Moysey Brio
 Associate Professor, Mathematics
 Associate Professor, Applied Mathematics  GIDP
 Member of the Graduate Faculty
Contact
 (520) 6214582
 Environment and Natural Res. 2, Rm. S407
 Tucson, AZ 85719
 brio@arizona.edu
Awards
 Visiting Scholar
 CNRS (French Center for Scientific Research), sabbatical at INSA, Rouen, France, Spring 2018
Interests
No activities entered.
Courses
202324 Courses

Dissertation
MATH 920 (Spring 2024) 
Formal Math Reasong+Wrtg
MATH 323 (Spring 2024) 
Independent Study
MATH 599 (Spring 2024) 
Research
MATH 900 (Spring 2024) 
Discrete Mathematics
MATH 243 (Fall 2023) 
Dissertation
MATH 920 (Fall 2023) 
Math Analysis Engineers
MATH 322 (Fall 2023) 
Research
MATH 900 (Fall 2023)
202223 Courses

Dissertation
MATH 920 (Spring 2023) 
Math Analysis Engineers
MATH 322 (Spring 2023) 
Numerical Analysis Pde
MATH 576B (Spring 2023) 
Dissertation
MATH 920 (Fall 2022) 
Honors Thesis
MATH 498H (Fall 2022) 
Math Analysis Engineers
MATH 322 (Fall 2022) 
Numerical Analysis Pde
MATH 576A (Fall 2022)
202122 Courses

Dissertation
MATH 920 (Spring 2022) 
Honors Thesis
MATH 498H (Spring 2022) 
Independent Study
MATH 599 (Spring 2022) 
Math Analysis Engineers
MATH 322 (Spring 2022) 
Dissertation
MATH 920 (Fall 2021) 
Math Analysis Engineers
MATH 322 (Fall 2021)
202021 Courses

Adv Applied Mathematics
MATH 422 (Summer I 2021) 
Adv Applied Mathematics
MATH 522 (Summer I 2021) 
Dissertation
MATH 920 (Spring 2021) 
Honors Thesis
MATH 498H (Spring 2021) 
Numerical Analysis Pde
MATH 576B (Spring 2021) 
Thesis
MATH 910 (Spring 2021) 
Dissertation
MATH 920 (Fall 2020) 
Honors Thesis
MATH 498H (Fall 2020) 
Independent Study
MATH 599 (Fall 2020) 
Math Analysis Engineers
MATH 322 (Fall 2020) 
Numerical Analysis Pde
MATH 576A (Fall 2020) 
Thesis
MATH 910 (Fall 2020)
201920 Courses

Independent Study
MATH 599 (Spring 2020) 
Partial Diff Equations
MATH 553B (Spring 2020) 
Research
MATH 900 (Spring 2020) 
Independent Study
MATH 599 (Fall 2019) 
Math Analysis Engineers
MATH 322 (Fall 2019) 
Partial Diff Equations
MATH 553A (Fall 2019) 
Research
MATH 900 (Fall 2019)
201819 Courses

Independent Study
MATH 599 (Spring 2019) 
Numerical Analysis Pde
MATH 576B (Spring 2019) 
Case Study Applied Math
MATH 586 (Fall 2018) 
Independent Study
MATH 599 (Fall 2018) 
Numerical Analysis Pde
MATH 576A (Fall 2018) 
Research Tutorial Group
MATH 596G (Fall 2018)
201718 Courses

Independent Study
MATH 599 (Spring 2018) 
Sports Analytics
MATH 367 (Spring 2018) 
Case Study Applied Math
MATH 586 (Fall 2017) 
Independent Study
MATH 599 (Fall 2017) 
Numerical Analysis
CSC 575A (Fall 2017) 
Numerical Analysis
MATH 575A (Fall 2017) 
Research Tutorial Group
MATH 596G (Fall 2017) 
Topics in Undergrad Math
MATH 396T (Fall 2017)
201617 Courses

Case Study Applied Math
MATH 586 (Spring 2017) 
Independent Study
MATH 599 (Spring 2017) 
Numerical Analysis Pde
MATH 576B (Spring 2017) 
Topics in Undergrad Math
MATH 396T (Spring 2017) 
Case Study Applied Math
MATH 586 (Fall 2016) 
Directed Research
MATH 392 (Fall 2016) 
Independent Study
MATH 599 (Fall 2016) 
Numerical Analysis Pde
MATH 576A (Fall 2016) 
Research Tutorial Group
MATH 596G (Fall 2016) 
Thesis
MATH 910 (Fall 2016)
201516 Courses

Case Study Applied Math
MATH 586 (Spring 2016) 
Directed Research
MATH 392 (Spring 2016) 
Independent Study
MATH 499 (Spring 2016) 
Independent Study
MATH 599 (Spring 2016) 
Numerical Analysis
MATH 575B (Spring 2016) 
Senior Capstone
BIOC 498 (Spring 2016)
Scholarly Contributions
Journals/Publications
 Brio, M. (2023). Localized eigenvectors on metric graphs. Journal of Mathematics and Computers in Simulation, 215, 352372.
 Brio, M. (2022). FDTD method with explicit noniterative and second order treatment for Kerr nonlinearities. IEEE Journal on Multiscale and Multiphysics Computational Techniques,, 7, 195199.
 Brio, M., Caputo, J., & Kravitz, H. (2022). Spectral solution of PDEs on networks. Applied Numerical Mathematics.
 Kano, P., Brio, M., & Bailey, J. (2021). "Optimal parameter selection in Week's method for numerical Laplace transform inversion based on machine learning". Journal of Algorithms and Computational Technology, 15, 122.
 Brio, M., Caputo, J., Gwirtz, K., Liu, J., & Maimistov, A. (2019). Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: Theoretical and numerical analysis. Wave Motion, 89, 4356.
 Brio, M., Nehls, J., Dineen, C., Poole, C., & Moloney, J. (2019). Secondorder, stable, fully anisotropic FDTD. Int. J. Num. Modeling:Electronic Networks, Devices and Fields, 32(2), e2521.
 Brio, M. (2015). Exponential timedifferencing with embedded RungeKutta adaptive step control. Journal of Computational Physics, 280, 579601.
 Brio, M. (2014). FiniteDifference TimeDomain simulatio n of spacetime cloak. Optics Express, 22(10), 1208712095.
 Liu, J., Brio, M., & Moloney, J. V. (2014). An overlapping Yee finitedifference timedomain method for material interfaces between anisotropic dielectrics and general dispersive or perfect electric conductor media. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 27(1), 2233.More infoAbstract: A novel stable anisotropic finitedifference timedomain (FDTD) algorithm based on the overlapping cells is developed for solving Maxwell's equations of electrodynamics in anisotropic media with interfaces between different types of materials, such as the interface between anisotropic dielectrics and dispersive medium or perfect electric conductor (PEC). The previous proposed conventional anisotropic FDTD methods suffer from the latetime instability due to the extrapolation of the field components near the material interface. The proposed anisotropic overlapping Yee FDTD method is stable, as it relies on the overlapping cells to provide the collocated field values without any interpolation or extrapolation. Our method has been applied to simulate electromagnetic invisibility cloaking devices with both anisotropic dielectrics and PEC included in the computational domain. Numerical results and eigenvalue analysis confirm that the conventional anisotropic FDTD method is weakly unstable, whereas our method is stable. Copyright © 2013 John Wiley & Sons, Ltd. Copyright © 2013 John Wiley & Sons, Ltd.
 Liu, J., Brio, M., & Moloney, J. V. (2014). Transformation optics based local mesh refinement for solving Maxwell's equations. Journal of Computational Physics, 258, 359370.More infoAbstract: In this paper, a novel local mesh refinement algorithm based on transformation optics (TO) has been developed for solving the Maxwell's equations of electrodynamics. The new algorithm applies transformation optics to enlarge a small region so that it can be resolved by larger grid cells. The transformed anisotropic Maxwell's equations can be stably solved by an anisotropic FDTD method, while other subgridding or adaptive mesh refinement FDTD methods require timespace field interpolations and often suffer from the latetime instability problem. To avoid small time steps introduced by the transformation optics approach, an additional application of the mapping of the material matrix to a dispersive material model is employed. Numerical examples on scattering problems of dielectric and dispersive objects illustrate the performance and the efficiency of the transformation optics based FDTD method. © 2013 Elsevier Inc.
 Pasenow, B., Dineen, C., Hader, J., Brio, M., Moloney, J. V., Koch, S. W., Chen, S. H., Becker, A., & JarońBecker, A. (2013). Anisotropic terahertz response from a strongfield ionized electronion plasma. Physical Review E  Statistical, Nonlinear, and Soft Matter Physics, 87(3).More infoAbstract: A microscopic theory is adapted to compute the timeresolved terahertz (THz) probe response of a dynamically evolving, strongfield ionized electronion plasma. The numerical solutions show that the relaxation of the initially highly anisotropic carrier distributions leads to a polarization dependent shorttime THz response. This THz polarization discrimination gradually vanishes as the plasma approaches a thermodynamic equilibrium configuration via electronelectron and electronion scattering. The detailed carrierrelaxation dynamics causes a strongly nonmonotonic timedevelopment of the THz absorption. © 2013 American Physical Society.
 Kano, P. O., Brio, M., Dostert, P., & Cain, J. (2012). DempsterShafer evidential theory for the automated selection of parameters for Talbot's method contours and application to matrix exponentiation. Computers and Mathematics with Applications, 63(11), 15191535.More infoAbstract: In this paper, the DempsterShafer theory of evidential reasoning is applied to the problem of optimal contour parameters selection in Talbot's method for the numerical inversion of the Laplace transform. The fundamental concept is the discrimination between rules for the parameters that define the shape of the contour based on the features of the function to invert. To demonstrate the approach, it is applied to the computation of the matrix exponential via numerical inversion of the corresponding resolvent matrix. Training for the DempsterShafer approach is performed on random matrices. The algorithms presented have been implemented in MATLAB. The approximated exponentials from the algorithm are compared with those from the rational approximation for the matrix exponential returned by the MATLAB expm function. © 2012 Elsevier Ltd. All rights reserved.
 Liu, J., Brio, M., & Moloney, J. V. (2012). Subpixel smoothing finitedifference timedomain method for material interface between dielectric and dispersive media. Optics Letters, 37(22), 48024804.More infoAbstract: In this Letter, we have shown that the subpixel smoothing technique that eliminates the staircasing error in the finitedifference timedomain method can be extended to material interface between dielectric and dispersive media by local coordinate rotation. First, we show our method is equivalent to the subpixel smoothing method for dielectric interface, then we extend it to a more general case where dispersive/dielectric interface is present. Finally, we provide a numerical example on a scattering problem to demonstrate that we were able to significantly improve the accuracy. © 2012 Optical Society of America.
 Kano, P. O., & Brio, M. (2010). Application of Post's formula to optical pulse propagation in dispersive media. Computers and Mathematics with Applications, 59(2), 629650.More infoAbstract: In this paper we consider the propagation of optical pulses in dielectric media with nontrivial dispersion relations. In particular, we implement Post's Laplace transform formula to invert in time the FourierLaplace space coefficients which arise from the joint space solution of the optical dispersive wave equation. Due to the inefficiency of a direct application of this formula, we have considered and present here two more efficient implementations. In the first, the GaverPost method, we utilize the well known Gaver functionals but store intermediate calculations to improve efficiency. The second, the BellPost method, involves an analytic reformulation of Post's formula such that knowledge of the dispersion relation and its derivatives are sufficient to invert the coefficients from Laplace space to time. Unlike other approaches to the dispersive wave equation which utilize a DebyeLorentzian assumption for the dispersion relation, our algorithm is applicable to general MaxwellHopkinson dielectrics. Moreover, we formulate the approach in terms of the FourierLaplace coefficients which are characteristic of a dispersive medium but are independent of the initial pulse profile. They thus can be precomputed and utilized when necessary in a realtime system. Finally, we present an illustration of the method applied to optical pulse propagation in a free space and in two materials with Coletype dispersion relations, room temperature ionic liquid (RTIL) hexafluorophosphate and brain white matter. From an analysis of these examples, we find that both methods perform better than a standard Postformula implementation. The BellPost method is the more robust of the two, while the GaverPost is more efficient at high precision and Post formula approximation orders. © 2009 Elsevier Ltd. All rights reserved.
 Liu, J., Brio, M., Zeng, Y., Zakharian, A. R., Hoyer, W., Koch, S. W., & Moloney, J. V. (2010). Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model. Journal of Computational Physics, 229(17), 59215932.More infoAbstract: In this paper we present a numerical method for solving a threedimensional coldplasma system that describes electron gas dynamics driven by an external electromagnetic wave excitation. The nonlinear Drude dispersion model is derived from the coldplasma fluid equations and is coupled to the Maxwell's field equations. The FiniteDifference TimeDomain (FDTD) method is applied for solving the Maxwell's equations in conjunction with the timesplit semiimplicit numerical method for the nonlinear dispersion and a physics based treatment of the discontinuity of the electric field component normal to the dielectricmetal interface. The application of the proposed algorithm is illustrated by modeling light pulse propagation and secondharmonic generation (SHG) in metallic metamaterials (MMs), showing good agreement between computed and published experimental results. © 2010 Elsevier Inc.
 Liu, J., Brio, M., & Moloney, J. V. (2009). A diagonal splitcell model for the overlapping Yee FDTD method. Acta Mathematica Scientia, 29(6), 16701676.More infoAbstract: In this paper, we present a nonorthogonal overlapping Yee method for solving Maxwell's equations using the diagonal splitcell model. When material interface is presented, the diagonal splitcell model does not require permittivity averaging so that better accuracy can be achieved. Our numerical results on optical force computation show that the standard FDTD method converges linearly, while the proposed method achieves quadratic convergence and better accuracy. © 2009 Wuhan Institute of Physics and Mathematics.
 Liu, J., Brio, M., & Moloney, J. V. (2009). Overlapping Yee FDTD method on nonorthogonal grids. Journal of Scientific Computing, 39(1), 129143.More infoAbstract: We propose a new overlapping Yee (OY) method for solving timedomain Maxwell's equations on nonorthogonal grids. The proposed method is a direct extension of the FiniteDifference TimeDomain (FDTD) method to irregular grids. The OY algorithm is stable and maintains secondorder accuracy of the original FDTD method, and it overcomes the latetime instability of the previous FDTD algorithms on nonorthogonal grids. Numerical examples are presented to illustrate the accuracy, stability, convergence and efficiency of the OY method. © 2008 Springer Science+Business Media, LLC.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Bibliography. Mathematics in Science and Engineering, 213(C), 273287.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Convergence theory for initial value problems. Mathematics in Science and Engineering, 213(C), 109144.More infoAbstract: The convergence theory for numerical methods approximating timedependent problems parallels the theory of ordinary differential equations (ODEs) where two types of behavior are studied, namely: (1) the finite time solution and (2) the longtime asymptotic behavior where the solution either passes through an initial transient state and sets into a steady state, or evolves into a periodic or chaotic motion, or escapes to infinity. We describe the notions of consistency, stability, local and global error estimates, resolution and order of accuracy, followed by LaxRichtmyer equivalence theorem. The rest of the chapter is devoted to practical implications of the convergence theory in terms of the resolution and error estimates together with von Neumann and CFL stability restrictions. © 2010 Elsevier Ltd.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Discretization methods. Mathematics in Science and Engineering, 213(C), 59108.More infoAbstract: Discretization of partial differential equations (PDEs) is based on the theory of function approximation, with several key choices to be made: an integral equation formulation, or approximate solution operator; the type of discretization, defined by the function subspace in which the solution is approximated; the choice of grids, e.g. regular versus irregular grids to conform to the geometry, or static versus solution adaptive grids. We explore some of the common approaches to the choice of form of the PDE and the spacetime discretization, leaving discussion of the grids for a later chapter. The goal is to introduce the reader to various forms of discretization and to illustrate the numerical performance of different methods. In particular, we will address how to choose a method that is accurate, robust and efficient for the problem at hand. © 2010 Elsevier Ltd.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Numerical boundary conditions. Mathematics in Science and Engineering, 213(C), 145174.More infoAbstract: Boundary conditions arise in the process of numerical implementation of given physical boundary conditions on the original physical or truncated domain due to restrictions imposed by computational resources. The latter situation is often encountered when an infinite domain is truncated or remapped onto a finite computational domain. Numerical implementation of the physical or numerical interface boundary conditions should preserve the accuracy and stability of the inner numerical method. The inaccuracies and instabilities created by numerical implementation of the interface and boundary conditions may be localized at the boundaries or interfaces, but more often they may propagate through out the whole computational domain. Examples include: (a) transparent boundary conditions for hyperbolic and dispersive systems, (b) Berenger's perfectly matched layer (PML) boundary conditions applied to Maxwell's equations (c) stability analysis in the presence of boundaries and interfaces, and (d) grid interfaces and material interfaces in semiconductor device simulation models, and finitedifference timedomain (FDTD) discretization of Maxwell's equations. © 2010 Elsevier Ltd.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Numerical grid generation. Mathematics in Science and Engineering, 213(C), 251272.More infoAbstract: In this chapter we overview the basic ideas behind numerical grid adaptation to geometrical features of the domain, its interfaces and boundaries, as well as adaptation to the solution features, such as high gradients, that are often employed to improve the accuracy and efficiency of the computation. Through representative examples we discuss implementation and stability issues for static and dynamic remapping grid generation methods. The last two sections describe Level Set and Front Tracking Methods for propagating moving interfaces and fronts. © 2010 Elsevier Ltd.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Overview of partial differential equations. Mathematics in Science and Engineering, 213(C), 157.More infoAbstract: Chapter 1 begins with some examples of partial differential equations in science and engineering and their linearization and dispersion equations. The concepts of wellposedness, regularity, and solution operator for systems of partial differential equations (PDE's) are discussed. Instabilities can arise from both numerical methods and from real physical instabilities. Some physical instabilities are described, including: (a) the distinction between convective and absolute instabilities, (b) the RayleighTaylor and KelvinHelmholtz instabilities in fluids, (c) wave breaking and gradient catastrophe in gas dynamics and in conservation laws, (d) modulational or Benjamin Feir instabilities and nonlinear Schrödinger related equations, (e) threewave resonant interactions and explosive instabilities associated with negative energy waves. Basic wave concepts are described (e.g. wavenumber surfaces, group velocity, wave action, wave diffraction, and wave energy equations). A project from semiconductor transport modeling is described. © 2010 Elsevier Ltd.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Preface. Mathematics in Science and Engineering, 213(C), vvii.
 Brio, M., Webb, G. M., & Zakharian, A. R. (2008). Problems with multiple temporal and spatial scales. Mathematics in Science and Engineering, 213(C), 175249.More infoAbstract: Problems with multiple disparate temporal and spatial scales are at the heart of most physical problems, as well as in mathematical modeling problems, in economics and social sciences. Numerical treatment usually allows one to deal with two, or at most, three disparate scales due to limitations of computer resources. Problems with different scales arise when the interest is in the description of the problem on the larger scales, for example the motion of weather fronts in the presence of fast gravity waves; the average trajectory of a fast rotating charged particle in a magnetic field, in which one averages over the fast gyroperiod of the particle; the behavior of society incorporating the dynamics of individuals; the role of singularities in solutions of differential equations or the role of geometry; etc. The methods of dealing with such problems are based on either resolving (if ever possible) or not resolving the fine scales. In the latter case, asymptotic analysis of the dominant terms is often used to separate scales or compute the cumulative effect of the small scale motion on the large scale dynamics. For problems where scales are weakly coupled, numerical treatment may often produce physical results as long as the choice of the method has remnants of the smaller scale behavior such as: damping, diffusion, dispersion, etc. For the second type of problem, underresolved numerical computations usually never produce physical results as cumulative effects are not of the form of the truncation errors, in addition to the fact that the details of the small scale motion are important and have to be accounted for on either a theoretical or experimental basis. In this Chapter we consider examples from stiff ODEs, longtime integrators, Hamiltonian systems, multisymplectic systems, hyperbolic conservation laws, Godunov methods, Riemann solvers with slope/flux limiters. © 2010 Elsevier Ltd.
 Kano, P., Barker, D., & Brio, M. (2008). Analysis of the analytic dispersion relation and density of states of a selected photonic crystal. Journal of Physics D: Applied Physics, 41(18).More infoAbstract: In this paper, we consider an anisotropic threedimensional photonic crystal for which an analytic solution to Maxwell's equations can be derived. We also discuss the reduction to a twodimensional variant which occurs for isotropic materials. For the twodimensional case, we integrate an analytic dispersion relation to accurately calculate the density of states. A comparison is made with a numerically determined density of states computed from the dispersion surface generated by the MIT photonicbands package (MPB). Close agreement between the analytic and numerical results is shown. Finally, we utilize the analytically calculated density of states to characterize the thermally pumped terahertz emission from the photonic crystal. © 2008 IOP Publishing Ltd.
 Hongbo, L. i., Brio, M., Schülzgen, A., Peyghambarian, N., & Moloney, J. V. (2007). Multimode interference in circular stepindex fibers studied with the mode expansion approach. Journal of the Optical Society of America B: Optical Physics, 24(10), 27072720.More infoAbstract: The mode expansion approach in vectorial form, using a complete set of guided modes of a circular stepindex fiber (SIF), is developed and applied to analyze multimode interference in multimode fibers (MMFs) for the first time, to the best of our knowledge. The complete set of guided modes of an SIF is defined based on its modal properties, and a suitable modal orthogonality relation is identified to evaluate the coefficients in a mode expansion. An algorithm, adaptive to incident fields, is then developed to systematically and efficiently perform mode expansion in highly MMFs. The mode expansion approach is successfully applied to investigate the modeselection properties of coreless fiber segments incorporated in multicore fiber lasers and the selfimaging in MMFs. © 2007 Optical Society of America.
 Sørensen, M., & Brio, M. (2007). The MaxwellLorentz model for optical pulses. European Physical Journal: Special Topics, 147(1), 253264.More infoAbstract: Dynamics of optical pulses, especially of ultra short femtosecond pulses, are of great technological and theoretical interest. The dynamics of optical pulses is usually studied using the nonlinear Schrödinger (NLS) equation model. While such approach workssurprisingly well for description of pulse propagation, at least in the femtosecond regime,the full system posses a wealth of new wave phenomena that we explore in this paper: envelope collapse regularization resulting in the orignal pulse splitting; development of infinite gradients in the carrier wave; existence of the stable top hat traveling wave solutions formed by a pair of kink antikink shaped optical waves. © EDP Sciences/Societé Italiana di Fisica/SpringerVerlag 2007.
 Zakharian, A. R., Brio, M., Dineen, C., & Moloney, J. V. (2006). Secondorder accurate FDTD space and time grid refinement method in three space dimensions. IEEE Photonics Technology Letters, 18(11), 12371239.More infoAbstract: We present an algorithm based on the finitedifference timedomain method for local refinement of a threedimensional computational grid in space and time. The method has secondorder accuracy in space and time as verified in the numerical examples. A number of test cases with material traverse normal to the grid interfaces were used to assess the long integration time stability of the algorithm. Resulting improvements in the computation time are discussed for a photonic crystal microcavity design that exhibits a sensitive dependence of the quality factor on subwavelength geometrical features. © 2006 IEEE.
 Gundu, K. M., Brio, M., & Moloney, J. V. (2005). A mixed highorder vector finite element method for waveguides: Convergence and spurious mode studies. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 18(5), 351364.More infoAbstract: In this article we construct and analyse the performance of various combinations of high order edge elements for the transverse field component with high order node elements for the longitudinal field component. The overall efficiency and mode resolution improve dramatically with high order elements. For example, quadratic/cubic elements provide accuracy of 104 in eigenvalue 1000 times faster and resolve the discontinuities in the field very accurately. In the mixed FEM for waveguides we observed that edge elements may produce spurious modes. When the order of the edge elements is equal or larger than the order of the node elements, spurious and degenerate modes are present. Any other combination of the orders did not produce spurious modes. We study the spurious modes from the divergence point of view. We also study the convergence properties of high order elements and compare the results with the theory. Copyright © 2005 John Wiley & Sons, Ltd.
 Sørensen, M. P., Webb, G. M., Brio, M., & Moloney, J. V. (2005). Kink shape solutions of the MaxwellLorentz system. Physical Review E  Statistical, Nonlinear, and Soft Matter Physics, 71(3).More infoAbstract: In the limit of high amplitude oscillating electromagnetic fields, a sequence of kink antikink shaped optical waves has been found in the Maxwell's equations coupled to a single Lorentz oscillator and with Kerr nonlinearity. The individual kinks and antikinks result from a traveling wave assumption and their stability has been assessed by numerical simulations. For typical physical parameter values the kink width is of the order of tens of femtoseconds. © 2005 The American Physical Society.
 Zakharian, A. R., Moloney, J. V., Dineen, C., & Brio, M. (2005). Application of the FiniteDifference TimeDomain method with local grid refinement to nanostructure design. Progress in Biomedical Optics and Imaging  Proceedings of SPIE, 5728, 154163.More infoAbstract: The FiniteDifference TimeDomain (FDTD) method is often a viable alternative to other computational methods used for the design of subwavelength components of photonic devices. We describe an FDTD based grid refinement method, which reduces the computational cell size locally, using a collection of nested rectangular grid patches. On each patch, a standard FDTD update of the electromagnetic fields is applied. At the coarse/fine grid interfaces the solution is interpolated, and consistent circulation of the fields is enforced on shared cell edges. Stability and accuracy of the scheme depend critically on the update scheme, space and time interpolation, and a proper implementation of flux conditions at mesh boundaries. Compared to the conformal grid refinement, the method enables better efficiency by using nonconformal grids around the region of interest and by refining both space and time dimensions, which leads to considerable savings in computation time. We discuss the advantages and shortcomings of the method and present its application to the problem of computation of a quality factor of a 3D photonic crystal microcavity.
 Gundu, K. M., Kano, P., Mafi, A., Moloney, J. V., & Brio, M. (2004). Numerical simulations of active microstructured fibers. Proceedings of the 4th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD '04, 6970.More infoAbstract: The development of a state of the art numerical simulator based on beam propagation method (BPM) is discussed. A mixed edgenode full vectorial finite element method (VFEM) is proposed for the numerical simulation of active microstructured fibers. The features of this simulator include mixed edgenode full VFEBPM; accurate treatment of curved surfaces for finite differences and a propagator based on numerical Laplace transform (NLT). The development of an algorithm for optimal parameter selection and automatic error control is also discussed. The main difficulty in the development of a VFEBPM is the instability due to the presence of complex modes.
 Kano, P., Kouznetsov, D., Moloney, J. V., & Brio, M. (2004). Slab delivery of incoherent pump light to doubleclad fiber amplifiers: Numerical simulations. IEEE Journal of Quantum Electronics, 40(9), 13011305.More infoAbstract: Compact fiber amplifiers with tapered slab delivery of the pump are analyzed numerically. A procedure for the selection of the structural parameters of this device is presented together with a simple asymptotic estimate for the efficiency. Simulations based on the scalar paraxial beam propagation equation demonstrate the validity of this estimate. © 2004 IEEE.
 Moloney, J. V., Zakharian, A., Dineen, C., & Brio, M. (2004). AMR FDTD solver for nanophotonic and plasmonic applications. Proceedings of SPIE  The International Society for Optical Engineering, 5451, 97104.More infoAbstract: Uniform, nonuniform and adaptive mesh refinement FDTD approaches to solving 3D Maxwell's equations are compared and contrasted. Specific applications of such schemes to optical memory, nanophotonics and plasmonics problems will be illustrated.
 Webb, G., Sørensen, M. P., Brio, M., Zakharian, A. R., & Moloney, J. V. (2004). Variational principles, Lie point symmetries, and similarity solutions of the vector Maxwell equations in nonlinear optics. Physica D: Nonlinear Phenomena, 191(12), 4980.More infoAbstract: The vector Maxwell equations of nonlinear optics coupled to a single Lorentz oscillator and with instantaneous Kerr nonlinearity are investigated by using Lie symmetry group methods. Lagrangian and Hamiltonian formulations of the equations are obtained. The aim of the analysis is to explore the properties of Maxwell's equations in nonlinear optics, without resorting to the commonly used nonlinear Schrödinger (NLS) equation approximation in which a high frequency carrier wave is modulated on long length and time scales due to nonlinear sideband wave interactions. This is important in femtosecond pulse propagation in which the NLS approximation is expected to break down. The canonical Hamiltonian description of the equations involves the solution of a polynomial equation for the electric field E, in terms of the canonical variables, with possible multiple real roots for E. In order to circumvent this problem, noncanonical Poisson bracket formulations of the equations are obtained in which the electric field is one of the noncanonical variables. Noether's theorem, and the Lie point symmetries admitted by the equations are used to obtain four conservation laws, including the electromagnetic momentum and energy conservation laws, corresponding to the space and time translation invariance symmetries. The symmetries are used to obtain classical similarity solutions of the equations. The traveling wave similarity solutions for the case of a cubic Kerr nonlinearity, are shown to reduce to a single ordinary differential equation for the variable y=E2, where E is the electric field intensity. The differential equation has solutions y=y(ξ), where ξ=zst is the traveling wave variable and s is the velocity of the wave. These solutions exhibit new phenomena not obtainable by the NLS approximation. The characteristics of the solutions depends on the values of the wave velocity s and the energy integration constant ε. Both smooth periodic traveling waves and nonsmooth solutions in which the electric field gradient diverges (i.e. solutions in which Eξ→∞ at specific values of E, but where E is bounded) are obtained. The traveling wave solutions also include a kinktype solution, with possible important applications in femtosecond technology. © 2003 Elsevier B.V. All rights reserved.
 Sørensen, M. P., Brio, M., Webb, G. M., & Moloney, J. V. (2002). Solitary waves, steepening and initial collapse in the MaxwellLorentz system. Physica D: Nonlinear Phenomena, 170(34), 287303.More infoAbstract: We present a numerical study of Maxwell's equations in nonlinear dispersive optical media describing propagation of pulses in one Cartesian space dimension. Dispersion and nonlinearity are accounted for by a linear Lorentz model and an instantaneous Kerr nonlinearity, respectively. The dispersion relation reveals various asymptotic regimes such as Schrödinger and KdV branches. Existence of solitontype solutions in the Schrödinger regime and light bullets containing few optical cycles together with dark solitons are illustrated numerically. Envelope collapse regimes of the Schrödinger equation are compared to the full system and an arrest mechanism is clearly identified when the spectral width of the initial pulse broadens beyond the applicability of the asymptotic behavior. We show that beyond a certain threshold the carrier wave steepens into an infinite gradient similarly to the canonical MajdaRosales weakly dispersive system. The weak dispersion in general cannot prevent the wave breaking with instantaneous or delayed nonlinearities. © 2002 Elsevier Science B.V. All rights reserved.
 Brio, M., Zakharian, A. R., & Webb, G. M. (2001). TwoDimensional Riemann Solver for Euler Equations of Gas Dynamics. Journal of Computational Physics, 167(1), 177195.More infoAbstract: We construct a Riemann solver based on twodimensional linear wave contributions to the numerical flux that generalizes the onedimensional method due to Roe (1981, J. Comput. Phys. 43, 157). The solver is based on a multistate Riemann problem and is suitable for arbitrary triangular grids or any other finite volume tessellations of the plane. We present numerical examples illustrating the performance of the method using both first and secondorderaccurate numerical solutions. The numerical flux contributions are due to onedimensional waves and multidimensional waves originating from the corners of the computational cell. Under appropriate CFL restrictions, the contributions of onedimensional waves dominate the flux, which explains good performance of dimensionally split solvers in practice. The multidimensional flux corrections increase the accuracy and stability, allowing a larger time step. The improvements are more pronounced on a coarse mesh and for large CFL numbers. For the secondorder method, the improvements can be comparable to the improvements resulting from a less diffusive limiter. © 2001 Academic Press.
 Webb, G. M., Brio, M., & Zank, G. P. (2001). Erratum: Steady MHD flows with an ignorable coordinate and the potential transonic flow equation (Journal of Plasma Physics (1994) 52 (141188)). Journal of Plasma Physics, 65(3), 255256.
 Webb, G. M., Zakharian, A. R., Brio, M., & Zank, G. P. (2001). Parametric instabilities and wave coupling in Alfvén simple waves. Journal of Plasma Physics, 66(3), 167212.More infoAbstract: The parametric instabilities in the largeamplitude Alfv́en wave (pump waves) were studied. The study was based on the formulation of a wave coupling for magnetohydrodynamic (MHD) waves in a nonuniform background flow. The smallamplitude MHD waves interact with the pump wave via wave coupling coefficients that were dependent on the gradient and time dependence of the background flow. The dispersion equations for different short and longwavelength eigenmodes, based on initial value problem, were utilized in discussing the wave stability phenomenon.
 Hunter, J. K., & Brio, M. (2000). Weak shock reflection. Journal of Fluid Mechanics, 410, 235261.More infoAbstract: We present numerical solutions of a twodimensional inviscid Burgers equation which provides an asymptotic description of the Mach reflection of weak shocks. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley for steady weakshock reflection. A theoretical analysis indicates that there is an expansion fan at the triple point, in addition to the three shocks. The supersonic patch is extremely small, and this work is the first time it has been resolved.
 Webb, G. M., Zakharian, A. R., Brio, M., & Zank, G. P. (2000). Nonlinear and threewave resonant interactions in magnetohydrodynamics. Journal of Plasma Physics, 63(5), 393445.More infoAbstract: Hamiltonian and variational formulations of equations describing weakly nonlinear magnetohydrodynamic (MHD) wave interactions in one Cartesian space dimension are discussed. For wave propagation in uniform media, the wave interactions of interest consist of (a) threewave resonant interactions in which highfrequency waves may evolve on long space and time scales if the wave phases satisfy the resonance conditions; (b) Burgers selfwave steepening for the magnetoacoustic waves, and (c) mean wave field effects, in which a particular wave interacts with the mean wave field of the other waves. The equations describe four types of resonant triads: slowfast magnetoacoustic wave interaction, Alfvenentropy wave interaction, Alfvenmagnetoacoustic wave interaction, and magnetoacousticentropy wave interaction. The formalism is restricted to coherent wave interactions. The equations are used to investigate the Alfvenwave decay instability in which a largeamplitude forward propagating Alfven wave decays owing to threewave resonant interaction with a backwardpropagating Alfven wave and a forwardpropagating slow magnetoacoustic wave. Exact solutions of the equations for Alfvenentropy wave interactions are also discussed.
 Zakharian, A. R., Brio, M., Hunter, J. K., & Webb, G. M. (2000). The von Neumann paradox in weak shock reflection. Journal of Fluid Mechanics, 422, 193205.More infoAbstract: We present a numerical solution of the Euler equations of gas dynamics for a weakshock Mach reflection in a halfspace. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agreement with the numerical solution of the unsteady transonic smalldisturbance equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weakshock Mach reflection. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.
 Webb, G. M., Zakharian, A., Brio, M., & Zank, G. P. (1999). Wave interactions in magnetohydrodynamics, and cosmicraymodified shocks. Journal of Plasma Physics, 61(2), 295346.More infoAbstract: Multiplescales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmicraymodified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a largescale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfven waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow, cosmicray squeezing instability effects, and damping due to the diffusing cosmic rays. In the most general case, the evolution equations also contain nonlinear wave interaction terms due to Burgers self wave steepening for the magnetoacoustic modes, resonant three wave interactions, and mean wave field interaction terms. The form of the wave interaction equations in the ideal MHD case is also discussed. Numerical simulations of the fully nonlinear cosmic ray MHD model equations are compared with spectral code solutions of the linear wave interaction equations for the case of perpendicular, cosmicraymodified shocks. The solutions are used to illustrate how the different wave modes can be generated by wave mixing, and the modification of the cosmic ray squeezing instability due to wave interactions. It is shown that the Alfven waves are coupled to the magnetoacoustic and entropy waves due to linear wave mixing, only in background flows with nonzero field aligned electric current and/or vorticity (i.e. if B · Δ × B ≠ 0 and/or B · Δ × u ≠ 0, where B and u are the magnetic field induction and fluid velocity respectively).
 Webb, G. M., Brio, M., & Zank, G. P. (1998). Lagrangian and Hamiltonian aspects of wave mixing in nonuniform media: Waves on strings and waves in gas dynamics. Journal of Plasma Physics, 60(2), 341382.More infoAbstract: Hamiltonian and Lagrangian perturbation theory is used to describe linear wave propagation in inhomogeneous media. In particular, the problems of wave propagation on an inhomogeneous string, and the propagation of sound waves and entropy waves in gas dynamics in one Cartesian space dimension are investigated. For the case of wave propagation on an inhomogeneous heavy string, coupled evolution equations are obtained describing the interaction of the backward and forward waves via wave reflection off gradients in the string density. Similarly, in the case of gas dynamics the backward and forward sound waves and the entropy wave interact with each other via gradients in the background flow. The wave coupling coefficients in the gasdynamical case depend on the gradients of the Riemann invariants R± and entropy S of the background flow. Coupled evolution equations describing the interaction of the different wave modes are obtained by exploiting the Hamiltonian and Poissonbracket structure of the governing equations. Both Lagrangian and Clebschvariable formulations are used. The similarity of the equations to equations obtained by Heinemann and Olbert describing the propagation of bidirectional Alfvén waves in the solar wind is pointed out.
 Webb, G. M., Ratkiewicz, R., Brio, M., & Zank, G. P. (1998). Multidimensional simple waves in gas dynamics. Journal of Plasma Physics, 59(3), 417460.More infoAbstract: A formalism for multidimensional simple waves in gas dynamics using ideas developed by Boillat is investigated. For simplewave solutions, the physical variables depend on a single function φ(r, t). The wave phase φ(r, t) is implicitly determined by an equation of the form(φ) = r·n(φ)λ(φ)t, where n(φ) denotes the normal to the wave front, λ is the characteristic speed of the wave mode of interest, r is the position vector, t is the time, and the function f(φ) determines whether the wave is a centred (f(φ) = 0) or a noncentred (f(φ) ≠ 0) wave. Examples are given of timedependent vortex waves, shear waves and sound waves in one or two space dimensions. The streamlines for the wave reduce to two coupled ordinary differential equations in which the wave phase φ plays the role of a parameter along the streamlines. The streamline equations are expressed in Hamiltonian form. The roles of Clebsch variables, Lagrangian variables, Hamiltonian formulations and characteristic surfaces are briefly discussed. © 1998 Cambridge University Press.
 Webb, G. M., Brio, M., Zank, G. P., & Story, T. (1997). Wavewave interactions in twofluid cosmicray hydrodynamics. Journal of Plasma Physics, 57(PART 3), 631676.More infoAbstract: Multiplescales perturbation methods are used to derive equations describing wavewave interactions in twofluid cosmicray hydrodynamics in one Cartesian space dimension. Two problems are considered: (a) the interaction between shortwavelength waves in a nonuniform largescale background flow, and (b) wave interactions between longwavelength waves propagating through a uniform background medium. The shortwavelength wave equations describe the interaction between the backward and forward thermal gas sound waves and the contact discontinuity eigenmode via 'wave mixing', in which the waves are reflected by gradients in the large scale background flow. The equations also contain quadratic wave interaction terms describing (i) the Burgers selfwave interaction term, (ii) meanwavefield interaction terms in which a specific wave interacts with the mean field of the other waves, and (iii) threewave resonant interactions that describe how a sound wave resonantly interacts with the contact discontinuity to generate a reverse sound wave. In the limit of no interaction between the cosmic rays and thermal gas, and for a uniform background state, the equations reduce to two coupled, integrodifferential Burgers equations derived by Majda and Rosales for resonantly interacting waves in adiabatic gasdynamics. The shortwavelength equations also contain squeezing instability terms associated with the largescale cosmicray pressure gradient, which were first investigated by Drury and Dorfi. A generalized waveaction equation, or canonical waveenergy equation, variational principles, and WKB analyses of the linearized equations are used to investigate the modification of the cosmicray squeezing instability by wave mixing. Coupled Burgers equations are also derived in the longwavelength regime that describe resonant wave interactions for weak, diffusively smoothed cosmicraymodified shocks. In the latter equations, the cosmicraymodified sound waves can resonantly interact with either the contact discontinuity or the cosmicray pressurebalance mode to generate a reverse sound wave.
 Webb, G. M., Zakharian, A., Brio, M., & Zank, G. P. (1997). Hamiltonian aspects of threewave resonant interactions in gas dynamics. Journal of Physics A: Mathematical and General, 30(12), 42274257.More infoAbstract: Equations describing threewave resonant interactions in adiabatic gas dynamics in one Cartesian space dimension derived by Majda and Rosales are expressed in terms of Lagrangian and Hamiltonian variational principles. The equations consist of two coupled integrodifferential Burgers equations for the backward and forward sound waves that are coupled by integral terms that describe the resonant reflection of a sound wave off an entropy wave disturbance to produce a reverse sound wave. Similarity solutions and conservation laws for the equations are derived using symmetry group methods for the special case where the entropy disturbance consists of a periodic sawtooth profile. The solutions are used to illustrate the interplay between the nonlinearity represented by the Burgers selfwave interaction terms and wave dispersion represented by the threewave resonant interaction terms. Hamiltonian equations in Fourier (p, t) space are also obtained where p is the Fourier space variable corresponding to the fast phase variable θ of the waves. The latter equations are transformed to normal form in order to isolate the normal modes of the system.
 Webb, G. M., Brio, M., & Zank, G. P. (1996). LieBäcklund symmetries of dispersionless, magnetohydrodynamic model equations near the triple umbilic point. Journal of Physics A: Mathematical and General, 29(16), 52095240.More infoAbstract: LieBäcklund symmetries and conservation laws are derived for weakly nonlinear magnetohydrodynamic (MHD) equations describing the interaction of the Alfvén and magnetoacoustic modes propagating parallel to the ambient magnetic field, in the parameter regime near the triple umbilic point, where the gas sound speed ag matches the Alfvén speed VA. The dispersive form of the equations can be expressed in Hamiltonian form and admit four Lie point symmetries and conservation laws associated with spacetranslation invariance (momentum conservation), time translation invariance (energy conservation), rotational invariance about the magnetic field B (helicity conservation), plus a further symmetry that is associated with accelerating wave similarity solutions of the equations. The main aim of the paper is a study of the symmetries and conservation laws of the dispersionless equations. The dispersionless equations are of hydrodynamic type and have three families of characteristics analogous to the slow, intermediate and fast modes of MHD and the Riemann invariants for each of these modes are given in closed form. The dispersionless equations are shown to be semiHamiltonian, and to possess two infinite families of symmetries and conservation laws. The analysis emphasizes the role of the Riemann invariants of the dispersionless equations and a hodograph transformation for a restricted version of the equations. © 1996 IOP Publishing Ltd.
 Webb, G. M., Brio, M., Zank, G. P., & Story, T. (1995). Contact and pressure balance structures in twofluid cosmicray hydrodynamics. Astrophysical Journal Letters, 442(2), 822846.More infoAbstract: The role of cosmicraymodified contact discontinuities and pressure balance structures in twofluid cosmicray hydrodynamics in one Cartesian space dimension are investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics. The fundamental wave modes of the twofluid cosmicray hydrodynamic equations in the longwavelength limit consist of the backward and forward propagating cosmicraymodified sound waves, with sound speed dependent on both the cosmicray and thermal gas pressures; the contact discontinuity; and a pressure balance mode in which the sum of the cosmic ray and thermal gas pressure perturbations is zero. The pressure balance mode, like the contact discontinuity is advected with the background flow. The interaction of the pressure balance mode with the contact discontinuity is investigated by means of the method of multiple scales. The thermal gas and cosmicray pressure perturbations satisfy a linear diffusion equation, and entropy perturbations arising from nonisentropic initial conditions for the thermal gas are frozen into the fluid. The contact discontinuity and pressure balance eigenmodes both admit nonzero entropy perturbations in the thermal gas, whereas the cosmicraymodified sound waves are isentropic. The total entropy perturbation is shared between the contact discontinuity and pressure balance eigenmodes, and examples are given in which there is a transfer of entropy between the two modes. In particular, Nwave type density disturbances are obtained which arise as a result of the entropy transfer between the two modes. A weakly nonlinear geometric optics perturbation expansion is used to study the long timescale evolution of the shortwavelength entropy wave and the thermal gas sound waves in a slowly varying, largescale background flow. The weakly nonlinear geometric optics expansion is also used to generalize previous studies of a squeezing instability for shortwavelength sound waves in the two fluid model, by including a weakly nonlinear wave steepening term that leads to shock formation, as well as the effect of long time and space dependence of the background flow. Implications of cosmicraymodified pressure balance structures and contact discontinuities in models of the interaction of traveling interplanetary shocks, and compression and rarefaction waves with the solar wind termination shock are briefly discussed.
 Brio, M., & Hunter, J. K. (1992). Asymptotic equations for conservation laws of mixed type. Wave Motion, 16(1), 5764.More infoAbstract: We derive canonical asymptotic equations for weakly nonlinear solutions of conservation laws of mixed type. When two real wave speeds coalesce and become complex, we obtain the transonic small disturbance equation which changes type from hyperbolic to elliptic. When the coefficient of the nonlinear term in the transonic small disturbance equation vanishes, we derive cubically nonlinear 2 × 2 asymptotic equations. These include canonical equations which are parabolic on a line in state space and strictly hyperbolic away from this line. © 1992.
 Brio, M., & Hunter, J. K. (1992). Mach reflection for the twodimensional Burgers equation. Physica D: Nonlinear Phenomena, 60(14), 194207.More infoAbstract: We study shock reflection for the two 2D Burgers equation. This model equation is an asymptotic limit of the Euler equations, and retains many of the features of the full equations. A von Neumann type analysis shows that the 2D Burgers equation has detachment, sonic, and Crocco points in complete analogy with gas dynamics. Numerical solutions support the detachment/sonic criterion for transition from regular to Mach reflection. There is also strong numerical evidence that the reflected shock in the 2D Burgers Mach reflection forms a smooth wave near the Mach stem, as proposed by Colella and Henderson in their study of the Euler equations. © 1992.
 Brio, M., & Wu, C. (1988). An upwind differencing scheme for the equations of ideal magnetohydrodynamics. Journal of Computational Physics, 75(2), 400422.More infoAbstract: Recently, upwind differencing schemes have become very popular for solving hyperbolic partial differential equations, especially when discontinuities exist in the solutions. Among many upwind schemes successfully applied to the problems in gas dynamics, Roe's method stands out for its relative simplicity and clarity of the underlying physical model. In this paper, an upwind differencing scheme of Roetype for the MHD equations is constructed. In each computational cell, the problem is first linearized around some averaged state which preserves the flux differences. Then the solution is advanced in time by computing the wave contributions to the flux at the cell interfaces. One crucial task of the linearization procedure is the construction of a Roe matrix. For the special case γ = 2, a Roe matrix in the form of a mean value Jacobian is found, and for the general case, a simple averaging procedure is introduced. All other necessary ingredients of the construction, which include eigenvalues, and a complete set of right eigenvectors of the Roe matrix and decomposition coefficients are presented. As a numerical example, we chose a coplanar MHD Riemann problem. The problem is solved by the newly constructed secondorder upwind scheme as well as by the LaxFriedrichs, the LaxWendroff, and the fluxcorrected transport schemes. The results demonstrate several advantages of the upwind scheme. In this paper, we also show that the MHD equations are nonconvex. This is a contrast to the general belief that the fast and slow waves are like sound waves in the Euler equations. As a consequence, the wave structure becomes more complicated; for example, compound waves consisting of a shock and attached to it a rarefaction wave of the same family can exist in MHD. © 1988.
 Brio, M. (1987). Propagation of weakly nonlinear magnetoacoustic waves. Wave Motion, 9(5), 455458.More infoAbstract: Using the reductive perturbation method we show that the modified Burgers equation governs the propagation of a weakly nonlinear slow magnetoacoustic wave near the equilibrium state with zero transverse magnetic field. © 1987.