On a Diffusive Version of the Lifschitz–Slyozov–Wagner Equation

This paper is concerned with the Becker–Döring (BD) system of equations and their relationship to the Lifschitz–Slyozov–Wagner (LSW) equations. A diffusive version of the LSW equations is derived from the BD equations. Existence and uniqueness theorems for this diffusive LSW system are proved. The major part of the paper is taken up with proving that solutions of the diffusive LSW system converge in the zero diffusion limit to solutions of the classical LSW system. In particular, it is shown that the rate of coarsening for the diffusive system converges to the rate of coarsening for the classical system.     

Symmetries of the Discrete Nonlinear Schrödinger Equation

The Lie algebra L(h) of point symmetries of a discrete analogue of the nonlinear Schrödinger equation (NLS) is described. In the continuous limit, the discrete equation is transformed into the NLS, while the structure of the Lie algebra changes: a contraction occurs with the lattice spacing h as the contraction parameter. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symmetry algebra of the NLS.     

On Error Control in the Numerical Solution of Reaction–Diffusion Equation

Computational Mathematics and Mathematical Physics - A novel method for deriving a posteriori error bounds for approximate solutions of reaction–diffusion equations is proposed. As a model...     

Generalized Darboux Transformation for the Discrete Kadomtsev–Petviashvili Equation with Self-Consistent Sources

We construct several types of Darboux transformations for the discrete Kadomtsev–Petviashvili equation with self-consistent sources (dKPwS) including the elementary Darboux transformation, the adjoint Darboux transformation, and the binary Darboux transformation. These Darboux transformations can be used to obtain some solutions of the dKPwS. We give some solutions explicitly.     

Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap

Theoretical and Mathematical Physics - We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave...     

A Discrete Limit Theorem for the Matsumoto Zeta‐Function in the Space of Meromorphic Functions

In the paper, a discrete limit theorem for the Matsumoto zetafunction in the space of meromorphic functions is proved.     

A Discrete Limit Theorem for the Matsumoto Zeta‐Function in the Space of Analytic Functions

In the paper, a discrete distribution of the Matsumoto zetafunction is considered. It is proved that the probability measure , converges weakly as .     

Wave Breaking of the Camassa–Holm Equation

This paper gives a new and direct proof for McKean’s theorem (McKean in Asian J. Math. 2:867–874, 1998) on wave breaking of the Camassa–Holm equation. The blow-up profile is also analyzed.     

Discrete stochastic approximations of the Mumford–Shah functional

We propose a new Γ-convergent discrete approximation of the Mumford–Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford–Shah functional in any dimension.     

Solitons and Other Solutions for the Nonlinear Convection–Diffusion–Reaction Equation with Power-Law Nonlinearity by the Extended Simplest Equation Method

Computational Mathematics and Modeling - Based on the extended simplest equation method, we construct solitons and other solutions for the nonlinear convection-diffusion-reaction equation with...     

Nonlocal Reductions of the Ablowitz–Ladik Equation

Our purpose is to develop the inverse scattering transform for the nonlocal semidiscrete nonlinear Schrödinger equation (called the Ablowitz–Ladik equation) with \(\mathcal{PT}\) symmetry. This includes the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data, and the fundamental analytic solutions. In addition, we study the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann–Hilbert problem, we derive the one- and two-soliton solutions for the nonlocal Ablowitz–Ladik equation. Finally, we prove the completeness relation for the associated Jost solutions. Based on this, we derive the expansion formula over the complete set of Jost solutions. This allows interpreting the inverse scattering transform as a generalized Fourier transform.     

The Stable Limit Cycles of a Generalized Lienard Equation

In this paper we give a series of sufficient conditions for the existence of autonomous systems with three singular points.     

Symmetry Algebra of the Benjamin–Ono Equation

The Lie algebra structure for symmetries of the Benjamin–Ono equation is completely described.     

Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation

We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.     

On the Well-Posedness of the Dissipative Kadomtsev–Petviashvili Equation

Mathematical Notes - The well-posedness of the initial-value problem associated with the dissipative Kadomtsev–Petviashvili equation in the case of two-dimensional space is studied. It is...