flow and a negative Sawada-Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlevé type: for the mCH equation, its reduction is of second ord…
This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted en…
The self-similarity in space and time (hereafter self-similarity), either deterministic or statistical, is characterized by similarity exponents and a function of scaled variable, called the scaling function. In the present paper, we address mainly the self-similarity in the limit of early stage, as~opposed to the latter one, and also consider the scaling functions that decay or grow algebraically, as~opposed to the rapidly decaying functions such as Gaussian or error function. In particular, in~the case of simple diffusion, our symmetry analysis shows a mathematical mechanism by which the rap…
We establish a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat equation of the form $(-Δ)^{β/2}(\log u)\leq C/t$, where $β\in (0,2)$. We also illustra…
Reduction operators of generalized Burgers equations are studied. A connection between these equations and potential fast diffusion equations with power nonlinearity -1 via reduction operators is established. Exact solutions of generalized Burgers equations are constructed using this connection and known solutions of the constant-coefficient potential fast diffusion equation.
A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in $n>1$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity red…
A detailed analysis of the invariant point transformations for the first four partial differential equations which belong to the Complex Burgers` Hierarchy is performed. Moreover, a detailed application of the reduction process through the Lie point symmetries is presented while we construct similarity solutions. We conclude that the differential equations of our consideration are reduced to first-order equations such as the Abel, Riccati and to a linearisable second-order differential equation by using similarity transformations.
We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this study is that from the application of the similarity transformations provided by the Lie point symmetries all the members of the family of the partial differential equations are reduced to second-order differential equations which are maximal symmetric and can be linearized.
A general formula in closed form to obtain exact similarity solutions of the Fokker-Planck equation with both time-dependent drift and diffusion coefficients was recently presented by Lin and Ho [ Ann. Phys. \textbf{327}, 386 (2012); J. Math. Phys. \textbf{54}, 041501 (2013)]. In this paper we extend the class of exact solutions by exploiting certain properties of the general formula.
