Blowup phenomena for a new periodic nonlinearly dispersive wave equation

We first establish the local well‐posedness for a new periodic nonlinearly dispersive wave equation. We then present a precise blowup scenario and several blowup results of strong solutions to the equation. The obtained blowup results on the equation improve considerably recent results on the Camassa‐Holm equation and the rod equation (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence and blow-up phenomena for a periodic 2-component Camassa–Holm equation

We first establish local well-posedness for a periodic 2-component Camassa?CHolm equation. We then present two global existence results for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation.     

Statistical Approach to Modulational Instability in Nonlinear Discrete Systems

We use a statistical approach to investigate the modulational instability (Benjamin-Feir instability) in several nonlinear discrete systems: the discrete nonlinear Schrodinger (NLS) equation, the Ablowitz-Ladik equation, and the discrete deformable NLS equation. We derive a kinetic equation for the two-point correlation function and use a Wigner-Moyal transformation to write it in a mixed space-wave-number representation. We perform a linear stability analysis of the resulting equation and discuss the obtained integral stability condition using several forms of the initial unperturbed spectrum (Lorentzian and δ-spectrum). We compare the results with the continuum limit (the NLS equation) and with previous results.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 56–63, July, 2005.     

Sign-definiteness of solution to inhomogeneous higher-order equation of mixed parabolic-hyperbolic type

We study solutions of a polycaloric equation and an equation of mixed parabolichyperbolic type of the second order. We prove the sign-definiteness of the solution in dependence of the right-hand side of the equation. Based on these results we study the sign-definiteness of a solution to a higher-order inhomogeneous equation of mixed parabolic-hyperbolic type in dependence on the right-hand side of the equation.     

Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions

In this paper, we propose complete radiation boundary conditions (CRBCs) for solutions of the convected Helmholtz equation with a uniform mean flow in a waveguide. We first study CRBCs for the Helmholtz equation in a waveguide. Noting that the convected Helmholtz equation is associated with the Helmholtz equation via the Prandtl–Glauert transformation, CRBCs for the convected Helmholtz equation is derived from CRBCs for the Helmholtz equation. We analyse well-posedness and convergence of approximate solutions satisfying CRBCs for the convected Helmholtz equation. In addition, simple numerical experiments will be presented to confirm the theoretical results.     

Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations.     

On the numerical solution of a complete two-dimensional hypersingular integral equation by the method of discrete singularities

We study the solvability of a complete two-dimensional linear integral equation with a hypersingular integral understood in the sense of the Hadamard principal value. We justify the convergence of a quadrature-type numerical method for the case in which the equation in question is uniquely solvable. We present an application of the results to the numerical solution of the Neumann boundary value problem on a plane screen for the Helmholtz equation by the surface potential method.     

Commutator identities on associative algebras,the non-Abelian Hirota difference equation and its reductions

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.     

Multidimensional exact solutions to a quasilinear parabolic equation with anisotropic heat conductivity

We prove invariance of a quasilinear parabolic equation with anisotropic heat conductivity in the three-dimensional coordinate space under some equivalence transformations and present some explicit formulas for these transformations. We consider nontrivial reductions of the equation to similar equations of less spatial dimension. Using these results, we construct new exact multidimensional solutions to the equation which depend on arbitrary harmonic functions.     

A note on solutions of an equation modelling arterial deformation

The derivation of exact solutions for a partial differential equation modelling arterial deformation in large arteries is considered. Amongst other results, we show that, for any values of the parameters appearing in the equation, solutions in terms of the first Painlevé transcendent can be obtained. This is in spite of the non-integrability of the equation. We also establish a connection, via an approximation of the equation under study by the Korteweg-de Vries equation, with the second Painlevé equation. Our results thus serve to further demonstrate the wide applicability and importance of the Painlevé equations.     

Geometrical Optics and Models of Computer Memory Fragmentation

We analyze a two-dimensional difference equation that arose in a stochastic model of memory fragmentation in computers. We use the ray method of geometrical optics, and other singular perturbation methods, to solve this equation asymptotically. The asymptotic limit corresponds to one of heavy system usages. We interpret the results from the ray method probabilistically.     

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

Hyers-Ulam stability for linear equations of higher orders

We show that, in the classes of functions with values in a real or complex Banach space, the problem of Hyers-Ulam stability of a linear functional equation of higher order (with constant coefficients) can be reduced to the problem of stability of a first order linear functional equation. As a consequence we prove that (under some weak additional assumptions) the linear equation of higher order, with constant coefficients, is stable in the case where its characteristic equation has no complex roots of module one. We also derive some results concerning solutions of the equation.     

Oscillation of delay differential equations on time scales

Consider the following equation: , where t is in a measure chain. We apply the theory of measure chains to investigate the oscillation and nonoscillation of the above equation on the basis of some well-known results. And in some sense, we show a method to unify the delay differential equation and delay difference equation.     

Asymptotic Analysis for a Class of Infinite Systems of First-Order PDE: Nonlinear Parabolic PDE in the Singular Limit

Abstract

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.     

Maximal regularity for non-autonomous stochastic evolution equations in UMD Banach spaces

A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partial differential equation.     

An averaging method applied to a duffing equation

We approximate a Duffing equation by an averaged system. We solve the system explicitely and draw bifurcation diagrams in dependence of the forcing term. We discuss the goodness of the averaging method, relative to the behavior of the solutions in dependence of involved parameters, by comparing with results obtained in [6,10] for the original Duffing equation.     

Properties of the Fast Diffusion Equation and Its Multidimensional Exact Solutions

We prove invariance of the fast diffusion equation in the two-dimensional coordinate space and give its reduction to a one-dimensional analog in the space variable. Using these results, we construct new exact multidimensional solutions which depend on arbitrary harmonic functions. As a consequence, we obtain new exact solutions to the well-known Liouville equation, the stationary analog of the fast diffusion equation with a linear source. We consider some generalizations to the case of systems of quasilinear parabolic equations.     

Local exact Lagrangian controllability of the Burgers viscous equation

We study and give the definition of the exact Lagrangian controllability of the viscous Burgers equation and prove a local result. We give similar results for the heat equation in dimension 1.