A characterization of complete integrability for partial differential equations of first order

We give a characterization of the notion of complete integrability for overdetermined systems of first order partial differential equations of real valued functions.Dedicated to the memory of Professor Masahisa Adachi     

Non-linear partial differential equations via vector fields on homogeneous Banach manifolds

The vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in this paper. From Banach Lie groups associated with the MKdV hierarchy of differential equations, we derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields. In the same way, we obtain homogeneous Banach manifolds of solutions to the sine-Gordon equation. The scattering and inverse scattering maps in this set-up are also discussed.     

Radial solutions of a class of iterated partial differential equations

We give some expansion formulas and the Kelvin principle for solutions of a class of iterated equations of elliptic type     

Solitary wave and other solutions for nonlinear heat equations

A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.     

Variational solutions for a class of fractional stochastic partial differential equations

In this Note we present new results regarding the existence, the uniqueness and the equivalence of two notions of variational solution related to a class of non autonomous, semilinear, stochastic partial differential equations defined on an open bounded domain $D?{\mathbb{R}}^d$. The equations we consider are driven by an infinite-dimensional noise derived from an $L^2(D)$-valued fractional Wiener process $W^{\mathrm{H}}$ with Hurst parameter $\mathrm{H}\in (\frac{1}{\gamma +1}\text{,}1)$, where $\gamma \in (0\text{,}1]$ denotes the Hölder exponent of the derivative of the nonlinearity that appears in the stochastic term. To cite this article: D. Nualart, P.-A. Vuillermot, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Exact travelling wave solutions for a class of nonlinear partial differential equations

In this paper, the tanh-method is improved by means of a proper rational transformation based upon a coupled projective Riccati equations. The ansatz can be applied to find more and new exact solutions of the partial differential equations with the aid of symbolic computation system, Maple. We choose an example, which includes φ⁴ equation, Klein–Gordon equation, Duffing equation, Landau–Ginburg–Higgs equation and Sine–Gordon equation, to illustrate the method.     

C n -almost automorphic solutions for partial neutral functional differential equations

In this work, we study the existence of C ⁿ -almost periodic solutions and C ⁿ -almost automorphic solutions (n?≥?1), for partial neutral functional differential equations. We prove that the existence of a bounded integral solution on ?⁺ implies the existence of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions. When the exponential dichotomy holds for the homogeneous linear equation, we show the uniqueness of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions.     

Stability of the analytic and numerical solutions for impulsive differential equations

This paper deals with the stability analysis of the analytic and numerical solutions of impulsive differential equations. In particular, the linear equation with variable coefficients and the nonlinear equation are considered. The stability conditions of the analytic solutions of these impulsive differential equations and the numerical solutions of the θ-methods are obtained. Finally, some numerical experiments are given.     

Existence of solutions of a class of multivalued partial differential equations

Conditions are given for the existence of solutions and the compactness of the set of solutions of the Darboux problem for the differential inclusion     

12.

Approximate solutions of abstract differential equations     

Emil Vitásek 《Applications of Mathematics》2007,52(2):171-183

The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

13.

Conditions for positivity of solutions of a class of dissipative partial differential equations     

M.V. Bartuccelli  S.A. Gourley  C.J. Woolcock 《Applied Mathematics Letters》1998,11(6):71-74

It is a difficult problem to establish useful results on positivity of solutions of semilinear dissipative partial differential equations containing differential operators of order higher than that of the Laplacian. Positivity results are of importance in some fields of applied mathematics, such as in mathematical biology, where positivity of solutions is often needed because of the modelling application. In this paper, we first obtain ladder estimates for a particular class of semilinear dissipative partial differential equations, through which one can find conditions which will at least ensure that solutions are eventually (i.e., asymptotically) positive.     

14.

On very weak entire solutions of nonlinear partial differential equations     

Alberto Fiorenza 《Annali di Matematica Pura ed Applicata》1998,175(1):211-221

Let us consider the variational equation in R ⁿ

Approximate solutions of abstract differential equations

The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Conditions for positivity of solutions of a class of dissipative partial differential equations

It is a difficult problem to establish useful results on positivity of solutions of semilinear dissipative partial differential equations containing differential operators of order higher than that of the Laplacian. Positivity results are of importance in some fields of applied mathematics, such as in mathematical biology, where positivity of solutions is often needed because of the modelling application. In this paper, we first obtain ladder estimates for a particular class of semilinear dissipative partial differential equations, through which one can find conditions which will at least ensure that solutions are eventually (i.e., asymptotically) positive.     

On very weak entire solutions of nonlinear partial differential equations

Let us consider the variational equation in R ⁿ

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).

Representation formulae and inequalities for solutions of a class of second order partial differential equations

Let be a possibly degenerate second order differential operator and let be its fundamental solution at ; here is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of on to satisfy the representation formula

We prove that (R) holds provided is superlinear, without any assumption on the behavior of at infinity. On the other hand, if satisfies the condition

then (R) holds with no growth assumptions on .

     

Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential

We consider Hamiltonian partial differential equations u_tt +|∂_x|u+ σu = f(u), x ∈ T, t ∈?, with periodic boundary conditions, where f(u) is a real-analytic function of the form f(u) = u⁵ + o(u⁵) near u = 0, σ ∈ (0, 1) is a fixed constant, and \mathbb{T}=?/\mathrm{2}\pi \mathbb{Z}T= R/2πZ. A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with σ ∈ (0, 1)\ ?. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.