linear stability of propagating phase boundaries in capillary fluids abstract dynamical phase changes occurring along planar traveling waves in capillary fluids are considered their multi dimensional stability which is encoded in the spectrum of a third order variational differential operator is addressed by two approaches on one hand an original energy method is used to show that these diffuse interfaces are weakly stable more precisely the spectrum of the involved operator is found to coincide with the imaginary axis on the other hand an evans function technique reveals a bifurcation phenomenon about zero the interpretation of which still needs some clarification it is the counterpart of the surface waves exhibited in an earlier work for the corresponding sharp interfaces

stability of large amplitude ekman hartmann boundary layers in mhd the case of ill prepared data in this paper we study an incompressible highly rotating fluid submitted to a high magnetic field between two planes with a dirichlet boundary condition we investigate the nonlinear stability of ekman hartmann boundary layers under a spectral assumption for general initial data this means that the data can be chosen as an arbitrary but smooth enough three dimensional divergence free vector field independent of the small parameter

renormalization group reduction of pulse dynamics in thermally loaded optical parametric oscillators we derive a perturbed parametrically forced nonlinear schrodinger equation to model pulse evolution in an optical parametric oscillator with absorption induced heating we apply both a rigorous renormalization group rg technique and the formal wilsonian renormalization group to obtain a low dimensional system of equations which captures the mutual interaction of pulses as well as their response to the thermally induced potential we compare the methodologies of the two approaches and find that the two reduced systems agree to leading order and compare well to simulations of the full equation predicting the formation of stably bound pulse pairs

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

erratum to on the complete intersection conjecture of murthy with an example in complete intersections j algebra 458 2016 156 170 abstract this is an erratum to 4 and as well to 3 this article provides examples to indicate inconsistencies in 3 4

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

on the number of generators of ideals in polynomial rings for an ideal i in a noetherian ring r let i be the minimal number of generators of i it is well known that there is a sequence of inequalities i i i i i 1 that are strict in general however murthy conjectured in 1975 that i i i for ideals in polynomial rings whose height equals i i the purpose of this article is to prove a stronger form of the conjecture in case the base field is infinite of characteristic different from 2 namely the equality i i i holds for any ideal i irrespective of its height

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

linear stability of solitary waves of the green naghdi equations we investigate the eigenvalue problem obtained from linearizing the green naghdi equations about solitary wave solutions unlike weakly nonlinear water wave models the physical system considered here has nonlinearity in its highest derivative term this results in more detailed asymptotic analysis of the eigenvalue problem in the presence of a large parameter combining the technique of singular perturbation with the evans function we show that for solitary waves of small amplitude the problem has no eigenvalues of positive real part and the evans function is nonvanishing everywhere except the origin this fact then leads to the linear stability of these solitary waves 2001 john wiley sons inc

young diagrams supercharacters of osp m n and modification rules the supercharacter of osp m n associated with an arbitrary young diagram is defined the distinction is made between osp m n standard and non standard supercharacters the corresponding modification rule which may be used to express a non standard supercharacter in terms of standard supercharacters is presented exemplified and proved this rule involves the removal of n 2 1 boundary strips from the young diagram in the case n 0 the rule reduces to the well known rule appropriate to o m for a non standard supercharacter corresponding to a typical irreducible representation of osp m n the modification yields a single typical standard supercharacter on the other hand for a non standard supercharacter corresponding to an atypical irreducible representation the rule yields a linear combination of atypical standard supercharacters

finite wavelength stability of capillary gravity solitary waves we consider the euler equations describing nonlinear waves on the free surface of a two dimensional inviscid irrotational fluid layer of finite depth for large surface tension bond number larger than 1 3 and froude number close to 1 the system possesses a one parameter family of small amplitude traveling solitary wave solutions we show that these solitary waves are spectrally stable with respect to perturbations of finite wave number in particular we exclude possible unstable eigenvalues of the linearization at the soliton in the long wavelength regime corresponding to small frequency and unstable eigenvalues with finite but bounded frequency arising from non adiabatic interaction of the infinite wavelength soliton with finite wavelength perturbations

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

internal modes of envelope solitons abstract we extend the concept of internal mode to envelope solitons and show that this mode is responsible for long lived weakly damped periodic oscillations of the soliton amplitude observed in numerical simulations we present analytical and numerical results for solitons of the generalized nonlinear schrodinger equation and analyze the example of the cubic quintic nonlinearity in more detail we obtain also analytical criteria for the existence and bifurcations of the soliton internal mode and calculate the rate of the radiation induced damping of the soliton oscillations induced by excitation of the internal mode

nonlinear stability of time periodic viscous shocks in order to understand the nonlinear stability of many types of time periodic travelling waves on unbounded domains one must overcome two main difficulties the presence of embedded neutral eigenvalues and the time dependence of the associated linear operator this problem is studied in the context of time periodic lax shocks in systems of viscous conservation laws using spatial dynamics and a decomposition into separate floquet eigenmodes it is shown that the linear evolution for the time dependent operator can be represented using a contour integral similar to that of the standard time independent case by decomposing the resulting green s distribution the leading order behavior associated with the embedded eigenvalues is extracted sharp pointwise bounds are then obtained which are used to prove that time periodic lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues the latter conditions hold for example for small oscillation time periodic waves that emerge through a supercritical hopf bifurcation from a family of time independent lax shocks of possibly large amplitude

approximation and attractivity properties of the degenerated ginzburg landau equation abstract we are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so called degenerated ginzburg landau equation takes the role of the classical ginzburg landau equation as the amplitude equation of the system this is the case when the relevant nonlinear terms vanish at the bifurcation point here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated ginzburg landau equation in this paper we restrict ourselves to a swift hohenberg kuramoto shivashinsky equation as a model for such a pattern forming system

on the stability of traveling waves in weighted l spaces abstract i present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations i am concerned with those problems in which the essential spectrum of the linearization about the wave contains 0 in these circumstances the stability results of d henry in lecture notes in mathematics vol 840 springer verlag new york 1981 and p bates and c jones dynamics rep 2 1989 1 38 are not applicable as the spectral information does not yield an exponential time decay estimate on the semigroup i show that under certain relatively easily computable conditions the wave is stable in polynomially weighted l spaces furthermore the rate at which the perturbation decays to the wave as t depends on the growth rate of the polynomial near infinity the method of proof uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates for the semigroup are derived through a new technique for estimating the resolvent

conditional stability of unstable viscous shock waves in compressible gas dynamics and mhd extending our previous work in the strictly parabolic case we show that a linearly unstable lax type viscous shock solution of a general quasilinear hyperbolic parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 3 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by mascia and zumbrun the main new difficulty in the hyperbolic parabolic case is to construct an invariant manifold in the absence of parabolic smoothing

asymptotic stability of the critical pulled front in a lotka volterra competition model abstract we prove that the critical pulled front of lotka volterra competition systems is nonlinearly asymptotically stable more precisely we show that perturbations of the critical front decay algebraically with rate t 3 2 in a weighted l space our proof relies on pointwise semigroup methods and utilizes in a crucial way that the faster decay rate t 3 2 is a consequence of the lack of an embedded zero of the evans function at the origin for the linearized problem around the critical front

stability of pulses in nonlinear optical fibers using phase sensitive amplifiers we consider the stability of soliton like pulses propagating in nonlinear optical fibers with periodically spaced phase sensitive amplifiers a situation where the averaged pulse evolution is governed by a fourth order nonlinear diffusion equation similar to the kuramoto sivashinsky or swift hohenberg equations a bifurcation and stability analysis of this averaged equation is carried out and in the limit of small amplifier spacing a steady state pulse solution is shown to be asymptotically stable furthermore both a saddle node bifurcation and a subcritical bifurcation from the zero solution are found analytical results are confirmed using the bifurcation software package auto the analysis provides evidence for the existence of stable pulse solutions for a wide range of parameter values including those corresponding to physically realizable soliton communications systems

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

pointwise green function bounds for shock profiles of systems with real viscosity following the pointwise semigroup approach of zh mz 1 we establish sharp pointwise green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic parabolic systems of conservation laws of dissipative type under the necessary assumptions z 1 z 3 z 4 of spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the connecting profile and hyperbolic stability of the corresponding ideal shock of the associated inviscid system with no additional assumptions on the structure or strength of the shock these bounds are used in a companion paper mz 2 to establish nonlinear stability of small amplitude lax shocks of symmetrizable hyperbolic parabolic systems

existence and stability of n pulses on optical fibers with phase sensitive amplifiers abstract the propagation of pulses in optical communication systems in which attenuation is compensated by phase sensitive amplifiers is investigated a central issue is whether optical fibers are capable of carrying several pieces of information at the same time in this paper multiple pulses are shown to exist for a fourth order nonlinear diffusion model due to kutz and co workers 1994 moreover criteria are derived for determining which of these pulses are stable the pulses arise in a reversible orbit flip a homoclinic bifurcation investigated here for the first time numerical simulations are used to study multiple pulses far away from the actual bifurcation point they confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

on the complete intersection conjecture of murthy abstract suppose a k x 1 x 2 x n is a polynomial ring over a field k and i is an ideal in a m p murthy conjectured that i i i 2 where denotes the minimal number of generators recently fasel 3 settled this conjecture affirmatively when k is an infinite perfect field with 1 2 k always we are able to do the same when k is an infinite field in fact we prove similar results for ideals i in a polynomial ring a r x that contains a monic polynomial and r is essentially smooth algebra over an infinite field k or r is a regular ring over a perfect field k

modification rules for orthosymplectic superalgebras modification rules are presented for finite dimensional graded tensor representations of osp m n with the representations specified by young supertableaux these rules relate non standard supertableaux to standard supertableaux via the removal of continuous boundary hooks all typical tensor representations are treated together with atypical representations which satisfy up to two atypically conditions

a stability index for detonation waves in majda s model for reacting flow abstract using evans function techniques we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in siam j math anal 32 2001 929 commun pure appl math 51 7 1998 797 yielding useful necessary conditions for stability here we carry out the analysis in the context of the majda model a simplified model for reacting flow the method is extended to the full navier stokes equations of reacting flow in g lyng one dimensional stability of detonation waves doctoral thesis indiana university 2002 g lyng k zumbrun stability of detonation waves preprint 2003 the resulting stability condition is satisfied for all nondegenerate i e spatially exponentially decaying weak and strong detonations of the majda model in agreement with numerical experiments of siam j sci statist comput 7u 1986 1059 and analytical results of commun math phys 204 3 1999 551 commun math phys 202 3 1999 547 for a related model of majda and rosales we discuss also the role in the znd limit of degenerate subalgebraically decaying weak detonation and for a modified bump type ignition function deflagration profiles as discussed in siam j math anal 24 1993 968 siam j appl math 55 1995 175 for the full equations

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

stability of periodic solutions of conservation laws with viscosity analysis of the evans function nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena including homoclinic pulse type and periodic solutions along with the standard heteroclinic shock or front type solutions here we investigate stability of periodic traveling waves within the abstract evans function framework established by r a gardner our main result is to derive a useful stability index analogous to that developed by gardner and zumbrun in the traveling front or pulse context giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period t as the traveling wave moreover we show that the periodic stability index has an interpretation analogous to that of the traveling front or pulse index in terms of well posedness of an associated riemann problem for an inviscid medium now to be interpreted as allowing a wider class of measure valued solutions or alternatively in terms of existence and nonsingularity of a local mass map from perturbation mass to potential time asymptotic t periodic states a closely related calculation yields also a complementary long wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period nt n we augment these analytical results with numerical investigations analogous to those carried out by brin in the traveling front or pulse case approximating the spectrum of the linearized operator about the wave

counting eigenvalues via the krein signature in infinite dimensional hamiltonian systems spectra of nonlinear waves in infinite dimensional hamiltonian systems are investigated we establish a connection via the krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave we apply our results to determine the effect of symmetry breaking on the spectral stability of nonlinear waves in weakly coupled nonlinear schrodinger equations

spectral stability of small shock waves this paper is the first in a series of articles that study the eigenvalue problem for small amplitude shock waves in systems of conservation laws with viscosity or relaxation the papers show in various contexts that in the zero amplitude limit appropriately scaled versions of the associated evans bundles converge to suspensions of evans bundles for fixed shock waves in related reduced systems with lower dimensional state space in this article the new geometric framework establishing this connection is introduced in the simplest context that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

multi hump pulses in systems with reflection and phase invariance abstract we investigate the existence and stability of standing and travelling multi hump waves in partial differential equations with reflection and phase symmetries we focus on 2 and 3 pulse solutions that arise near bi foci and apply our results to the complex cubic quintic ginzburg landau equation

stability criterion for bright solitary waves of the perturbed cubic quintic schro dinger equation abstract the stability of the bright solitary wave solution to the perturbed cubic quintic schrodinger equation is considered it is shown that in a certain region of parameter space these solutions are unstable with the instability being manifested as a small positive eignevalue furthermore it is shown that in the complimentary region of parameter space there are no small unstable eigenvalues the proof involves a novel calculation of the evans function which is of interest in its own right as a consequence of the eigenvalue calculation it is additionally shown that n bump bright solitary waves bifurcate from the primary wave

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

travelling waves in a singularly perturbed sine gordon equation we determine the linearised stability of travelling front solutions of a perturbed sine gordon equation this equation models the long josephson junction using the rcsj model for currents across the junction and includes surface resistance for currents along the junction the travelling waves correspond to the so called fluxons and their linear stability is determined by calculating the evans function surface resistance corresponds to a singular perturbation term in the governing equation which specifically complicates the computation of the corresponding evans function both the flow of quasi particles across and along the junction stabilise the waves

exact solutions of the one dimensional quintic complex ginzburg landau equation abstract exact solitary wave solutions of the one dimensional quintic complex ginzburg landau equations are obtained using a method derived from the painleve test for integrability these solutions are expressed in terms of hyperbolic functions and include the pulses and fronts found by van saarloos and hohenberg we also find previously unknown sources and sinks the emphasis is put on the systematic character of the method which breaks away from approaches involving somewhat ad hoc ansatze

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

spectral stability analysis for periodic traveling wave solutions of nls and cgl perturbations abstract we consider cnoidal traveling wave solutions to the focusing nonlinear schrodinger equation nls that have been shown to persist when the nls is perturbed to the complex ginzburg landau equation cgl we show that while these periodic traveling waves are spectrally stable solutions of nls with respect to periodic perturbations they are unstable with respect to bounded perturbations furthermore we use an argument based on the fredholm alternative to find an instability criterion for the persisting solutions to cgl

spectral transverse instability of solitary waves in korteweg fluids the motion of korteweg fluids is governed by the euler korteweg model which admits planar solitary waves for nonmonotone pressure laws such as the van der waals law below critical temperature in an earlier work with danchin descombes and jamet it was shown by variational arguments and numerical computations that some of these solitary waves are stable in one space dimension the purpose here is to study their stability with respect to transverse perturbations in several space dimensions by evans functions techniques and rouche s theorem it is shown that transverse perturbations of large wave length always destabilize solitary waves in the euler korteweg model whereas energy estimates show that perturbations of short wave length tend to stabilize them

modification rules for orthosymplectic superalgebras ii atypical representations for pt i see ibid vol 19 no 3 p 321 7 1986 modification rules are presented for all finite dimensional atypical representations of osp m n with the representations specified by young supertableaux these rules relate non standard supertableaux to standard supertableaux via the removal of continuous boundary hooks

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

combustion waves in hydraulically resistant porous media in a special parameter regime abstract in this paper we study the stability of fronts in a reduction of a well known pde system that is used to model the combustion in hydraulically resistant porous media more precisely we consider the original pde system under the assumption that one of the parameters of the model the lewis number is chosen in a specific way and with initial conditions of a specific form for a class of initial conditions then the number of unknown functions is reduced from three to two for the reduced system the existence of combustion fronts follows from the existence results for the original system the stability of these fronts is studied here by a combination of energy estimates and numerical evans function computations and nonlinear analysis when applicable we then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the lewis number the fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level

variational method how it can generate false instabilities when the variational method is applied to nonlinear evolution equations for determining solitary wave dynamics it is possible for the method to predict the pulse to be unstable when in fact it is stable we determine the necessary conditions for this to occur as well as give sufficient conditions for avoiding such false instabilities we also discuss the general problem of applying the method to a general evolution equation

the semistrong limit of multipulse interaction in a thermally driven optical system abstract we consider the semistrong limit of pulse interaction in a thermally driven parametrically forced nonlinear schrodinger tdnls system modeling pulse interaction in an optical cavity the tdnls couples a parabolic equation to a hyperbolic system and in the semistrong scaling we construct pulse solutions which experience both short range tail tail interactions and long range thermal coupling we extend the renormalization group rg methods used to derive semistrong interaction laws in reaction diffusion systems to the hyperbolic parabolic setting of the tdnls system a key step is to capture the singularly perturbed structure of the semigroup through the control of the commutator of the resolvent and a re scaling operator the rg approach reduces the pulse dynamics to a closed system of ordinary differential equations for the pulse locations