We identify two new traveling waves of the Holling-Tanner model with weak diffusion. One connects two constant states; at one of them, the model is undefined. The other connects a constant state to a periodic wave train. We exploit the multi-scale structure of the Holling-Tanner model in the weak diffusion limit. Our analysis uses geometric singular perturbation theory, compactification and the blow-up method.

}, journal={DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B}, author={Manukian, Vahagn and Schecter, Stephen}, year={2021}, month={Oct} } @article{ozbag_schecter_2018, title={Stability of combustion waves in a simplified gas-solid combustion model in porous media}, volume={376}, ISSN={["1471-2962"]}, DOI={10.1098/rsta.2017.0185}, abstractNote={We study the stability of the combustion waves that occur in a simplified model for injection of air into a porous medium that initially contains some solid fuel. We determine the essential spectrum of the linearized system at a travelling wave. For certain waves, we are able to use a weight function to stabilize the essential spectrum. We perform a numerical computation of the Evans function to show that some of these waves have no unstable discrete spectrum. The system is partly parabolic, so the linearized operator is not sectorial, and the weight function decays at one end. We use an extension of a recent result about partly parabolic systems that are stabilized by such weight functions to show nonlinear stability.}, number={2117}, journal={PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES}, author={Ozbag, Fatih and Schecter, Stephen}, year={2018}, month={Apr} } @article{lin_schecter_2016, title={Stability of Concatenated Traveling Waves}, volume={28}, ISSN={["1572-9222"]}, DOI={10.1007/s10884-015-9428-z}, number={3-4}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao and Schecter, Stephen}, year={2016}, month={Sep}, pages={867–896} } @article{de maesschalck_schecter_2016, title={The entry-exit function and geometric singular perturbation theory}, volume={260}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2016.01.008}, abstractNote={For small ε>0, the system x˙=ε, z˙=h(x,z,ε)z, with h(x,0,0)<0 for x<0 and h(x,0,0)>0 for x>0, admits solutions that approach the x-axis while x<0 and are repelled from it when x>0. The limiting attraction and repulsion points are given by the well-known entry–exit function. For h(x,z,ε)z replaced by h(x,z,ε)z2, we explain this phenomenon using geometric singular perturbation theory. We also show that the linear case can be reduced to the quadratic case, and we discuss the smoothness of the return map to the line z=z0, z0>0, in the limit ε→0.}, number={8}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={De Maesschalck, Peter and Schecter, Stephen}, year={2016}, month={Apr}, pages={6697–6715} } @article{lin_schecter_2015, title={Stability of concatenated traveling waves: Alternate approaches}, volume={259}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2015.04.015}, abstractNote={We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is asymptotically stable. As in [14], [24], [25], we show that the sequence of traveling waves is itself asymptotically stable: as t→∞, the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Our proof is similar to that of [14] in that it is based on spatial dynamics, Laplace transform, and exponential dichotomies, but it incorporates a number of modifications.}, number={7}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao and Schecter, Stephen}, year={2015}, month={Oct}, pages={3144–3177} } @article{ghazaryan_manukian_schecter_2015, title={Travelling waves in the Holling-Tanner model with weak diffusion}, volume={471}, ISSN={["1471-2946"]}, DOI={10.1098/rspa.2015.0045}, abstractNote={For a wide range of parameters, we study travelling waves in a diffusive version of the Holling–Tanner predator–prey model from population dynamics. Fronts are constructed using geometric singular perturbation theory and the theory of rotated vector fields. We focus on the appearance of the fronts in various singular limits. In addition, periodic travelling waves of relaxation oscillation type are constructed using a recent generalization of the entry–exit function.}, number={2177}, journal={PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES}, author={Ghazaryan, Anna and Manukian, Vahagn and Schecter, Stephen}, year={2015}, month={May} } @article{chapiro_marchesin_schecter_2014, title={Combustion waves and Riemann solutions in light porous foam}, volume={11}, ISSN={["1793-6993"]}, DOI={10.1142/s021989161450009x}, abstractNote={ We prove the existence of traveling waves, and identify the wave sequences appearing in Riemann solutions, for a system of three evolutionary partial differential equations that models combustion of light porous foam under air injection. }, number={2}, journal={JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS}, author={Chapiro, G. and Marchesin, D. and Schecter, S.}, year={2014}, month={Jun}, pages={295–328} } @article{schecter_xu_2014, title={Morse theory for Lagrange multipliers and adiabatic limits}, volume={257}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2014.08.018}, abstractNote={Given two Morse functions $f, μ$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M \times {\mathbb R}$ which sends $(x, η)$ to $f(x) + ημ(x)$. Take a product metric on $M \times {\mathbb R}$, and rescale its ${\mathbb R}$-component by a factor $λ^2$. We show that generically, for large $λ$, the Morse-Smale-Witten chain complex is isomorphic to the one for $f$ and the metric restricted to ${μ^{-1}(0)}$, with grading shifted by one. On the other hand, let $λ\to 0$, we obtain another chain complex, which is geometrically quite different but has the same homology as the singular homology of $μ^{-1}(0)$ and the isomorphism between them is provided by the homotopy by varying $λ$. Our proofs contain both the implicit function theorem on Banach manifolds and geometric singular perturbation theory.}, number={12}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Schecter, Stephen and Xu, Guangbo}, year={2014}, month={Dec}, pages={4277–4318} } @article{ghazaryan_schecter_simon_2013, title={GASLESS COMBUSTION FRONTS WITH HEAT LOSS}, volume={73}, ISSN={["0036-1399"]}, DOI={10.1137/110854540}, abstractNote={For a model of gasless combustion with heat loss, we use geometric singular perturbation theory to show existence of traveling combustion fronts. We show that the fronts are nonlinearly stable in an appropriate sense if an Evans function criterion, which can be verified numerically, is satisfied. For a solid reactant and exothermicity parameter that is not too large, we verify numerically that the criterion is satisfied.}, number={3}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Ghazaryan, Anna and Schecter, Stephen and Simon, Peter L.}, year={2013}, pages={1303–1326} } @article{ghazaryan_latushkin_schecter_2013, title={Stability of Traveling Waves in Partly Parabolic Systems}, volume={8}, ISSN={["1760-6101"]}, DOI={10.1051/mmnp/20138503}, abstractNote={We review recent results on stability of traveling waves in partly parabolic reaction-diffusion systems with stable or marginally stable equilibria. We explain how attention to what are apparently mathematical technicalities has led to theorems that allow one to convert spectral calculations, which are used in the sciences and engineering to study stability of a wave, into detailed, theoretically-based information about the behavior of perturbations of the wave.}, number={5}, journal={MATHEMATICAL MODELLING OF NATURAL PHENOMENA}, author={Ghazaryan, A. and Latushkin, Y. and Schecter, S.}, year={2013}, pages={31–47} } @article{schecter_taylor_2012, title={DAFERMOS REGULARIZATION OF A DIFFUSIVE-DISPERSIVE EQUATION WITH CUBIC FLUX}, volume={32}, ISSN={["1553-5231"]}, DOI={10.3934/dcds.2012.32.4069}, abstractNote={We study existence and spectral stability of stationary solutions of the Dafermos regularization of a much-studied diffusive-dispersive equation with cubic flux. Our study includes stationary solutions that corresponds to Riemann solutions consisting of an undercompressive shock wave followed by a compressive shock wave. We use geometric singular perturbation theory (1) to construct the solutions, and (2) to show that asmptotically, there are no large eigenvalues, and any order-one eigenvalues must be near $-1$ or a certain number $\lambda^*$. We give numerical evidence that $\lambda^*$ is also $-1$. Finally, we use pseudoexponential dichotomies to show that in a space of exponentially decreasing functions, the essential spectrum is contained in Re$ \lambda \le -\delta <0 $.}, number={12}, journal={DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS}, author={Schecter, Stephen and Taylor, Monique Richardson}, year={2012}, month={Dec}, pages={4069–4110} } @article{schecter_sourdis_2010, title={Heteroclinic Orbits in Slow-Fast Hamiltonian Systems with Slow Manifold Bifurcations}, volume={22}, ISSN={["1040-7294"]}, DOI={10.1007/s10884-010-9171-4}, number={4}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Schecter, Stephen and Sourdis, Christos}, year={2010}, month={Dec}, pages={629–655} } @article{ghazaryan_latushkin_schecter_2010, title={STABILITY OF TRAVELING WAVES FOR A CLASS OF REACTION-DIFFUSION SYSTEMS THAT ARISE IN CHEMICAL REACTION MODELS}, volume={42}, ISSN={["0036-1410"]}, DOI={10.1137/100786204}, abstractNote={Stability results are proved for traveling waves in a class of reaction-diffusion systems that arise in chemical reaction models. The class includes systems in which there is no diffusion in some equations. A weight function that decays exponentially at one end is required to stabilize the essential spectrum. Perturbations of the wave in $H^1$ or $BUC$ that are small in both the weighted norm and the unweighted norm are shown to stay small in the unweighted norm and to decay exponentially to a shift of the traveling wave in the weighted norm. Perturbations that are in addition small in the $L^1$ norm decay algebraically to a shift of the wave in the $L^\infty$ norm. A decomposition of the variables that yields a triangular structure for the linearization at one end of the wave is exploited to prove the results. An application to exothermic-endothermic reactions is given.}, number={6}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Ghazaryan, Anna and Latushkin, Yuri and Schecter, Stephen}, year={2010}, pages={2434–2472} } @article{ghazaryan_latushkin_schecter_souza_2010, title={Stability of Gasless Combustion Fronts in One-Dimensional Solids}, volume={198}, ISSN={["1432-0673"]}, DOI={10.1007/s00205-010-0358-y}, number={3}, journal={ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS}, author={Ghazaryan, Anna and Latushkin, Yuri and Schecter, Stephen and Souza, Aparecido J.}, year={2010}, month={Dec}, pages={981–1030} } @article{schecter_szmolyan_2009, title={Persistence of Rarefactions under Dafermos Regularization: Blow-Up and an Exchange Lemma for Gain-of-Stability Turning Points}, volume={8}, ISSN={["1536-0040"]}, DOI={10.1137/080715305}, abstractNote={We construct self-similar solutions of the Dafermos regularization of a system of conservation laws near structurally stable Riemann solutions composed of Lax shocks and rarefactions, with all waves possibly large. The construction requires blowing up a manifold of gain-of-stability turning points in a geometric singular perturbation problem as well as a new exchange lemma to deal with the remaining hyperbolic directions.}, number={3}, journal={SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS}, author={Schecter, Stephen and Szmolyan, Peter}, year={2009}, pages={822–853} } @article{manukian_schecter_2009, title={Travelling waves for a thin liquid film with surfactant on an inclined plane}, volume={22}, ISSN={["1361-6544"]}, DOI={10.1088/0951-7715/22/1/006}, abstractNote={We show the existence of travelling wave solutions for a lubrication model of surfactant-driven flow of a thin liquid film down an inclined plane, in various parameter regimes. Our arguments use geometric singular perturbation theory.}, number={1}, journal={NONLINEARITY}, author={Manukian, Vahagn and Schecter, Stephen}, year={2009}, month={Jan}, pages={85–122} } @article{schecter_2008, title={Exchange lemmas 1: Deng's lemma}, volume={245}, ISSN={["0022-0396"]}, DOI={10.1016/j.jde.2007.08.011}, abstractNote={Deng's lemma gives estimates on the behavior of solutions of ordinary differential equations in the neighborhood of a partially hyperbolic equilibrium. We prove a generalization in which "partially hyperbolic equilibrium" is replaced by "normally hyperbolic invariant manifold."}, number={2}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Schecter, Stephen}, year={2008}, month={Jul}, pages={392–410} } @article{schecter_2008, title={Exchange lemmas 2: General Exchange Lemma}, volume={245}, ISSN={["0022-0396"]}, DOI={10.1016/j.jde.2007.10.021}, abstractNote={Exchange lemmas are used in geometric singular perturbation theory to track flows near normally hyperbolic invariant manifolds. We prove a General Exchange Lemma, and show that it implies versions of existing exchange lemmas for rectifiable slow flows and loss-of-stability turning points.}, number={2}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Schecter, Stephen}, year={2008}, month={Jul}, pages={411–441} } @article{da mota_schecter_2006, title={Combustion fronts in a porous medium with two layers}, volume={18}, ISSN={["1572-9222"]}, DOI={10.1007/s10884-006-9019-0}, number={3}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Da Mota, J. C. and Schecter, S.}, year={2006}, month={Jul}, pages={615–665} } @article{schecter_2006, title={Eigenvalues of self-similar solutions of the dafermos regularization of a system of conservation laws via geometric singular perturbation theory}, volume={18}, ISSN={["1572-9222"]}, DOI={10.1007/s10884-005-9000-3}, number={1}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Schecter, Stephen}, year={2006}, month={Jan}, pages={53–101} } @article{schecter_plohr_marchesin_2004, title={Computation of Riemann solutions using the Dafermos regularization and continuation}, volume={10}, DOI={10.3934/dcds.2004.10.965}, abstractNote={We present a numerical method, based on the Dafermos regularization, for computing a one-parameter family of Riemann solutions of a system of conservation laws. The family is obtained by varying either the left or right state of the Riemann problem. The Riemann solutions are required to have shock waves that satisfy the viscous profile criterion prescribed by the physical model. The system is not required to satisfy strict hyperbolicity or genuine nonlinearity; the left and right states need not be close; and the Riemann solutions may contain an arbitrary number of waves, including composite waves and nonclassical shock waves. The method uses standard continuation software to solve a boundary-value problem in which the left and right states of the Riemann problem appear as parameters. Because the continuation method can proceed around limit point bifurcations, it can sucessfully compute multiple solutions of a particular Riemann problem, including ones that correspond to unstable asymptotic states of the viscous conservation laws.}, number={4}, journal={Discrete and Continuous Dynamical Systems}, author={Schecter, S. and Plohr, B. J. and Marchesin, D.}, year={2004}, pages={965–986} } @article{schecter_2004, title={Existence of Dafermos profiles for singular shocks}, volume={205}, DOI={10.1016/j.ide.2004.06.013}, number={1}, journal={Journal of Differential Equations}, author={Schecter, S.}, year={2004}, pages={185–210} } @article{marchesin_schecter_2003, title={Oxidation heat pulses in two-phase expansive flow in porous media}, volume={54}, ISSN={["0044-2275"]}, DOI={10.1007/pl00012634}, number={1}, journal={ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK}, author={Marchesin, D and Schecter, S}, year={2003}, month={Jan}, pages={48–83} } @article{lin_schecter_2003, title={Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws}, volume={35}, ISSN={["1095-7154"]}, DOI={10.1137/S0036141002405029}, abstractNote={In contrast to a viscous regularization of a systemof n conservation laws, a Dafermos regularization admits many self-similar solutions of the form u = u( X T ). In particular, it is known in many cases that Riemann solutions of a system of conservation laws have nearby self-similar smooth solutions of an associated Dafermos regularization. We refer to these smooth solutions as Riemann-Dafermos solutions. In the coordinates x = X , t =l nT , Riemann-Dafermos solutions become stationary, and their time-asymptotic stability as solutions of the Dafermos regularization can be studied by linearization. We study the stability of Riemann-Dafermos solutions near Riemann solutions consisting of n Lax shock waves. We show, by studying the essential spectrumof the linearized systemin a weighted function space, that stability is determ ined by eigenvalues only. We then use asymptotic methods to study the eigenvalues and eigenfunctions. We find there are fast eigenvalues of order 1 and slow eigenvalues of order 1. The fast eigenvalues correspond to eigenvalues of the viscous profiles for the individual shock waves in the Riemann solution; these have been studied by other authors using Evans function methods. The slow eigenvalues are related to inviscid stability conditions that have been obtained by various authors for the underlying Riemann solution.}, number={4}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Lin, XB and Schecter, S}, year={2003}, pages={884–921} } @article{schecter_2002, title={Traveling-wave solutions of convection-diffusion systems by center manifold reduction}, volume={49}, ISSN={["0362-546X"]}, DOI={10.1016/s0362-546x(01)00097-9}, abstractNote={This paper analyzes the existence of smooth trajectories through singular points of differential algebraic equations, or DAEs, arising from traveling wave solutions of a degenerate convection-diffusion model. The DAE system can be written in the quasilinear form A(x)x′ = b(x). In this setting, singularities are displayed when the matrix A(x) undergoes a rank change. The singular hypersurface may be smoothly crossed by trajectories in a finite time if x* is a geometric singularity satisfying certain directional conditions. The basis of our analysis is a two-phase fluid flow model in one spatial dimension with dissipative mechanism involved.}, number={1}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Schecter, S}, year={2002}, month={Apr}, pages={35–59} } @article{schecter_2002, title={Undercompressive shock waves and the Dafermos regularization}, volume={15}, ISSN={["0951-7715"]}, DOI={10.1088/0951-7715/15/4/318}, abstractNote={For a system of conservation laws in one space dimension, we identify all structurally stable Riemann solutions that include only shock waves. Shock waves are required to satisfy the viscous profile criterion for a given viscosity (B(u)ux)x. Undercompressive shock waves are allowed. We also show that all such Riemann solutions have nearby smooth solutions of the Dafermos regularization with the given viscosity.}, number={4}, journal={NONLINEARITY}, author={Schecter, S}, year={2002}, month={Jul}, pages={1361–1377} } @article{schecter_marchesin_2001, title={Geometric singular perturbation analysis of oxidation heat pulses for two-phase flow in porous media}, volume={32}, ISSN={["0100-3569"]}, DOI={10.1007/bf01233667}, number={3}, journal={BOLETIM DA SOCIEDADE BRASILEIRA DE MATEMATICA}, author={Schecter, S and Marchesin, D}, year={2001}, month={Nov}, pages={237–270} } @article{schecter_1999, title={Codimension-one Riemann solutions: Classical missing rarefaction cases}, volume={157}, ISSN={["0022-0396"]}, DOI={10.1006/jdeq.1998.3590}, abstractNote={Abstract This paper is the third in a series that undertakes a systematic investigation of Riemann solutions of systems of two conservation laws in one spatial dimension. Sixty-three codimension-one degeneracies of such solutions have been identified at which strict hyperbolicity is maintained. In this paper, 18 of the degeneracies (9 pairs), constituting the most classical degeneracies, are studied in detail. Precise conditions for a codimension-one degeneracy are identified in each case, as are conditions for folding of the Riemann solution surface, which can occur in 4 of the cases. Such folding gives rise to local multiplicity or nonexistence of Riemann solutions.}, number={2}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Schecter, S}, year={1999}, month={Sep}, pages={247–318} } @article{marchesin_plohr_schecter_1997, title={An organizing center for wave bifurcation in multiphase flow models}, volume={57}, DOI={10.1137/s0036139995280683}, abstractNote={We consider a one-parameter family of nonstrictly hyperbolic systems of conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter. In this paper we use singularity theory and bifurcation theory of dynamical systems, including Melnikov's method, to find all nearby shock waves that are admissible according to the viscous profile criterion. We use these results to construct a unique solution of the Riemann problem for each left and right state and parameter value in a neighborhood of the unperturbed shock wave solution; together with previous numerical work, this construction completes the solution of the three-phase flow model. In the bifurcation analysis, the unperturbed shock wave acts as an organizing center for the waves appearing in Riemann solutions.}, number={5}, journal={SIAM Journal on Applied Mathematics}, author={Marchesin, D. and Plohr, B. J. and Schecter, S.}, year={1997}, pages={1189–1215} }