Closed form Solutions to Some Nonlinear equations by a Generalized Cole-Hopf Transformation
Mayer Humi
备注：
The paper is a merger of arXiv:1308.5965 and arXiv:1308.0858 with a new section in which I apply the generalized Cole-Hopf transformation to find analytical solutions to Painleve III equation
In the first part of this paper we linearize and solve the Van der Pol and Lienard equations with some additional nonlinear terms by the application of a generalized form of Cole-Hopf transformation. We then show that the same transformation can be used to linearize Painleve III equation for certain combinations of its parameters. Finally we linearize new forms of Burger's and related convective equations with higher order nonlinearities.
Differential equations with convective terms such as the Burger's equation appear in many applications and have been the subject of intense research. In this paper we use a generalized form of Cole-Hopf transformation to relate the solutions of some of these nonlinear equations to the solutions of linear equations. In particular we consider generalized forms of Burger's equation and second order nonlinear ordinary differential equations with convective terms which can represent steady state one-dimensional convection.
In this short letter we show that the one dimensional non linear Schrödinger equation(NLS)can be solved by a Hopf-Cole transformation which converts it to the Burgers equation in turbulence.
In this paper we propose some linearizability tests of partial difference equations on a quad-graph given by one point, two points and generalized Hopf-Cole transformations. We apply the so obtained tests to a set of nontrivial examples.
We introduce a hybrid Cole-Hopf-Darboux transformation to relate solutions of nonlinear and linear second order differential equations and derive a sufficient condition for this correspondence. In particular we show that solutions of some nonlinear second order equations are related to the special functions of mathematical physics through this transformation. These nonlinear equations can be viewed as the "class of special nonlinear equations" which correspond to the linear differential equations which define the special functions of mathematical physics.
In case of the KP hierarchy where the dependent variable takes values in an (arbitrary) associative algebra A, it is known that there are solutions which can be expressed in terms of quasideterminants of a Wronski matrix which solves the linear heat hierarchy. We obtain these solutions without the help of quasideterminants in a simple way via solutions of matrix KP hierarchies (over A) and by use of a Cole-Hopf transformation. For this class of exact solutions we work out a correspondence with 'weakly nonassociative' algebras.
We consider systems, which conserve the particle number and are described by Schrödinger equations containing complex nonlinearities. In the case of canonical systems, we study their main symmetries and conservation laws. We introduce a Cole-Hopf like transformation both for canonical and noncanonical systems, which changes the evolution equation into another one containing purely real nonlinearities, and reduces the continuity equation to the standard form of the linear theory. This approach allows us to treat, in a unifying scheme, a wide variety of canonical and noncanonical nonlinear syste…
Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a Frenet null frame by a gauge transformation that belongs to the little group contained in the Lorentz group $SO(2,1)$ and provides an analog of the Hasimoto transformation (relating a parallel frame to a Frenet frame for curves in Euclidean space). The Cartan structure equations of the transformed frame are shown to encode a hereditary recursion operator givin…
We consider the stochastic heat equation $\partial_tZ= \partial_x^2 Z - Z \dot W$ on the real line, where $\dot W$ is space-time white noise. $h(t,x)=-\log Z(t,x)$ is interpreted as a solution of the KPZ equation, and $u(t,x)=\partial_x h(t,x)$ as a solution of the stochastic Burgers equation. We take $Z(0,x)=\exp\{B(x)\}$ where $B(x)$ is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist $0< c_1\le c_2 <\infty$ such that $c_1t^{2/3}\le \Var (\log Z(t,x))\le c_2 t^{2/3}.$ Analogous results are obtained for some mom…
In this paper, we show that a variant of the discrete Burgers equation can be obtained through the Cole--Hopf transformation to a generalized discrete diffusion equation corresponding to the correlated random walk, which is also known as a generalization of the well known random walk. By applying the technique called ultradiscretization, we obtain the generalized ultradiscrete diffusion equation, the ultradiscrete Cole--Hopf transformation and a variant of the ultradiscrete Burgers equation. Moreover, we show that the resulting ultradiscrete Burgers equation yields cellular automata which can …
Miura transform is known as the transformation between Korteweg de-Vries equation and modified Korteweg de-Vries equation. Its formal similarity to the Cole-Hopf transform has been noticed. This fact sheds light on the logarithmic type transformations as an origin of certain kind of nonlinearity in the soliton equations. In this article, based on the logarithmic representation of operators in infinite-dimensional Banach spaces, a structure common to both Miura and Cole-Hopf transforms is discussed. In conclusion, the Miura transform is generalized as the transform in abstract Banach spaces, an…
We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a "universal" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transfo…
We show that a generalized Asymmetric Exclusion Process called ASEP(q,j) introduced by Carinci, Giardina, Redig and Sasamoto converges to the Cole-Hopf solution to the KPZ equation under weak asymmetry scaling.
We study infinitesimal gauge transformations of an equivariant noncommutative principal bundle as a braided Lie algebra of derivations. For this, we analyse general $K$-braided Hopf and Lie algebras, for $K$ a (quasi)triangular Hopf algebra of symmetries, and study their representations as braided derivations. We then study Drinfeld twist deformations of braided Hopf algebras and of Lie algebras of infinitesimal gauge transformations. We give examples coming from deformations of abelian and Jordanian type. In particular we explicitly describe the braided Lie algebra of gauge transformations of…
We linearize and solve the Van der Pol equation (with additional nonlinear terms) by the application of a generalized form of Cole-Hopf transformation. We classify also Lienard equations with low order polynomial coefficients which can be linearized by this transformation.
A hierarchy of normalized classes of generalized Burgers equations is studied. The equivalence groupoids of these classes are computed. The equivalence groupoids of classes of linearizable generalized Burgers equations are related to those of the associated linear counterparts using the Hopf--Cole transformation.
We show that one can define through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete Cole-Hopf transformation. If the equation is linearizable the symmetry provides the linearizing transformation. At the end we present few examples of applications for equations defined on four lattice points.
In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of fractional Darboux transformations which can be recognized also as generalized Cole-Hopf transformations. We apply these transformations to find Schrodinger equations with isospectral potentials and to the linearization of some new classes of nonlinear partial differential equations.
This work introduces a new notion of solution for the KPZ equation, in particular, our approach encompasses the Cole-Hopf solution. We set in the context of the distribution theory the proposed results by Bertini and Giacomin from the mid 90's. This new approach provides a pathwise notion of solution as well as a structured approximation theory. The developments are based on regularization arguments from the theory of distributions.
From a 'discrete' functional zero curvature equation, functional representations of (matrix) Burgers and potential KP (pKP) hierarchies (and others), as well as corresponding Backlund transformations, can be obtained in a surprisingly simple way. With their help we show that any solution of a Burgers hierarchy is also a solution of the pKP hierarchy. Moreover, the pKP hierarchy can be expressed in the form of an inhomogeneous Burgers hierarchy. In particular, this leads to an extension of the Cole-Hopf transformation to the pKP hierarchy. Furthermore, these hierarchies are solved by the soluti…
Admissible point transformations between Burgers equations with linear damping and time-dependent coefficients are described and used in order to exhaustively classify Lie symmetries of these equations. Optimal systems of one- and two-dimensional subalgebras of the Lie invariance algebras obtained are constructed. The corresponding Lie reductions to ODEs and to algebraic equations are carried out. Exact solutions to particular equations are found. Some generalized Burgers equations are linearized to the heat equation by composing equivalence transformations with the Hopf-Cole transformation.
We compare two ways of force terms generating in the model of nonrelativistic particle moving in the presence of constant field force $\vec{F}$. First of them uses the twist-deformed acceleration-enlarged Newton-Hooke quantum space-times while the second one incorporates the doubly enlarged Newton-Hooke transformations of classical space. Particulary, we find the conditions for which the both treatments provide the same force terms.
This paper is devoted to studying the Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which is not well-defined in the classical sense and shall be understood by using paracontrolled distribution method introduced in \cite{GIP15}. By a new characterization of weighted Hölder space and Zvonkin's transformation we prove some new a priori estimates, and therefore, establish the global well-posedness for singular HJB equations. As an application, the global well-posedness for KPZ equations on the real line in polynomial weighted Hölder spaces is obtained without using Cole…
