Analytic solutions of the Riemann problem in relativistic hydrodynamics and their numerical applications

Symmetries and Riemann Invariants

To establish a relation between two approaches to the construction of Riemann k-wave solutions of hydrodynamic-type systems, namely the symmetry reduction method and the generalized method of characteristics.

关于非线性薛定谔流体动力学的反聚焦问题的严格渐近分析

Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics

本文利用Whitham调制理论和Riemann-Hilbert方法，研究了反聚焦非线性薛定谔方程在一般阶梯初始数据下的长时间渐近行为，揭示了六个初始不连续问题的分类及各区域的主项和误差估计。

Conditional symmetries and Riemann invariants for inhomogeneous hydrodynamic-type systems

A new approach to the solution of quasilinear nonelliptic first-order systems of inhomogeneous PDEs in many dimensions is presented. It is based on a version of the conditional symmetry and Riemann invariant methods. We discuss in detail the necessary and sufficient conditions for the existence of rank-2 and rank-3 solutions expressible in terms of Riemann invariants. We perform the analysis using the Cayley-Hamilton theorem for a certain algebraic system associated with the initial system. The problem of finding such solutions has been reduced to expanding a set of trace conditions on wave ve…

The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations

Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation $%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases $%N=2,3$ and N=4 for which the conservation laws, Lax type representations and bi-Hamiltonian structures are analyzed in detail. We also show that the case N=2 is equivalent to a generalized Hunter-Saxton dynamical system, whose integrability …

Exact solutions with singularities to ideal hydrodynamics of inelastic gases

We construct a large family of exact solutions to the hyperbolic system of 3 equations of ideal granular hydrodynamics in several dimensions for arbitrary adiabatic index $γ$. In dependence of initial conditions these solutions can keep smoothness for all times or develop singularity. In particular, in the 2D case the singularity can be formed either in a point or along a line. For $γ=-1$ the problem is reduced to the system of two equations, related to a special case of the Chaplygin gas. In the 1D case this system can be written in the Riemann invariant and can be treated in a standard way. …

Analytic Poisson brackets on rational functions on the Riemann sphere and their generalization

We consider a hierarchy of Poisson structures defined on rational functions on the Riemann sphere. This hierarchy is originated in the theory of the integrable Camassa-Holm equation associated with the Krein's string spectral problem. Previously the proof of Jacobi identity was obtained by reducing the bracket to canonical Darboux coordinates. The main result of this note is a direct proof of the Jacobi identity. It turns out that the direct proof of the Jacobi identity is far from trivial. We also give an example of another hierarchy of Poisson brackets and construct Darboux coordinates for i…

多物种驱动扩散系统的水力学：一个双物种排除过程研究

Hydrodynamics of Multi-Species Driven Diffusive Systems with Open Boundaries: A Two-Tasep Study

本文研究了双物种排除过程，通过分析黎曼问题揭示了多物种驱动扩散系统中相分离现象的关键机制。

Fluid dynamics solutions obtained from the Riemann invariant approach

The generalized method of characteristics is used to obtain rank-2 solutions of the classical equations of hydrodynamics in (3+1) dimensions describing the motion of a fluid medium in the presence of gravitational and Coriolis forces. We determine the necessary and sufficient conditions which guarantee the existence of solutions expressed in terms of Riemann invariants for an inhomogeneous quasilinear system of partial differential equations. The paper contains a detailed exposition of the theory of simple wave solutions and a presentation of the main tool used to study the Cauchy problem. A s…

Solutions of first-order quasilinear systems expressed in Riemann invariants

We present a new technique for constructing solutions of quasilinear systems of first-order partial differential equations, in particular inhomogeneous ones. A generalization of the Riemann invariants method to the case of inhomogeneous hyperbolic and elliptic systems is formulated. The algebraization of these systems enables us to construct certain classes of solutions for which the matrix of derivatives of the unknown functions is expressible in terms of special orthogonal matrices. These solutions can be interpreted as nonlinear superpositions of $k$ waves (or $k$ modes) in the case of hype…

Analytic and Algebraic Deformations of Super Riemann Surfaces

By analytic deformations of complex structures, we mean perturbations of the Dolbeault operator. By algebraic deformations of complex structures, we mean deformations of holomorphic glueing data. For complex manifolds there is, infinitesimally, a correspondence between these two types deformations. In this article we argue that an analogous correspondence holds between the analytic and algebraic deformations of a super Riemann surface.

Relative energy gap for harmonic maps of Riemann surfaces into real analytic Riemannian manifolds

We extend the well-known Sacks-Uhlenbeck energy gap result (1981) for harmonic maps from closed Riemann surfaces into closed Riemannian manifolds from the case of maps with small energy (thus near a constant map), to the case of harmonic maps with high absolute energy but small energy relative to a reference harmonic map.

Analytical solution to the Riemann problem of 1D elastodynamics with general constitutive laws

Under the hypothesis of small deformations, the equations of 1D elastodynamics write as a 2 x 2 hyperbolic system of conservation laws. Here, we study the Riemann problem for convex and nonconvex constitutive laws. In the convex case, the solution can include shock waves or rarefaction waves. In the nonconvex case, compound waves must also be considered. In both convex and nonconvex cases, a new existence criterion for the initial velocity jump is obtained. Also, admissibility regions are determined. Lastly, analytical solutions are completely detailed for various constitutive laws (hyperbola,…

2-TASEP 的流体动力学行为

Hydrodynamic behavior of the 2-TASEP

本文研究双向运动并交换位置的双物种 2-TASEP 的流体动力学极限，证明其方程属于 Temple 类，求得稀疏波与激波解、完整求解 Riemann 问题，并用 Monte Carlo 模拟验证理论预测。

Liouville canonical form for compatible nonlocal Poisson brackets of hydrodynamic type, and integrable hierarchies

We solve the problem of reducing to the simplest and convenient for our purposes, canonical form for an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature in order to get an effective construction of the integrable hierarchies related to all these compatible Poisson brackets.

Analytic Continuation of Bernoulli Numbers, a New Formula for the Riemann Zeta Function, and the Phenonmenon of Scattering of Zeros

The method analytic continuation of operators acting integer n-times to complex s-times (hep-th/9707206) is applied to an operator that generates Bernoulli numbers B_n (Math. Mag. 70(1), 51 (1997)). B_n and Bernoulli polynomials B_n(s) are analytic continued to B(s) and B_s(z). A new formula for the Riemann zeta function zeta(s) in terms of nested series of zeta(n) is derived. The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an interesting phenonmenon of `scatterings' of the zeros of B_s(z) is observed.

Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem

A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of such solutions leads to a notion of local monodromies of difference equations. It is shown that in the continuous limit they converge to the monodromy matrices of differential equations. New type of isomonodromic deformations of difference equations with elliptic coefficients changing the periods of elliptic curves is constructed.

A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems

We study the asymptotic behavior of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear PDE's. Our method is based on the Deift-Zhou nonlinear steepest descent method in which the given Riemann-Hilbert problem localizes to small neighborhoods of stationary phase points. In their original work, Deift and Zhou only considered analytic phase functions. Subsequently Varzugin extended the Deift-Zhou method to a certain restricted class of non-analytic phase functions. In this paper, we extend Varzugin's method to a substantially more general class of non-analy…

Lax pairs for the equations describing compatible nonlocal Poisson brackets of hydrodynamic type, and integrable reductions of the Lame equations

The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demonstrate the integrability of the equations for some especially important partial classes of compatible nonlocal Poisson brackets of hydrodynamic type, in particular, for the most important case when one of the compatible Poisson brackets is local and also for the case when one of…