The Riemann-Hilbert approach to obtain critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems

通过 k-Riemann 不变量对双曲系统的微分约束

Differential constraints for hyperbolic systems through k-Riemann invariants

本文基于微分约束法和 k-Riemann 不变量，提出求解一维非齐次拟线性双曲系统精确波解的约化方法，并应用于带源项 Euler 系统以刻画稀疏波和求解 Riemann 问题。

描述冷等离子体的双曲-双曲-椭圆型系统的适定性

Well-posedness for an hyperbolic-hyperbolic-elliptic system describing cold plasmas

本文证明了描述磁场中无碰撞等离子体运动的双曲-双曲-椭圆型PDEs系统的适定性，其证明结合了逐点估计与椭圆部分的bootstrap型论证。

Path Integral Solution of Linear Second Order Partial Differential Equations II. Elliptic, Parabolic and Hyperbolic Cases

A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial differential equations with Dirichlet/Neumann boundary conditions. The construction is checked by evaluating several known kernels for regions with planar and spherical boundaries. Some new calculational techniques are introduced.

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

In this paper we employ a "direct method" in order to obtain rank-k solutions of any hyperbolic system of first order quasilinear differential equations in many dimensions. We discuss in detail the necessary and sufficient conditions for existence of these type of solutions written in terms of Riemann invariants. The most important characteristic of this approach is the introduction of specific first order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method of solving multi-dimensional systems of PDEs. We have demonstrated th…

Hyperbolic Formulas in Elliptic Cauchy Problems

We study the Cauchy problem for the Laplace equation in a cylindrical domain with data on a part of it's boundary which is a cross-section of the cylinder. On reducing the problem to the Cauchy problem for the wave equation in a complex domain and using hyperbolic theory we obtain explicit formulas for the solution, thus developing the classical approach of Hans Lewy (1927).

Conditional Symmetries and Riemann Invariants for Hyperbolic Systems of PDEs

This paper contains an analysis of rank-k solutions in terms of Riemann invariants, obtained from interrelations between two concepts, that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of PDEs in many dimensions. A variant of the conditional symmetry method for obtaining this type of solutions is proposed. A Lie module of vector fields, which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints, is constructed. It is shown that th…