local solution for the kadomtsev petviashvili equation in r2 abstract in this paper we obtain results about local existence uniqueness regularity and continuous dependence on the initial data for the solution of the kadomtsev petviashvili equation with positive dispersion in r 2 and initial data in sobolev spaces of order s 3 our method uses an approximated problem where the theory of quasi linear evolution equations is applicable

interactions of two dimensionally localized pulses of the regularized long wave equation abstract interactions between positive and or negative cylindrical bell shaped pulses are investigated numerically for a two dimensional regularized long wave equation overtaking collisions of two same signed pulses are found to be fairly elastic head on collisions between positive and negative pulses are inelastic and their overall features are analogous to those in the one dimentional case some properties of collision processes are discussed in terms of the conversed quantities

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints hamiltoniens sousquadratiques conditions aux limites infinies et donnees d explosion hamiltoniens superquadratiques formulation viscosite des conditions aux limites le probleme ergodique commande stochastique optimale avec contraintes d etat

on stabilizability and sampling for infinite dimensional systems some questions either motivated by or related to time sampling regarding the stabilizability of infinite dimensional or distributed parameter systems are considered in particular these include the characteristics of those systems which are finite rank stabilizable and the preservation of stabilizability under sampling a pathological class of systems which when sampled may be rendered unstabilizable by arbitrarily small perturbations in the sampling rate is identified the approach taken in establishing these results is based upon a spectral decomposition of the infinitesimal generator of the underlying open loop semigroup the results obtained apply to control systems with dynamics described by broad classes of partial and functional differential equations that arise frequently in applications

on the inequality f x d 2u ge f u g u du q we consider fully nonlinear degenerate elliptic equations with zero and first order terms we provide a priori upper bounds and characterize the existence of entire subsolutions under growth conditions on the lower order coefficients which extend the classical keller osserman condition for semilinear equations

harnack inequality for nonlinear elliptic equations with strong absorption abstract we develop harnack inequalities for two different types of equations first we consider a fully nonlinear uniformly elliptic equation related to the pucci s maximal and the minimal operators next we consider a quasilinear equation related to the p laplacian in both cases we consider lower order terms of keller osserman type although the equations considered are quite different we employ a unified method to approach both problems and the results we find are similar

stabilization of infinte dimensional systems with periodic feedback gains and sampled output abstract a novel control technique periodic output feedback is investigated in the context of infinite dimensional linear dynamic systems with unbounded input and output operators this control scheme differs from classical control schemes in that the output of the system in question need not be monitored continuously but only monitored at sampling instants it is shown that for systems satisfying an analog of detectability and stabilizability periodic output feedback provides a stabilization scheme which is robust under perturbations of the sampling time sequence and is easily implementable

implicit spectral methods for wave propagation problems abstract the numerical solution of a non linear wave equation can be obtained by using spectral methods to resolve the unknown in space and the standard crank nicolson differencing scheme to advance the solution in time we have analyzed iterative techniques for solving the non linear equations that arise from such implicit time stepping schemes for the k dv and the k p equations we derived predictor corrector method that retain the full accuracy of the implicit method with minimal stability restrictions on the size of the time step some numerical examples show the propagation of interacting solitons

a note on the liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the pucci operators and the keller osserman condition this paper is devoted to the study of the viscosity solutions of begin array l mathbb f rm d 2 u u x f 0 end array in the whole space mathbb r rm n under suitable structural assumptions on mathbb f involving the pucci extremal operators for the leading part and the keller osserman condition on the zeroth order term by means a kind of liouville method we prove that uniqueness of solutions holds therefore a unique growth at infinity is compatible with the structure of the equations furthermore the knowledge of these growth at infinity of solutions is not required a priori due to the presence of a strong absorption term the liouville theorem proved is independent on the dimension

robustness with respect to sampling for stabilization of riesz spectral systems we suppose that a continuous time feedback is input output stabilizing for an infinite dimensional system we address the question of whether the sampled data controller obtained by applying idealized sample and hold to this continuous time feedback is also input output stabilizing if the sampling time is small enough this question has been previously addressed for fairly general systems under various conditions in this note we restrict our attention to riesz spectral systems for which we generalize the existing results specifically we give two relatively simple conditions which combined are sufficient for the sampled data controller to be stabilizing the first condition is a spectrum decomposition for the open loop system generator which by itself is necessary but not sufficient for the system to be stabilizable by sampled data control the second is a summability condition relating the real part of the spectrum of the generator and the expansion coefficients for the input and feedback operators

algorithmic framework for group analysis of differential equations and its application to generalized zakharov kuznetsov equations abstract in this paper we explain in more details the modern treatment of the problem of group classification of systems of partial differential equations pdes from the algorithmic point of view more precisely we revise the classical lie algorithm of construction of symmetries of differential equations describe the group classification algorithm and discuss the process of reduction of systems of pdes to systems of equations with smaller number of independent variables in order to construct invariant solutions the group classification algorithm and reduction process are illustrated by the example of the generalized zakharov kuznetsov gzk equations of form u t f u x x x g u x y y h u x 0 as a result a complete group classification of the gzk equations is performed and a number of new interesting nonlinear invariant models which have non trivial invariance algebras are obtained lie symmetry reductions and exact solutions for two important invariant models i e the classical and modified zakharov kuznetsov equations are constructed the algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear pdes

entire solutions of fully nonlinear elliptic equations with a superlinear gradient term in this paper we consider second order fully nonlinear operators with an additive superlinear gradient term like in the pioneering paper of brezis for the semilinear case we obtain the existence of entire viscosity solutions defined in all the space without assuming global bounds a uniqueness result is also obtained for special gradient terms subject to a convexity concavity type assumption where superlinearity is essential and has to be handled in a different way from the linear case

high order difference methods for waves in discontinuous media second fourth and sixth order one step methods have been constructed for the solution of wave propagation problems the method is based on the first order system formulation of the wave equation and uses a staggered grid both in space and time the method is applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity the behavior of the truncation error has been investigated for one dimensional problems and stability criteria have been derived for one and two dimensional cases

sampled data boundary feedback control of 1 d parabolic pdes abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d linear parabolic systems on bounded domains it is shown that the continuous time boundary feedback applied in a sample and hold fashion guarantees closed loop exponential stability provided that the sampling period is sufficiently small two different continuous time feedback designs are considered the reduced model design and the backstepping design the obtained results provide stability estimates for weighted 2 norms of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

on positive solutions of fully nonlinear degenerate lane emden type equations abstract we prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power like zero order perturbations in bounded domains the principal part of such equations is either p k d 2 u or p k d 2 u some sort of truncated laplacians given respectively by the smallest and the largest partial sum of k eigenvalues of the hessian matrix new phenomena with respect to the semilinear case occur moreover for p k we explicitly find the critical exponent p of the power nonlinearity that separates the existence and nonexistence range of nontrivial solutions with zero dirichlet boundary condition

holder estimates for degenerate elliptic equations with coercive hamiltonians we prove a priori estimates and regularity results for some quasi linear degenerate elliptic equations arising in optimal stochastic control problems our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally holder continuous and solutions to be locally lipschitz continuous we also give an existence result for the associated dirichlet problem

existence through convexity for the truncated laplacians we study the dirichlet problem on a bounded convex domain of mathbb r n with zero boundary data for truncated laplacians mathcal p k pm which are degenerate elliptic operators for k n defined by the upper and respectively lower partial sum of k eigenvalues of the hessian matrix we establish a necessary and sufficient condition theorem 1 in terms of the flatness of domains for existence of a solution for general inhomogeneous term this result in particular shows that the strict convexity of the domain is sufficient for the solvability of the dirichlet problem the result and related ideas are applied to the solvability of the dirichlet problem for the operator mathcal p k with lower order term when the domain is strictly convex and the existence of principal eigenfunctions for the operator mathcal p 1 an existence theorem is presented with regard to the principal eigenvalue for the dirichlet problem with zero th order term for the operator mathcal p 1 a nonexistence result is established for the operator mathcal p k with first order term when the domain has a boundary portion which is nearly flat furthermore when the domain is a ball we study the dirichlet problem with a constant inhomogeneous term and a possibly sign changing first order term and the associated eigenvalue problem

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

the dirichlet problem for the convex envelope the convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation pde in this article we study a modified problem the dirichlet problem for the underlying pde the main result is an optimal regularity result differentiability c1 regularity of the boundary data implies the corresponding result for the solution in the interior despite the fact that the solution need not be continuous up to the boundary secondary results are the characterization of the convex envelope as i the value function of a stochastic control problem and ii the optimal underestimator for a class of nonlinear elliptic pdes

indirect sampled data control with sampling period adaptation it is known that if a continuous time feedback system is exponentially stable then the corresponding sampled data system obtained by sample hold discretisation with constant sampling period is also exponentially stable provided that the sampling period 0 is sufficiently small in general it is difficult to estimate how small the sampling period has to be in order to achieve the stability of the sampled data system in this article we present an adaptive mechanism for adjusting the sampling period this mechanism has the properties that for every initial state i the adaptation of the sampling period terminates after finitely many time steps and ii the state of the adaptive sampled data system is integrable and converges to zero as time goes to infinity

cylindrical quasi solitons of the zakharov kuznetsov equation abstract evolutions and interactions of two dimensional solitary waves of the zakharov kuznetsov equation are investigated numerically formations of cylindrical bell shaped pulses are observed in the initial value problems a single bell shaped pulse propagates stably without any deformation like a soliton two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision after a collision of two pulses the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples so that the collision process is slightly inelastic generated ripples are very small for center to center collision of similar pulses the cylindrically symmetric pulses of the zakharov kuznetov equation are thus found to behave approximately soliton like some properties of collision process are interpreted in terms of the conservation laws

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

boundary behavior of k convex solutions for singular k hessian equations abstract we discuss the existence and boundary behavior of k convex solution to the singular k hessian problem s k d 2 u x b x f u x x u x 0 x where s k d 2 u k 1 2 n is the k hessian operator r n n 2 is a smooth bounded strictly convex domain here the weight function b x is not necessarily bounded on another interest is that f u as u 0 our approach mainly relies on karamata s regular variation theory and the construction of suitable sub and super solutions

boundary sampled data feedback stabilization for parabolic equations abstract the aim of this work is to design an explicit finite dimensional boundary feedback controller of sampled data form for locally exponentially stabilizing the equilibrium solutions to semilinear parabolic equations the feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation this stabilizing procedure is applicable for any sampling rate and when the sampling period tends to zero the feedback converges to certain feedback designed for stabilizing the parabolic equations with continuous time boundary feedback control

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

keller osserman type conditions for some elliptic problems with gradient terms in this paper we consider the elliptic boundary blow up problems u g u f u in u on where is a smooth bounded domain and the functions f and g are increasing and continuous our main concern will be to prove both existence and nonexistence of nonnegative solutions depending on new integral conditions of keller osserman type involving f and g we show in particular that the problem with a minus sign may have solutions inclusive for some functions g with slightly superquadratic growth at infinity that is somehow not expected we also obtain uniqueness of nonnegative solutions in some cases

boundary blow up solutions to the k hessian equation with a weakly superlinear nonlinearity abstract by constructing new sub and super solutions we are concerned with determining values of for which there exist k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 and is a smooth bounded strictly convex domain in r n n 2 we suppose that the nonlinearity behaves like u k ln u as u which is more complex and difficult to deal with than the nonlinearity grows like u p with p k or faster at infinity further several new results of nonexistence global estimates and estimates near the boundary for the solutions are also given

nonrobustness of closed loop stability for infinite dimensional systems under sample and hold it is a well known principle for finite dimensional systems that applying sampled and hold in the feedback loop around a stabilizing state feedback or dynamic controller results in a stable sampled data feedback control system if the sampling period is small enough the principle extends to infinite dimensional systems with compact state feedback if either the input operator is bounded or the given state space system is analytic in this paper we give an example for which this principle fails but which nevertheless satisfies certain necessary conditions arising in sampled data control of infinite dimensional systems

removable singularities for fully nonlinear elliptic equations we obtain a removability result for the fully nonlinear uniformly elliptic equations f d 2 u f u 0 the main theorem states that every solution to the equation in a punctured ball without any restrictions on the behaviour near the centre of the ball is extendable to the solution in the entire ball provided the function f satisfies certain sharp conditions depending on f previously such results were known for linear and quasilinear operators f in comparison with the semi or quasilinear theory the techniques for the fully nonlinear equations are new and based on the use of the viscosity notion of generalised solution rather than the distributional or the weak solutions

the multi symplectic fourier pseudospectral method for solving two dimensional hamiltonian pdes in this paper the multi symplectic fourier pseudospectral msfp method is generalized to solve two dimensional hamiltonian pdes with periodic boundary conditions using the fourier pseudospectral method in the space of the two dimensional hamiltonian pde 2d hpde the semi discrete system obtained is proved to have semi discrete multi symplectic conservation laws and a global symplecticity conservation law then the implicit midpoint rule is employed for time integration to obtain the msfp method for the 2d hpde the fully discrete multi symplectic conservation laws are also obtained in addition the proposed method is applied to solve the zakharov kuznetsov zk equation and the kadomtsev petviashvili kp equation numerical experiments on soliton solutions of the zk equation and the kp equation show the high accuracy and effectiveness of the proposed method

stabilization and structural assignment of dirichlet boundary feedback parabolic equations a parabolic equation defined on a bounded domain is considered with input acting on the boundary through the dirichlet b c expressed as a specified finite dimensional feedback of the solution the free system zero b c is assumed throughout to be unstable two main results are established first we provide in 2 a novel proof that fully solves the corresponding boundary feedback stabilization problem thereby removing an annoying assumption required by the technique of appl math optim 6 1980 pp 201 220 under certain natural algebraic conditions based on the finitely many unstable eigenvalues we establish the existence of general boundary vectors for which the corresponding feedback semigroup decays exponentially to zero in the l 2 omega operator norm or more generally in the h s omega operator norm 0 leqq s frac 1 2 however most of the paper is devoted to the second problem structural or spectral assignment which is a natural question relevant to the selfadj

a conservative fourth order stable finite difference scheme for the generalized rosenau kdv equation in both 1d and 2d abstract in the present work a conservative fourth order stable finite difference scheme is proposed to solve the generalized rosenau kdv equation in both 1d and 2d the existence uniqueness and mass and energy conservations of the numerical solution are proved using the discrete energy method the new scheme is convergent with o 2 h 4 and unconditionally stable numerical experiments are carried out to show that the scheme is efficient and reliable

nonlinear elliptic equations in rn without growth restrictions on the data abstract we show existence and regularity of solutions in rn to nonlinear elliptic equations of the form div a x du g x u when is just a locally integrable function under appropriate growth conditions on a and g but not on roughly speaking in the model case p u u s 1u with p 2 1 n existence of a nonnegative solution in rn is guaranteed for every nonnegative l1loc rn if and only if s p 1

stabilization of distributed parameter systems by piecewise polynomial control in this paper we consider the stabilization of continuous time distributed parameter systems by piecewise polynomial controls we obtain necessary and sufficient conditions for stabilization by piecewise polynomial controls

output feedback stabilization for heat equations with sampled data controls abstract in this paper we build up an output feedback law to stabilize a sampled data controlled heat equation with a potential in a bounded domain the feedback law abides the following rules first we divide equally the time interval 0 into infinitely many disjoint time periods and divide each time period into three disjoint subintervals second for each time period we observe a solution over an open subset of in the first subinterval take sample from outputs at one time point of the first subinterval add a time invariant output feedback control over another open subset of in the second subinterval let the equation evolve free in the last subinterval thus the corresponding feedback control is of sampled data our feedback law has the following advantages the sampling period which is the length of the above time period can be arbitrarily taken the feedback law has an explicit expression in terms of the sampling period the behaviors of the norm of the feedback law when the sampling period goes to zero or infinity are clear the construction of the feedback law is based on two kinds of approximate null controllability for heat equations one has time invariant controls while another has impulse controls the studies of the aforementioned controllability with time invariant controls need a new observability inequality for heat equations built up in the current work

solvability of nonlinear elliptic equations with gradient terms abstract we study the solvability in the whole euclidean space of coercive quasi linear and fully nonlinear elliptic equations modeled on u g u f u u 0 where f and g are increasing continuous functions we give conditions on f and g which guarantee the availability or the absence of positive solutions of such equations in r n our results considerably improve the existing ones and are sharp or close to sharp in the model cases in particular we completely characterize the solvability of such equations when f and g have power growth at infinity we also derive a solvability statement for coercive equations in general form

the kp equation with quasiperiodic initial data abstract the kadomtsev petviashvili kp equation is known to admit exact quasiperiodic solutions that can be written in terms of riemann theta functions with a finite number of phases in each solution in this paper we propose a method to solve the initial value problem for the kp equation for initial data taken from this class of quasiperiodic functions

time compact high order difference methods for wave propagation we construct one step explicit difference methods for solution of wave propagation problems with fourth order accuracy in both space and time by using a principle that can be generalized to arbitrary order we use the first order system form and a staggered grid the fourth order accuracy in time is obtained by transferring time derivatives in the truncation error to space derivatives discontinuous coefficients corresponding to interfaces between different materials are considered as a special case of variable coefficients and the method is applied across the discontinuities the accuracy is much improved compared to second order methods even for this type of problem a certain norm is shown to be conserved ensuring good accuracy even for long time integration

numerical methods for nonlinear second order hyperbolic partial differential equations i time linearized finite difference methods for 1 d problems abstract a three parameter family of finite difference methods for one dimensional hyperbolic equations with damping and nonlinearities based on the introduction of a new dependent variable and three implicitness parameters for the time discretization and the diffusion and reaction terms is presented and shown to result in two time level semi discrete equations second order accurate spatial discretizations and time linearization of the nonlinear source terms are shown to result in three time level linear finite difference equations which are second order accurate in time whenever the reaction and diffusion terms are allocated in the proportion 1 2 1 to the past current and future time levels second order accurate in time time linearized three point compact operator methods for the spatial derivatives which only involve three time levels and three grid points and are fourth order accurate in space are also presented and their linear stability analyzed an exhaustive comparison between analytical and numerical results indicates that the accuracy of time linearized methods increases as the time step and grid spacing are decreased and is nearly independent of the order of the spatial discretization when there is a sufficient number of grid points to resolve the wave structure and the best accuracy is achieved when the three implicitness parameters are equal to 1 2 it is also shown that the accuracy of time linearized methods is a strong function of the wave speed as the wave speed is increased and the wave equation tends to a parabolic one the methods are still three time level techniques which may not preserve the positivity of the solution whereas a crank nicolson discretization of the corresponding parabolic equation does preserve this property provided that there are no mathematical incompatibilities between the initial and the boundary conditions it is also shown that for one dimensional wave equations with monotonic kinetics and time dependent forcing there are very few differences between second and first order time accurate discretizations of the diffusion terms provided that the time step and grid size are sufficiently small and the discretization of the reaction terms plays a less important rule than that of the diffusion terms

multi symplectic spectral discretizations for the zakharov kuznetsov and shallow water equations abstract the time evolution of multi symplectic pdes with periodic boundary conditions in one and two space dimensions is considered we introduce the idea of a multi symplectic fourier transform for systems on periodic domains leading to a semi discretization on fourier space and to the concept of multi symplecticity on fourier space the spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it a discrete wave model a by product of the semi discretization is that it leads automatically to a system of hamiltonian odes in time when truncated we show that the one dimensional shallow water equations and the two dimensional zakharov kuznetsov equation are multi symplectic and derive spectral discretizations for these systems and present numerical experiments

generalized sampled data feedback control of distributed parameter systems abstract in the sampled data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled data control for finite dimensional systems the necessary conditions are also known to be sufficient we show that this equivalence extends to the infinite dimensional case if the underlying semigroup is analytic however for general systems the necessary conditions are not sufficient nor are the sufficient conditions necessary we prove this by a single example with a free parameter one choice of parameter shows that the necessary conditions are too weak and another choice shows that the sufficient conditions are too strong

local discontinuous galerkin methods for two classes of two dimensional nonlinear wave equations in this paper we develop analyze and test local discontinuous galerkin dg methods to solve two classes of two dimensional nonlinear wave equations formulated by the kadomtsev petviashvili kp equation and the zakharov kuznetsov zk equation our proposed scheme for the kadomtsev petviashvili equation satisfies the constraint from the pde which contains a non local operator and at the same time has the local property of the discontinuous galerkin methods the scheme for the zakharov kuznetsov equation extends the previous work on local discontinuous galerkin method solving one dimensional nonlinear wave equations to the two dimensional case l2 stability of the schemes is proved for both of these two nonlinear equations numerical examples are shown to illustrate the accuracy and capability of the methods

stability and convergence properties of forced infinite dimensional discrete time lur e systems incremental stability and convergence properties for forced infinite dimensional discrete time lur e systems are addressed lur e systems have a linear and nonlinear component and arise as the fe

a conservative spectral method for several two dimensional nonlinear wave equations a conservative spectral method is proposed to solve several two dimensional nonlinear wave equations the conventional fast fourier transform is used to approximate the spatial derivatives and a three level difference scheme with a free parameter is to advance the solution in time our time discretization is semi implicit in the sense that the linear terms are treated implicitly while the nonlinear terms are evaluated only by previous time levels and thus treated explicitly however the cost of the algorithm is no greater than that of a fully explicit method because the linear boundary value problem that must be solved at each time step is almost trivial in a spectral spatial discretization a linear stability analysis shows that the method leads to a less restrictive stability condition than the corresponding explicit one the method is conservative and the ratio of the numerical dispersion to the physical dispersion is of the order o t2 applications of our method to the kadomtsev petviashvili and the zakharov kuznetsov equations exhibit excellent results

wave collapse and instability of solitary waves of a generalized kadomtsev petviashvili equation abstract the solutions of a family of generalized kadomtsev petviasvili equation in two diemnsions u t u m u x u xxx x 2 u yy are studied with appropriate boundary conditions and constraints on the initial data for m in a dense set m 0 and for 2 1 if m 4 then the solution can blow up in finite time we show that if 1 m 4 a solitary wave so exists which is unstable if m 4 3 in this and related problems th e instability of solitary waves is associated with onset of wave collapse we give numerical evidence to support this

cylindrical solitary pulses in a two dimensional stabilized kuramoto sivashinsky system abstract by linearly coupling a generalized zakharov kuznetsov equation alias the two dimensional 2d benney equation to an extra linear dissipative equation a 2d extension of a recently proposed stabilized kuramoto sivashinsky system is developed the model applies to the description of surface waves on 2d liquid layers in various physical settings the extra equation provides for the stability of the zero state in the system thus paving a way to the existence of stable 2d localized solitary pulses sps a perturbation theory based on a family of cylindrical solitons existing in the conservative counterpart of the system is developed by treating dissipation and gain in the model as small perturbations it is shown that the system may select two steady state solitons from the continuous family provided by the conservative counterpart of which the one with larger amplitude is expected to be stable numerical simulations support the analytical predictions quite well additionally it is found that a shallow quasi one dimensional 1d trough is attached to the stable sp if the integration domain is not very large and an explanation to this feature is proposed stable double humped bound states of two pulses are found too

stability of infinite dimensional sampled data systems suppose that a static state feedback stabilizes a continuous time linear infinite dimensional control system we consider the following question if we construct a sampled data controller by applying an idealized sample and hold process to a continuous time stabilizing feedback will this sampled data controller stabilize the system for all sufficiently small sampling times here the state space x and the control space u are hilbert spaces the system is of the form x t ax t bu t where a is the generator of a strongly continuous semigroup on x and the continuous time feedback is u t fx t the answer to the above question is known to be yes if x and u are finite dimensional spaces in the infinite dimensional case if f is not compact then it is easy to find counterexamples therefore we restrict attention to compact feedback we show that the answer to the above question is yes if b is a bounded operator from u into x moreover if b is unbounded we show that the answer yes remains correct provided that the semigroup generated by a is analytic we use the theory developed for static state feedback to obtain analogous results for dynamic output feedback control

the existence and asymptotic behavior of boundary blow up solutions to the k hessian equation abstract in this paper we consider the existence and asymptotic behavior of k convex solution to the boundary blow up k hessian problem s k d 2 u h x f u for x u x as dist x 0 where k 1 2 n s k d 2 u is the k hessian operator is a smooth bounded strictly convex domain in r n n 2 h c is positive in but is not necessarily bounded on and f is a smooth positive function that satisfies the so called keller osserman condition further results are obtained for the special case that is a ball our approach to show the existence and asymptotic behavior exploits the method of sub and super solutions and karamata regular variation theory

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

sampled data boundary feedback control of 1 d linear transport pdes with non local terms abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d inhomogeneous linear transport partial differential equations on bounded domains with constant velocity and non local terms it is shown that the emulation design based on the recently proposed continuous time boundary feedback designed by means of backstepping guarantees closed loop exponential stability provided that the sampling period is sufficiently small it is also shown that contrary to the parabolic case a smaller sampling period implies a faster convergence rate with no upper bound for the achieved convergence rate the obtained results provide stability estimates for the sup norm of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

conservative finite difference schemes for the generalized zakharov kuznetsov equations this paper is concerned with the construction of conservative finite difference schemes by means of discrete variational method for the generalized zakharov kuznetsov equations and the numerical solvability of the two dimensional nonlinear wave equations a finite difference scheme is proposed such that mass and energy conservation laws associated with the generalized zakharov kuznetsov equations hold our arguments are based on the procedure that d furihata has recently developed for real valued nonlinear partial differential equations numerical results are given to confirm the accuracy as well as validity of the numerical solutions and then exhibit remarkable nonlinear phenomena of the interaction and behavior of pulse wave solutions

on a k hessian equation with a weakly superlinear nonlinearity and singular weights abstract consider the existence nonexistence and global estimates of k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 is a smooth bounded strictly convex domain in r n n 2 and h x is a positive weight function which is singular near the boundary we first give the existence and nonexistence results of k convex solution to the above boundary blow up problem on a larger range of h and then we show that there is a k convex solution provided that h x grows fast near and u k ln u grows slow at it turns out that this case is more difficult to handle than the case in which h x grows slow near and u k ln u grows fast at this needs some new ingredients in the arguments

a finite difference method for the korteweg de vries and the kadomtsev petviashvili equations a linearized implicit finite difference method for the korteweg de vries equation is proposed and straightforwardly extended to the kadomtsev petviashvili equation we investigate the order of accuracy of the method and prove the method to be unconditionally linearly stable the numerical experiments for the korteweg de vries and the kadomtsev petviashvili equations are carried out with various conditions numerical results for the collision of two lump type solitary wave solutions to the kadomtsev petviashvili equation are also reported

input output stability of sampled data systems the authors study the l sub p input output stability of a continuous time controller using the usual arrangement of periodic sampling and zero order hold it is noted that even if the hybrid system is exponentially stable this arrangement does not yield l sub p 1

time compact difference methods for wave propagation in discontinuous media in b gustafsson and e mossberg siam j sci comput 26 2004 pp 259 271 a fourth order one step method was constructed for the solution of wave propagation problems the method is based on the first order system form of the pde and uses a staggered grid both in space and time the method was also applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity in this paper we will analyze a model problem from acoustics and demonstrate the theoretical foundation for this behavior furthermore we shall present more detailed numerical experiments which confirm the theoretical results

on the energetic galerkin boundary element method applied to interior wave propagation problems we consider two dimensional interior wave propagation problems with vanishing initial and mixed boundary conditions reformulated as a system of two boundary integral equations with retarded potential these latter are then set in a weak form based on a natural energy identity satisfied by the solution of the differential problem and discretized by the related energetic galerkin boundary element method numerical results are presented and discussed