Long Time Asymptotics Behavior of the Focusing Nonlinear Kundu-Eckhaus Equation

The authors study the Cauchy problem for the focusing nonlinear KunduEckhaus(KE for short) equation and construct the long time asymptotic expansion of its solution in fixed space-time cone with C(x₁, x2, v₁, v₂) = {(x, t) ∈ R²: x = x₀ + vt,x₀ ∈ [x₁, x₂], v ∈ [v₁, v₂]}. By using the inverse scattering transform, Riemann-Hilbert approach and ■ steepest descent method, they obtain the lone...     

Estimates for the Locally Periodic Homogenization of the Riemann–Hilbert Problem for a Generalized Beltrami Equation

Differential Equations - The homogenization method for differential operators based on an asymptotic expansion in a small parameter is widely used in mathematics and physics. In addition to the...     

Two Boundary-Value Problems for the Cauchy–Riemann Equation in a Sector

By reflections, we obtain the Schwarz?CPoisson formula in a sector with angle ${\vartheta=\frac{\pi}{n},\,n\in \mathbb{N}}$ , which is a generalization of the corresponding result obtained by Begehr and Vaitekhovich (Funct Approx 40(2):251?C282, 2009). Especially, boundary behaviors at corner points are discussed in detail. Then we consider the Schwarz and Dirichlet boundary-value problems (BVPs) for the Cauchy?CRiemann equation, and expressions of solution and the condition of solvability are explicitly obtained.     

Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

Journal of Nonlinear Science - We analyze the large-n behavior of soliton solutions of the integrable focusing nonlinear Schrödinger equation with associated spectral data consisting of a...     

The Nonlinear Steepest Descent Method to Long-Time Asymptotics of the Coupled Nonlinear Schrödinger Equation

The Riemann–Hilbert problem for the coupled nonlinear Schrödinger equation is formulated on the basis of the corresponding \(3\times 3\) matrix spectral problem. Using the nonlinear steepest descent method, we obtain leading-order asymptotics for the Cauchy problem of the coupled nonlinear Schrödinger equation.     

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

Collapse Rate of Solutions of the Cauchy Problem for the Nonlinear Schrödinger Equation

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms.

     

Asymptotics of the Fredholm Determinant Associated with the Correlation Functions of the Quantum Nonlinear Schrödinger Equation

The correlation functions of the quantum nonlinear Schrödinger equation can be presented in terms of the Fredholm determinant. An explicit expression for this determinant is found for large time and long distance. Bibliography: 6 titles.     

On the nonvanishing of abstract Cauchy–Riemann cohomology groups

In this paper we prove infinite dimensionality of some local and global cohomology groups on abstract Cauchy–Riemann manifolds.     

A Riemann–Hilbert Type Problem for a Singularly Perturbed Cauchy–Riemann Equation with a Singularity in the Coefficient

Differential Equations - We consider the Riemann–Hilbert problem for a singularly perturbed system of partial differential equations of the Cauchy–Riemann type. Using the Lomov...     

Large Time Asymptotics for the BBM–Burgers Equation

We study large time asymptotics of solutions to the BBM–Burgers equation

Blow-up of Solutions of the Cauchy Problem for a Nonlinear Schrödinger Evolution Equation

The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.     

Wellposedness of a Nonlinear, Logarithmic Schr?dinger Equation of Doebner–Goldin Type Modeling Quantum Dissipation

This paper is concerned with the modeling and analysis of quantum dissipation phenomena in the Schr?dinger picture. More precisely, we do investigate in detail a dissipative, nonlinear Schr?dinger equation somehow accounting for quantum Fokker–Planck effects, and see how it is drastically reduced to a simpler logarithmic equation via a nonlinear gauge transformation in such a way that the physics underlying both problems keeps unaltered. From a mathematical viewpoint, this allows for a more easily achievable analysis regarding the local wellposedness of the initial-boundary value problem. This simplification requires the performance of the polar (modulus argument) decomposition of the wavefunction, which is rigorously attained (for the first time to the best of our knowledge) under quite reasonable assumptions.     

Asymptotics for the Korteweg-de Vries-Burgers Equation

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut＋uux-uxx＋uxxx=0,x∈R,t〉0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u0 ∈H^s （R）∩L^1 （R）, where s 〉 -1/2, then there exists a unique solution u （t, x） ∈C^∞ （（0,∞）;H^∞ （R）） to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u（t）=t^-1/2fM（（·）t^-1/2）＋0（t^-1/2） as t →∞, where fM is the self-similar solution for the Burgers equation. Moreover if xu0 （x） ∈ L^1 （R）, then the asymptotics are true u（t）=t^-1/2fM（（·）t^-1/2）＋O（t^-1/2-γ） where γ ∈ （0, 1/2）.     

Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

On a finite segment [0, l], we consider the differential equation

On the Approximate Functional Equation of Riemann zeta-Function

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Asymptotics of the Riemann–Hilbert Problem for the Somov Model of Magnetic Reconnection of Long Shock Waves

Mathematical Notes - We consider the Riemann–Hilbert problem in a domain of complicated shape (the exterior of a system of cuts), with the condition of growth of the solution at infinity....     

The Existence of Solutions for a Nonlinear First-Order Differential Equation Involving the Riemann–Liouville Fractional-Order and Nonlocal Condition

In this paper, we study necessary conditions for the existence and uniqueness of continuous solution for a nonlocal boundary value problem with nonlinear term involving Riemann–Liouville fractional derivative. Our results are based on Schauder fixed point theorem and the Banach contraction principle fixed point theorem. Examples illustrating the obtained results are also presented.