A polynomial–exponential equation related to the Ramanujan–Nagell equation

For any given odd prime p and a fixed positive integer D prime to p, we study the equation \(x^2+D^m=p^n\) in positive integers x, m and n. We use a classical work of Dem’janenko in 1965 on a certain quadratic Diophantine equation together with some results concerning the existence of primitive divisors of Lucas sequences to examine our equation when D is a product of \(p-1\) and a square.     

Analogues to Fermat primes related to Pell��s equation

The coefficients of some weight 3 modular forms give reason to study primes of the form p = 2x ² ? 1 = 2dy ² + 1. If x _a , y _a are the positive solutions of Pell??s equation x ² ? dy ² = 1, given by ${x_a + y_a \sqrt{d} = (x_1 + y_1 \sqrt{d})^a}The coefficients of some weight 3 modular forms give reason to study primes of the form p = 2x ² − 1 = 2dy ² + 1. If x _a , y _a are the positive solutions of Pell’s equation x ² − dy ² = 1, given by x_a + y_a ?d = (x₁ + y₁ ?d)^a{x_a + y_a \sqrt{d} = (x_1 + y_1 \sqrt{d})^a}, and if p_a = 2 x_a² - 1{p_a = 2 x_a^2 - 1} is prime, then a = 2 ^m is a power of 2. So there are analogues to the Fermat numbers 2 ^a + 1.     

Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞

Here, all solutions of the form u=r^kf() to the p-harmonic equation, div(|u|^p–2u)=0, (p>2) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions u=r^kf() of the limit equation for p=, u _x ² u_xx+2u_xu_yu_xy+u _y ² u_yy=2, are constructed, some of which have a strong singularity at the origin (Theorem 2).     

Sobolev–Morrey spaces related to an ultraparabolic equation

Let us consider the class of hypoelliptic operators

A Gamma-convergence approach to the Cahn–Hilliard equation

We study the asymptotic dynamics of the Cahn–Hilliard equation via the “Gamma-convergence” of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an H ¹-version of a conjecture by De Giorgi, namely, the slope of the Allen–Cahn functional with respect to the H ⁻¹-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. We confirm this conjecture in the case of constant multiplicity of the limiting interface. Finally, under suitable conditions for which the conjecture is true, we prove that the limiting dynamics for the Cahn–Hilliard equation is motion by Mullins–Sekerka law. Partially supported by a Vietnam Education Foundation graduate fellowship.     

Compressible Navier–Stokes approximation to the Boltzmann equation

Even though the system of the compressible Navier–Stokes equations is not a limiting system of the Boltzmann equation when the Knudsen number tends to zero, it is the second order approximation by applying the Chapman–Enskog expansion. The purpose of this paper is to justify this approximation rigorously in mathematics. That is, if the difference between the initial data for the compressible Navier–Stokes equations and the Boltzmann equation is of the second order of the Knudsen number, so is the difference between two solutions for all time. The analysis is based on a refined energy method for a fluid-type system using the techniques for the system of viscous conservation laws.     

On traveling wave solutions to combined KdV–mKdV equation and modified Burgers–KdV equation

In this work, we established exact solutions for some nonlinear evolution equations. The extended tanh method was used to construct solitary and soliton solutions of nonlinear evolution equations. The extended tanh method presents a wider applicability for handling nonlinear wave equations.     

Invariant sets and solutions to the generalized thin film equation

The invariant sets and the solutions of the 1 2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set E0 = {u : ux = vxF(u), uy = vyF(u)}, where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the l l-dimensional nonlinear evolution equations.     

On variational methods to a generalized Emden–Fowler equation

In the present paper, the variational principle to the boundary value problems for a generalized Emden-Fowler equation is given and some existence results of solutions are obtained by using the critical point theory.     

Standing waves to discrete vector nonlinear Schrödinger equation

The purpose of this paper is to study a class of periodic discrete vector nonlinear Schrödinger equation. By using the critical point theory for strongly indefinite problems developed by Ding (Interdisciplinary Mathematical Sciences, World Scientific, Hackensack, NJ, 2007), we prove the existence of non-trivial standing waves for the vector equation with periodic or asymptotically periodic nonlinearities.     

Picard and Chazy solutions to the Painlevé VI equation

We study the solutions of a particular family of Painlevé VI equations with parameters and , for . We show that in the case of half-integer , all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVI equation for any such that . As an application, we classify all the algebraic solutions. For half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions. Received: 23 February 1999 / Accepted: 10 January 2001 / Published online: 18 June 2001     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

A probabilistic approach to the Yang–Mills heat equation

We construct a parallel transport U in a vector bundle E, along the paths of a Brownian motion in the underlying manifold, with respect to a time dependent covariant derivative ∇ on E, and consider the covariant derivative ∇₀U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U⁻¹∇₀U of this covariant derivative has quadratic variation twice the Yang–Mills energy density (i.e., the square norm of the curvature 2-form) integrated along the Brownian motion, and that the drift of such processes vanishes if and only if ∇ solves the Yang–Mills heat equation. A monotonicity property for the quadratic variation of U⁻¹∇₀U is given, both in terms of change of time and in terms of scaling of U⁻¹∇₀U. This allows us to find a priori energy bounds for solutions to the Yang–Mills heat equation, as well as criteria for non-explosion given in terms of this quadratic variation.     

Reduction of the multidimensional d’Alembert equation to two-dimensional equations

We give a classification of maximal subalgebras of rankn−1 for the extended Poincaré algebra , which is realized on the set of solutions of the d'Alembert equation . These subalgebras are used for constructing anzatses that reduce this equation to differential equations with two invariant variables. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 651–662, June, 1994.     

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.