Stability of solutions of first-order impulsive partial differential equations

Theorems on stability and asymptotic stability of solutions of impulsive partial differential equations of first order are proved. These results are obtained via the method of differential inequalities and via the method of Lyapunov functions.     

Conformal invariance and integration of first-order differential equations

This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differential equations, and expresses the differential equations by the equations of a Birkhoff system or a generalized Birkhoff system. If the infinitesimal generators are those of a Noether symmetry, the conserved quantity can be obtained by using the Noether theory of the Birkhoff system or the generalized Birkhoff system.     

Solution multistability in first-order nonlinear differential delay equations

The dependence of solution behavior to perturbations of the initial function (IF) in a class of nonlinear differential delay equations (DDEs) is investigated. The structure of basins of attraction of multistable limit cycles is investigated. These basins can possess complex structure at all scales measurable numerically although this is not necessarily the case. Sensitive dependence of the asymptotic solution to perturbations in the initial function is also observed experimentally using a task specific electronic analog computer designed to investigate the dynamics of an integrable first-order DDE.     

A class of nonlinear relativistic partial differential equations in elementary particle theory

In mathematical approaches to elementary particle theory, the equation [² - ²/t²]=m² ;+g ³ has been of interest [1,2]; it describes a quartically self-coupled neutral scalar meson field. This paper applies the decomposition method [3-6] to obtain accurate non-perturbative timedevelopment of the field for this equation, or variations involving other nonlinear interactions, without the use of cutoff functions or truncations.     

Lie symmetry and conserved quantity of a system of first-order differential equations

This paper focuses on studying the Lie symmetry and a conserved quantity of a system of first-order differential equations. The determining equations of the Lie symmetry for a system of first-order differential equations, from which a kind of conserved quantity is deduced, are presented. And their general conclusion is applied to a Hamilton system, a Birkhoff system and a generalized Hamilton system. Two examples are given to illustrate the application of the results.     

Symmetry and first integrals of first-order system of differential equations

The defining equation is constructed for an algebra of infinitesimal Lie-symmetry-group operators of a first-order system of ordinary differential equations; some properties of the algebra are formulated; and the relation of the algebra with conservation laws is discussed. A similar approach is applied to a Hamilton system of general form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 53–56, February, 1977.     

Critical nonlinearities in partial differential equations

The paper is devoted to the role of critical nonlinearities in the framework of the theory of global solvability of nonlinear partial differential equations. In particular, a new approach to blow-up problems for solutions of nonlinear partial differential equations is suggested.     

Generalized symmetries of nonlinear partial differential equations

We propose a rigorous definition of the generalized infinitesimal symmetries using the notion of k-vector fields, and we derive an algorithm for their determination. We show that Bäcklund transformations between evolution equations are differential operators having prescribed symmetries.     

The Gibbs measures and partial differential equations

We investigate the connections of the Gibbs measures, which appear in Euclidean Field Theory, and the corresponding partial differential equations of Classical Euclidean Field Theory.     

Finite-dimensional behavior in dissipative partial differential equations

Dissipative partial differential equations have applications throughout the sciences: models of turbulence in fluids, chemical reactions, and morphogenesis in biology can all be written in a general form which allows them to be subjected to a unified analysis. Recent results on these equations show that in many cases they are not as complex as they initially appear, and can be converted into a set of ordinary differential equations. However, most of the relevant references present a bewildering array of terms which can obscure the simple underlying ideas. The main purpose of this paper is to introduce this terminology, motivated by several major results, slowly and by example. Detailed proofs are omitted, but it is hoped that this approach will give a good understanding of and intuitive feel for the subject without recourse to technicalities. Nevertheless, sufficient mathematical detail is included to allow application of these results to many examples. (c) 1995 American Institute of Physics.     

Spontaneous chiral symmetry-breaking and partial differential equations

Using a modified normal-product quantization as suggested by Lowenstein and Zimmermann which incorporates B. Lee's resummation problem from the very beginning into the BPHZ renormalization, we discuss the derivation of partial differential equations of the Callan-Symanzik type. In addition to the renormalization group and the Callan-Symanzik equation which are the appropriate differential-equations for the high energy limit (the overall zero-mass preasymptotic theory), we find a third partial differential equation which allows discussion of the spontaneous broken symmetry limit m_π→0 to all orders of the new perturbation theory. In this treatment there is no necessity for loop summation; all the interesting questions are delegated to ordinary perturbation problems.     

Whitney singularities of solutions of partial differential equations

We classify singularities of multiple-valued solutions of higher order systems of partial differential equations in the case when projections of the corresponding integral manifolds of the Cartan distribution in the jet space to the space of independent variables have Whitney singularities.     

Painlevé classification of semilinear partial differential equations

The Painlevé-type classification of first degree second-order polynomial PDEs is performed.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.     

Reduction of systems of first-order differential equations via Λ-symmetries

The notion of λ-symmetries, originally introduced by C. Muriel and J.L. Romero, is extended to the case of systems of first-order ODEs (and of dynamical systems in particular). It is shown that the existence of a symmetry of this type produces a reduction of the differential equations, restricting the presence of the variables involved in the problem. The results are compared with the case of standard (i.e. exact) Lie-point symmetries and are also illustrated by some examples.     

Probabilistic models for stochastic elliptic partial differential equations

Mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively. Yet, additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities, referred to as physical requirements, have not been examined systematically.     

Reparameterization for statistical state estimation applied to differential equations

The ensemble Kalman filter is a widely applied data assimilation technique useful for improving the forecast of computational models. The main computational cost of the ensemble Kalman filter comes from the numerical integration of each ensemble member forward in time. When the computational model involves a partial differential equation, the degrees of freedom of the solution in the discretization of the spatial domain are oftentimes used for the representation of the state of the system, and the filter is applied to this state vector. We propose a method of approximating the state of a partial differential equation in a representation space developed separately from the numerical method. This representation space represents a reparameterization of the state vector and can be chosen to retain desirable physical features of the solutions. We apply the ensemble Kalman filter to this representation of the state, and numerically demonstrate that acceptable results are obtained with substantially smaller ensemble sizes.