Classical Lax Research Articles

On a new class of non-dynamical ABCD algebras for classical and quantum integrable systems

We consider classical and quantum non-dynamical quadratic abcd Lax algebras with classical and quantum gl(n)⊗gl(n)-valued abcd-tensors satisfying a set of quadratic non-dynamical Yang-Baxter-type equations generalizing those of Fredel and Maillet [1]. We establish a relation of some of these equations with the so-called “semi-dynamical” Yang-Baxter equations of [2]. We show that the linearization of the corresponding quadratic structures lead to linear tensor structures with the classical gl(n)⊗gl(n)-valued r-matrices satisfying usual “permuted” classical Yang-Baxter equations [1,3–5]. We consider example of our construction associated with the deformed Zn-graded r-matrix of [10–12] and explicitly construct the corresponding abcd-tensors — both classical and quantum. Small n examples are also considered in some details.

A new compact scheme-based Lax–Wendroff method for high fidelity simulations

The classical Lax–Wendroff (LW) method employs explicit second order central difference schemes to solve partial differential equations. Recently, the authors in Sengupta et al. (2023) have identified numerical parameter ranges in performing large eddy simulation (LES) using this classical LW method. In a bid to improve these numerical parameter ranges for high fidelity computations, a new non-uniform grid based compact scheme for LW method (NUCLW) is developed here. This new method uses sixth order non-uniform compact scheme (NUC6) for the first and higher order derivatives. Reported global spectral analysis confirms substantial improvement in the resolution and accuracy of NUCLW scheme over the classical LW scheme. A significant advantage of the present scheme is its ability to compute solutions for non-uniform, structured grids, and that does not suffer degradation in accuracy. The potential of the NUCLW scheme for high accuracy computations is demonstrated by solving 2D incompressible Navier–Stokes equations for the benchmark unsteady flow inside a square lid driven cavity problem for a subcritical Reynolds number of 3200 and a post-critical Reynolds number of 10,000 and simulating the instability of the Taylor–Green vortex problem. These demonstrate the ability of the NUCLW scheme to solve the Navier–Stokes equations for high fidelity simulations such as LES/DNS with enhanced ranges of numerical parameters with respect to enhanced accuracy and faster computations.

Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux

We study the Riemann problem for a new model on immiscible vertical two-phase flow under point injection. The point injection is modeled by a Dirac [Formula: see text]-source term as well as by a spatially discontinuous flux function, which defines two fluxes, one on each side of the discontinuity. The solutions comprise up to three wave groups: downward waves, a stationary shock and upward waves. Owning the interplay between the Dirac [Formula: see text]-source and the discontinuous flux, there is no standard entropy condition for the stationary shock (flux’s connections). An entropy condition was deduced based on impinging characteristics and perturbation of the constant solution. This condition leads to shocks that do not satisfy the classical Lax’s conditions — even for arbitrarily small shocks — and may also have no viscous profile. The Rankine–Hugoniot condition — embedding the Dirac [Formula: see text]-source — and the entropy condition are geometrically represented by “Flux Projections” that support the analytical method proposed in this paper. We then obtain a [Formula: see text] analytic solution for all initial value problems. We verify the entropy condition using a Lagrangian–Eulerian scheme recently introduced in the literature, which is based in the new concept of no-flow curves. Analytic and numerical solutions fit.

Universal 1-loop divergences for integrable sigma models

We present a simple, new method for the 1-loop renormalization of integrable σ-models. By treating equations of motion and Bianchi identities on an equal footing, we derive ‘universal’ formulae for the 1-loop on-shell divergences, generalizing case-by-case computations in the literature. Given a choice of poles for the classical Lax connection, the divergences take a theory-independent form in terms of the Lax currents (the residues of the poles), assuming a ‘completeness’ condition on the zero-curvature equations. We compute these divergences for a large class of theories with simple poles in the Lax connection. We also show that ℤT coset models of ‘pure-spinor’ type and their recently constructed η- and λ-deformations are 1-loop renormalizable, and 1-loop scale-invariant when the Killing form vanishes.

On the origin of the correspondence between classical and quantum integrable theories

If we start from certain functional relations as definition of a quantum integrable theory, then we can derive from them a linear integral equation. It can be extended, by introducing dynamical variables, to become an equation with the form of Marchenko's. Then, we derive from the latter a classical (differential) Lax pair. We exemplify our method by focusing on the massive version of the ODE/IM (Ordinary Differential Equations/Integrable Models) correspondence from Quantum sine-Gordon (sG) with many moduli/masses to the classical sinh-Gordon (shG) equation, so describing, in a particular case, some super-symmetric gauge theories and the AdS3 strong coupling scattering amplitudes/Wilson loops. Yet, we present it in a way which reveals its generality of application. In fact, we give some hints on how it works for spin chains.

Fifth order AWENO finite difference scheme with adaptive numerical diffusion for Euler equations

In solving hyperbolic conservation laws using the fifth-order characteristic-wise alternative WENO finite-difference scheme (AWENO) with Z-type affine-invariant nonlinear Ai-weights, the classical local Lax–Friedrichs flux (LLF) is modified with an adaptive numerical diffusion (ND) coefficient to form an adaptive LLF flux (LLF-M). The adaptive ND coefficient depends nonlinearly on the local scale-independent smoothness measures on the pressure, density, dilation, and vorticity. The feature sensor combines the well-known Durcos’ sensor on the dilation and vorticity of the velocity field and Jameson’s sensor on the density and pressure. Based on the measure of the feature sensor, the ND coefficient of the LLF-M flux is designed to transit smoothly and quickly from a set minimum to the maximum, the local spectral radius of the eigenvalues of the Jacobian of the flux. Hence, the modified AWENO scheme improves the resolution of small-scale structures due to a substantial reduction of excessive dissipation while capturing discontinuities essentially non-oscillatory (ENO-property). The performance of the improved AWENO scheme is validated in one- and two-dimensional benchmark problems with discontinuities and smooth small-scale structures. The results show that the new adaptive LLF-M flux improves resolution, captures fine-scale structures, and is robust in a long-term simulation.

Lax–Wendroff Schemes with Polynomial Extrapolation and Simplified Lax–Wendroff Schemes for Dispersive Waves: A Comparative Study

One of the features of Boussinesq-type models for dispersive wave propagation is the presence of mixed spatial/temporal derivatives in the partial differential system. This is a critical point in the design of the time marching strategy, as the cost of inverting the algebraic equations arising from the discretization of these mixed terms may result in a nonnegligible overhead. In this paper, we propose novel approaches based on the classical Lax–Wendroff (LW) strategy to achieve single-step high-order schemes in time. To reduce the cost of evaluating the complex correction terms arising in the Lax–Wendroff procedure for Boussinesq equations, we propose several simplified strategies which allow to reduce the computational time at fixed accuracy. To evaluate these qualities, we perform a spectral analysis to assess the dispersion and damping error. We then evaluate the schemes on several benchmarks involving dispersive propagation over flat and nonflat bathymetries, and perform numerical grid convergence studies on two of them. Our results show a potential for a CPU reduction between 35 and 40% to obtain accuracy levels comparable to those of the classical RK3 method.

Initial Data Identification for the One-Dimensional Burgers Equation

In this article, we study the problem of identification for the 1-D Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of nonbackward uniqueness of the Burgers equation, there may exist multiple initial data leading to the same given target. In articles “Initial data identification in conservation laws and Hamilton-Jacobi equations” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arXiv:1903.06448</i> , 2019) and “The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SIAM Journal on Mathematical Analysis</i> , vol. 52, the authors fully characterize the set of initial data leading to a given target using the classical Lax–Hopf formula. In this article, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigating systems of conservation laws in 1-D, where the classical Lax–Hopf formula no more holds. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wavefront tracking algorithm.

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a rarefaction wave; (iii) there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves. These solutions are self-similar. Furthermore, for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data, there always exists a unique delta shock for at least a short time. It could be prolonged to a global solution. Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass (particle). Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified. This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases, that is strictly hyperbolic, and whose characteristics are both genuinely nonlinear. We also discuss possible physical interpretations and applications of these new solutions.

Stability of Classical Shock Fronts for Compressible Hyperelastic Materials of Hadamard Type

This paper studies the uniform and weak Lopatinski\u{\i} conditions associated to classical (Lax) shock fronts of arbitrary amplitude for compressible hyperelastic materials of Hadamard type in several space dimensions. Thanks to the seminal works of Majda (Mem. Amer. Math. Soc. 43 (1983), no. 281; Mem. Amer. Math. Soc. 41 (1983), no. 275) and M\'etivier (Trans. Amer. Math. Soc. 296 (1986), pp. 431-479; Comm. Partial Diff. Eqs. 15 (1990), no. 7, pp. 983-1028), the uniform Lopatinski\u{\i} condition ensures the local-in-time, multidimensional, nonlinear stability of such fronts. The stability function (also called Lopatinski\u{\i} determinant) for shocks of arbitrary amplitude in this large class of hyperelastic materials is computed explicitly. This information is used to establish the conditions for uniform and weak shock stability in terms of the parameters of the shock and of the elastic moduli of the material.

Existence Theorem for Noninstantaneous Impulsive Evolution Equations

In this note, the variational form of the classical Lax–Milgram theorem is used for the divulgence of variational structure of the first-order noninstantaneous impulsive linear evolution equation. The existence and uniqueness of the weak solution of the problem is obtained. In future, this constructive theory can be used for the corresponding semilinear problems.

Quantum Lax Pairs via Dunkl and Cherednik Operators

We establish a direct link between Dunkl operators and quantum Lax matrices $\mathcal L$ for the Calogero--Moser systems associated to an arbitrary Weyl group $W$ (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix $\mathcal A$ so that $\mathcal L, \mathcal A$ form a quantum Lax pair. Moreover, such an $\mathcal A$ can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of $W$, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D'Hoker-Phong and Bordner-Corrigan-Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic $BC_n$ case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem.

On the Coupling of VEM and BEM in Two and Three Dimensions

This paper introduces and analyzes the combined use of the virtual element method (VEM) and the boundary element method (BEM) to numerically solve linear transmission problems in two and three dimensions. As a model, we consider an elliptic equation in divergence form holding in an annular domain coupled with the Laplace equation in the corresponding unbounded exterior region, together with transmission conditions on the interface and a suitable radiation condition at infinity. We employ the usual primal formulation in the bounded region, and combine it, by means of the Costabel and Han approach, with the boundary integral equation method in the exterior domain. As a consequence, and in addition to the original unknown of the model, its normal derivative in two dimensions, and both its normal derivative and its trace in the three-dimensional case, are introduced as auxiliary nonvirtual unknowns. Moreover, for the latter case, a new and more suitable variational formulation for the coupling is introduced. In turn, the main ingredients required by the discrete analyses include the virtual element subspaces for the domain unknowns, explicit polynomial subspaces for the boundary unknowns, and suitable projection and interpolation operators that allow us to define the corresponding discrete bilinear forms. Then, as for the continuous formulations, the classical Lax--Milgram lemma is employed to derive the well-posedness of our coupled VEM/BEM scheme. A priori error estimates in the energy and weaker norms, and corresponding rates of convergence for the solution as well as for a fully computable projection of its virtual component, are provided. Finally, a couple of numerical examples in two dimensions illustrating the performance of the derived VEM/BEM schemes are reported.

On R-matrix valued Lax pairs for Calogero–Moser models

This article is devoted to the study of R-matrix-valued Lax pairs for N-body (elliptic) Calogero–Moser models. Their matrix elements are given by quantum R-matrices of Baxter–Belavin type. For the widely known Krichever’s Lax pair with spectral parameter is reproduced. First, we construct the R-matrix-valued Lax pairs for Calogero–Moser models associated with classical root systems. For this purpose we study generalizations of the D’Hoker–Phong Lax pairs. It appeared that in the R-matrix-valued case the Lax pairs exist in special cases only. The number of quantum spaces (on which R-matrices act) and their dimension depend on the values of coupling constants. Some of the obtained classical Lax pairs admit straightforward extension to the quantum case. In the end we describe a relationship of the R-matrix-valued Lax pairs to Hitchin systems defined on bundles with nontrivial characteristic classes over elliptic curve. We show that the classical analogue of the anisotropic spin exchange operator entering the R-matrix-valued Lax equations is reproduced in these models.

Variational approach to differential equations with not instantaneous impulses

In this note we introduce the concept of a weak solution for a linear equation with not instantaneous impulses. We use the classical Lax–Milgram Theorem to reveal the variational structure of the problem and get the existence and uniqueness of weak solutions as critical points. This will allow us in the future to deal with the corresponding nonlinear problems.

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

Expansion shock waves in regularized shallow-water theory.

We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularized shallow-water equations that include the Benjamin-Bona-Mahony and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.

Trigonometric version of quantum–classical duality in integrable systems

We extend the quantum–classical duality to the trigonometric (hyperbolic) case. The duality establishes an explicit relationship between the classical N-body trigonometric Ruijsenaars–Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles while the velocities of classical particles are proportional to the eigenvalues of the spin chain Hamiltonians (residues of the properly normalized transfer matrix). In the rational version of the duality, the action variables of the Ruijsenaars–Schneider model are equal to the twist parameters with some multiplicities defined by quantum (occupation) numbers. In contrast to the rational version, in the trigonometric case there is a splitting of the spectrum of action variables (eigenvalues of the classical Lax matrix). The limit corresponding to the classical Calogero–Sutherland system and quantum trigonometric Gaudin model is also described as well as the XX limit to free fermions.

Traveling waves for conservation laws with cubic nonlinearity and BBM type dispersion

Scalar conservation laws with non-convex fluxes have shock wave solutions that violate the Lax entropy condition. In this paper, such solutions are selected by showing that some of them have corresponding traveling waves for the equation supplemented with dissipative and dispersive higher-order terms. For a cubic flux, traveling waves can be calculated explicitly for linear dissipative and dispersive terms. Information about their existence can be used to solve the Riemann problem, in which we find solutions for some data that are different from the classical Lax–Oleinik construction. We consider dispersive terms of a BBM type and show that the calculation of traveling waves is somewhat more intricate than for a KdV-type dispersion. The explicit calculation is based upon the calculation of parabolic invariant manifolds for the associated ODE describing traveling waves. The results extend to the p-system of one-dimensional elasticity with a cubic stress–strain law.

General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters

Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.