Class Of Nonlinear Hyperbolic Systems Research Articles

On the global solubility of the Cauchy problem for hyperbolic Monge–Ampére systems

This paper is devoted to the global solubility of the Cauchy problem for a class of non-linear hyperbolic systems of two first-order equations with two independent variables. This class contains quasilinear systems. The problem has a unique maximal (with respect to inclusion) many-valued solution, which possesses a completeness property. Namely, characteristics of various families lying on such a solution and converging to the corresponding boundary point have infinite length.

Robust boundary iterative learning control for a class of nonlinear hyperbolic systems with unmatched uncertainties and disturbance

Abstract In this paper, the robust boundary iterative learning control for the output tracking and disturbance attenuation of the 2 × 2 nonlinear hyperbolic system is addressed. Since the measurement limitation, the control and measurement are implemented at the same boundary of the system and the disturbance is not necessary to be estimated, which makes the iterative learning control be easy in implementation and low in measurement cost. By using the characteristic method, the robust convergence with respect to iteration-varying uncertainties arising from initial states shift, external disturbances, model plants uncertainties and disturbed reference trajectories is analyzed without any model reduction, rigorously. It is shown that the robust convergence bound is continuously dependent on the bounds of the iteration-varying uncertainties. Furthermore, to implement the proposed iterative learning control, the actuator dynamic is considered, also. Finally, with the actuator dynamic, two examples are given to demonstrate the effectiveness of the proposed iterative learning control strategy for the 2 × 2 nonlinear hyperbolic system.

Existence and uniqueness results for the pressureless Euler–Poisson system in one spatial variable

We study the Euler-Poisson system describing the evolution of a fluid without pressure e¤ect and, more generally, also treat a class of nonlinear hyperbolic systems with an analogous structure. We investigate the initial value problem by generalizing a method first introduced by LeFloch in 1990 and based on Volpert’s product and Lax’s explicit formula for scalar conservation laws. We establish several existence and uniqueness results when one component of the system (the density) is measure-valued and the second one (the velocity) has bounded variation. Existence is proven for general initial data, while uniqueness is guaranteed only when the initial data does not generate rarefaction centers. Our proof proceeds by solving first a nonconservative version of the problem and constructing solutions with bounded variation, while the solutions of the Euler-Poisson system is then deduced by differentiation.

High-frequency nonlinear optics: from the nonlinear Schrödinger approximation to ultrashort-pulses equations

After a brief introduction and physical motivation, we show how the nonlinear Schrödinger (NLS) equation can be derived from a general class of nonlinear hyperbolic systems. Its purpose is to describe the behaviour of high-frequency oscillating wave packets over a large time-scale that requires us to take into account diffractive effects. We then show that the NLS approximation fails for short pulses and propose some alternative models, including a modified Schrödinger equation with improved frequency dispersion. It turns out that these models have better properties and are quite accurate for short pulses. For ultrashort pulses, however, they must also be abandoned for more complex approaches. We give the main steps for such an analysis and explain one striking fact about ultrashort pulses: their dynamics in dispersive media is linear.

Global solvability of the Cauchy characteristic problem for one class of nonlinear second order hyperbolic systems

The Cauchy characteristic problem in the light cone of the future for one class of nonlinear hyperbolic systems of the second order is considered. The existence and uniqueness of global solution of this problem is proved.

Global entropy solutions to a variant of the compressible Euler equations

DiPerna [R.J. DiPerna, Global solutions to a class of nonlinear hyperbolic systems of equations, Comm. Rat. Pure Appl. Math. 26 (1973) 1–28] use the Glimm’s scheme method to obtain a global weak solution to the Euler equations of one-dimensional, compressible fluid flow with 1 < γ < 3 , while in this work, we use the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor [P.L. Lions, B. Perthame, P.E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996) 599–638; P.L. Lions, B. Perthame, E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p -system, Comm. Math. Phys. 163 (1994) 415–431] to obtain the existence of global entropy solutions to the system with a uniform amplitude bound.

An alternative procedure for approximating hyperbolic systems of conservation laws

This work presents an alternative numerical procedure for simulating a class of nonlinear hyperbolic systems, using Glimm's method for advancing in time. The standard procedure to implement this methodology suffers from the disadvantage of requiring a complete solution of the associated Riemann problem—a task, in general, not easily reached. The alternative procedure introduced in this article consists in approximating the solution of the associated Riemann problem by piecewise constant functions always satisfying the jump condition—thus circumventing the difficulty of solving the Riemann problem and giving rise to an approximation easier to implement with lower computational cost. In order to illustrate the good performance of the alternative methodology proposed, two problems are considered—namely the transport of a pollutant in the atmosphere and the dynamics of the filling up of a rigid porous medium, modeled under a mixture theory viewpoint. Comparison with the standard procedure, employing the complete solution of the associated Riemann problem for implementing Glimm's scheme, has shown good agreement.

Nonlinear hyperbolic equations in surface theory: Integrable discretizations and approximation results

A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this difference scheme. These results are applied to hyperbolic systems of differential–geometric origin, like the sine–Gordon equation describing the surfaces of the constant negative Gaussian curvature (K–surfaces). In particular, we prove the convergence of discrete K–surfaces and their Backlund transformations to their continuous counterparts. This puts on a firm basis the generally accepted belief (which however remained unproved untill this work) that the classical differential geometry of integrable classes of surfaces and the classical theory of transformations of such surfaces may be obtained from a unifying multi–dimensional discrete theory by a refinement of the coordinate mesh–size in some of the directions. E–Mail: bobenko@math.tu-berlin.de E–Mail: matthes@math.tu-berlin.de E–Mail: suris@sfb288.math.tu-berlin.de

On a class of nonlinear hyperbolic systems

On a class of nonlinear hyperbolic systems

Linearization method to stability analysis for nonlinear hyperbolic systems

It is shown that the linearization principle is true for a class of nonlinear hyperbolic systems with two independent variables, time and space. To keep clear exposition, our proof is based on the Saint Venant equation. More results can be proven by using the Lyapunov function candidate that we give here.

On a Class of Nonlinear Hyperbolic Systems with Nonlocal Boundary Conditions

We study the existence, uniqueness and some regularity properties of solutions to a nonlinear hyperbolic problem.

Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems

In this paper, we study the long wave approximation for quasilinear symmetric hyperbolic systems. Using the technics developed by Joly-Métivier-Rauch for nonlinear geometrical optics, we prove that under suitable assumptions the long wave limit is described by KdV-type systems. The error estimate if the system is coupled appears to be better. We apply formally our technics to Euler equations with free surface and Euler-Poisson systems. This leads to new systems of KdV-type.

Uniqueness via the adjoint problems for systems of conservation laws

AbstractWe prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws that includes in particular the p‐system of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy Condition, or WEC for short, introduced in this paper. The main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution. For the proof of uniqueness, we derive an existence result (respectively a uniqueness result) for the backward (respectively forward) adjoint problem associated with the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients. © 1993 John Wiley &amp; Sons, Inc.

Global solutions to a class of nonlinear hyperbolic systems of equations

Communications on Pure and Applied MathematicsVolume 26, Issue 1 p. 1-28 Article Global solutions to a class of nonlinear hyperbolic systems of equations† Ronald J. Diperna, Ronald J. Diperna Brown UniversitySearch for more papers by this author Ronald J. Diperna, Ronald J. Diperna Brown UniversitySearch for more papers by this author First published: January 1973 https://doi.org/10.1002/cpa.3160260102Citations: 52 † The results of this paper are contained in a doctoral thesis submitted to the Graduate School of Arts and Sciences of New York University. Reproduction in whole or in part is permitted for any purpose of the United States Government. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume26, Issue1January 1973Pages 1-28 RelatedInformation