Sense Of Lax Research Articles

Exchange of conserved quantities, shock loci and Riemann problems

AbstractSystems of conservation laws admitting extensions, such as entropy density/flux functions, generate related systems obtained by exchanging the extension with one of the constituent equations. Often if not always, the smooth solutions of the two systems coincide, and weak solutions of one system containing only small discontinuities are approximate weak solutions of the other. The adiabatic approximation for the Euler system illustrates the utility of this procedure.Such an exchange of conserved quantities preserves hyperbolicity and genuine non‐linearity in the sense of Lax. On the other hand, the topological structure of the shock locus of a point in phase space and the solvability of Riemann problems in the large can be strongly affected. A discussion of when and how this occurs is given here.In this paper the exchange of conserved quantities is conveniently described by a simple homotopy in an extended version of the usual ‘symmetric variables’. A dynamical system in phase space is constructed, the trajectories of which describe the Hugoniot locus of a fixed point in phase space at each state of the homotopy. The appearance of critical points for this dynamical system is identified with the alteration of the topological structure of the Hugoniot locus by the exchange of conserved quantities. Copyright © 2001 John Wiley & Sons, Ltd.

A Note on the Riemann Problem for General n × n Conservation Laws

We analyze the structure of the general solution of the Riemann problem for a strictly hyperbolic system of conservation laws whose characteristic fields are neither genuinely non-linear nor linearly degenerate in the sense of Lax.

La otredad: ¿real o inventada? El caso de las brujas

The purpose of this paper is two-fold. On the one hand, it offers a general analysis of stigmas (a person has one when, in virtue of its belonging to a certain group, such as that of women, homosexuals, etc., he or she is subjugated or persecuted). On the other hand, I argue that stigmas are “invented”. More precisely, I claim that they are not descriptive of real inequalities. Rather, they are socially created, or invented in a lax sense, in so far as the real differences to which they refer are socially valued or construed as negative, and used to justify social inequalities (that is, the placing of a person in the lower positions within an economic, cultural, etc., hierarchy), or persecutions. Finally, I argue that in some cases, such as that of the witch persecution of the early modern times, we find the extreme situation in which a stigma was invented in the strict sense of the word, that is, it does not have any empirical content.

Derivatives of Wronskians with applications to families of special Weierstrass points

Let π : X ⟶ S \pi : \mathcal {X} \longrightarrow S be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over C \mathbb {C} . On every such a family, suitable derivatives “along the fibers” (in the sense of Lax) of the relative wronskian, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the g ( g + 1 ) / 2 g(g+1)/2 -th tensor power of the relative canonical bundle of the family itself. The geometrical meaning of such sections is discussed: the zero schemes of the ( k − 1 ) (k-1) -th derivative ( k ≥ 1 k\geq 1 ) of a relative wronskian correspond to families of Weierstrass Points (WP’s) having weight at least k k . The locus in M g M_{g} , the coarse moduli space of smooth projective curves of genus g g , of curves possessing a WP of weight at least k k , is denoted by w t ( k ) wt(k) . The fact that w t ( 2 ) wt(2) has the expected dimension for all g ≥ 2 g\geq 2 was implicitly known in the literature. The main result of this paper hence consists in showing that w t ( 3 ) wt(3) has the expected dimension for all g ≥ 4 g\geq 4 . As an application we compute the codimension 2 2 Chow ( Q Q -)class of w t ( 3 ) wt(3) for all g ≥ 4 g\geq 4 , the main ingredient being the definition of the k k -th derivative of a relative wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension 2 2 Chow ( Q Q -)classes in M 4 M_{4} ( g ≥ 4 g\geq 4 ), corresponding to varieties of curves having a point P P with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.

Lax representations and integrable time discretizations of the DDKdV, DDmKdV, and DDHOKdV

From a proper 2 × 2 discrete isospectral problem, a new differential-difference integrable equation in Lax sense is proposed by a discrete zero curvature equation. The DDKdV (differential-difference DdV equation) proposed by Ohta and Hirota and DDCDGKS (DD Caudrey-Dodd-Gibbon-Kotera-Sawada equation) are rederived. Some other new discrete KdV equations, discrete mKdV equations and discrete high order KdV equations which converge to the corresponding continuous soliton equations in the continuum limit are obtained. Integrable time discretizations of the DDKdV, DDmKdV (differential-difference mKdV equation) and DDHOKdV (differential-difference high order KdV equations) are given.

Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws

We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, nonoscillatory central scheme, a natural extension of the one-dimensional second-order central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408--448]. As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the computation of the exact Jacobians can be avoided. Moreover, the central scheme is genuinely multidimensional in the sense that it does not necessitate dimensional splitting. We prove that the scheme satisfies the scalar maximum principle, and in the more general context of systems, our proof indicates that the scheme is positive (in the sense of Lax and Liu [CFD Journal, 5 (1996), pp. 1--24]). We demonstrate the application of our central scheme to several prototype two-dimensional Euler problems. Our numerical experiments include the resolution of shocks oblique to the computational grid; they show how our central scheme solves with high resolution the intricate wave interactions in the so-called double Mach reflection problem [J. Comput. Phys., 54 (1988), pp. 115--173] without following the characteristics; and finally we report on the accurate ray solutions of a weakly hyperbolic system [J. Comput. Appl. Math., 74 (1996), pp. 175--192], rays which otherwise are missed by the dimensional splitting approach. Thus, a considerable amount of simplicity and robustness is gained while achieving stability and high resolution.

On the Integrability Properties of a Perturbed Nonlinear Schrödinger Field Equation

The integrability properties of a perturbed nonlinear Schrödinger equation are studied using Painleve analysis and Laxs method. We distinguish between integrability in the Painleve sense and integrability in the Lax sense. The solution of the perturbed equation, with the perturbing term depending only on time, is found by direct integration. It is seen that the perturbing nonlinear inhomogeneity can split the fundamental soliton into pulses at high values of the perturbation strength.

On the regularity of one-dimensional elastic waves in an incompressible isotropic material

Wave motion is considered in an incompressible isotropic material which is defined by quasi-linear hyperbolic system of four equations for which two characteristic fields are linearly degenerate in the sense of Lax. For the remaining characteristic fields a singificant nonlinearity is assumed. The behavior of derivatives of solution is investigated along these two different types of characteristic fields. The effect of the system nonlinearity shows itself in the unboundedness of derivatives with the limited solution along the essentially nonlinear fields.

Global existence of solutions to nonlinear hyperbolic systems of conservation laws

We introduce a new method of constructing solutions to the Cauchy problem for nonlinear hyperbolic systems of conservation laws in one space dimension. We consider systems which are strictly hyperbolic and genuinely nonlinear in the sense of Lax [lo]. We present the method here in the setting of systems of two conservation laws, Ut + G(U), F 0, --oo 0,

Eigenfunctions of Operator-Valued Analytic Functions

This paper is a sequel to [2], whose primary purposes are to clarify and generalize the concept introduced there of an eigenfunction of an inner function, and to answer questions raised there concerning the equivalence of several possible forms of the definition. A new definition, proposed here, leads to a complete characterization of the eigenfunctions of Potapov inner functions of normal operators, and the result is more satisfactory than [2, Theorem 3.4], although the latter is used strongly in the proof.Let be an inner function in the sense of Lax; i.e., is almost everywhere (a.e.) a unitary operator on a separable Hilbert space and belongs weakly to the Hardy class H2. An analytic function q (which will have to be a scalar inner function) was defined to be an eigenfunction of if the set of z in the disk {z: |z| ≦ 1} for which is invertible is a set of linear measure 0 on the circle {z: |z| = 1}.

A spectral theory for inner functions

Let ! be an inner function in the sense of Lax; i.e., W(ei0) is almost everywhere a unitary operator on a separable Hilbert space 4 and ? belongs weakly to the Hardy class H2. Inner functions arise from subspaces of H2 invariant under the right shift operator (invariant subspaces) and from bounded operators on X' in ways to be specified. We are interested in finding canonical forms for and invariants of inner functions, and where an inner function comes from an operator on X in relating the invariants of the inner function to those of the operator. For basic definitions and notations consult [5, particularly Chapter VI], or [15].

Generally Unconditionally Stable Difference Operators

u(x, 0) = uo(x), where P(D) is a partial differential operator of order p with respect to x with constant matrix coefficients. We shall study consistent difference operators of the form (2) v(x, t + k) = Eh,kv(x, t) = f(kP(ah))v(x, t), where f(z) is a rational function, h and k are the mesh-widths in x and t, respectively, and where ahV = (ah,lV, , ah,V) with NOhzv(x) = (ih)<'Eljjj, d(j)v(x + jhel) is consistent with Dv = (i7'av/0x1,, i-1a/v9xd) and the ah,N, I = 1, ***, d, have the same real-valued symbol d(rq) = i-'Ej d(j exp (iji). We say that such an operator is generally unconditionally stable if it is unconditionally stable for all correctly posed initial-value problems. Depending on whether the correctness of (1) is interpreted in Petrowsky's sense or in the sense of Lax and Richtmyer, the stability is understood to be in the sense of Forsythe and Wasow or Lax and Richtmyer, respectively. It is proved that in both cases the necessary and sufficient condition for general unconditional stability is