Nonautonomous soliton solutions for a nonintegrable Korteweg–de Vries equation with variable coefficients by the variational approach

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.     

一类非局部Cahn-Hilliard方程弱解的存在唯一性

研究一类对流非局部Cahn-Hilliard方程的Neumann问题.通过一致Schauder估计和Leray-Schauder不动点定理,得到了该问题经典解的存在唯一性.进而,利用弱收敛方法得到了该问题弱解的存在唯一性.     

Nonvanishing at spatial extremity solutions of the defocusing nonlinear Schrödinger equation

We show local existence of certain type of solutions for the Cauchy problem of the defocusing nonlinear Schrödinger equation with pure power nonlinearity, in various cases of open sets, unbounded or bounded. These solutions do not vanish at the boundary or at infinity. We also show, in certain cases, that these solutions are unique and global.     

Strong solutions to 1D compressible Navier‐Stokes/Allen‐Cahn system with free boundary

This paper is concerned with a diffuse interface model for two‐phase flow of compressible fluids with a type of free boundary. We establish the existence and uniqueness of global strong solutions of a coupled Navier‐Stokes/Allen‐Cahn system in 1D.     

Global strong solutions to the Shliomis system for ferrofluids in a bounded domain

We consider the partial differential equations proposed by Shliomis to model the dynamics of an incompressible viscous ferrofluid submitted to an external magnetic field. The Shliomis system consists of the incompressible Navier‐Stokes equations, the magnetization equations, and the magnetostatic equations. The magnetization equations is of Bloch type, and no regularizing term is added. We prove the global existence of unique strong solution to the initial boundary value problem for the system in a bounded domain, with the small initial data and external magnetic field but without any restrictions on the physical parameters. The novelty of the analysis is to introduce a linear combination of magnetic fields.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton–Jacobi–Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear–quadratic model.     

Stationary solitons of a three-wave model generated by Type II second-harmonic generation in quadratic media

In this paper, we prove the uniqueness of stationary standing wave solutions of an optical model generated by Type II Second Harmonic Generation (SHG) with behaviors tending to zero at infinity under certain conditions on parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established.     

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L^∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.     

Reduction of the n-dimensional static vacuum Einstein equation and generalized Schwarzschild solutions

We consider the static vacuum Einstein spacetime when the spatial factor is conformal to a n-dimensional pseudo-Euclidean space. The most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described. We obtain the entire set of solutions of the reduced system, where the classical Schwarzschild solution arises as a particular solution. In addition, we show that the Riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature.