boundary effect on asymptotic behaviour of solutions to the p system with linear damping we consider the asymptotic behaviour of solutions to the p system with linear damping on the half line r 0 vt ux 0 ut p v x u with the dirichlet boundary condition u x 0 0 or the neumann boundary condition ux x 0 0 the initial date v0 u0 x has the constant state v u at x l hsiao and t p liu commun math phys 143 1992 599 605 have shown that the solution to the corresponding cauchy problem behaves like diffusion wave and k nishihara j differential equations131 1996 171 188 137 1997 384 395 has proved its optimal convergence rate our main concern in this paper is the boundary effect in the case of null dirichlet boundary condition on u the solution v u is proved to tend to v 0 as t tends to infinity its optimal convergence rate is also obtained by using the green function of the diffusion equation with constant coefficients in the case of null neumann boundary condition on u v 0 t is conservative and v 0 t v0 0 by virtue of the first equation so that v x t is expected to tend to the diffusion wave v x t connecting v0 0 and v in fact the solution v u x t is proved to tend to v x t 0 in the special case v0 0 v the optimal convergence rate is also obtained however this is not known in the case v0 0 v

high accuracy numerical methods for the gardner ostrovsky equation the gardner ostrovsky equation also known as the extended rotation modified korteweg de vries kdv equation describes weakly nonlinear internal oceanic waves under the influence of earth rotation high accuracy numerical methods are needed to follow with precision the long time evolution of the solutions of this equation with the additional difficulty that the numerical methods have to conserve very accurately several invariants of the solutions including the mass and the energy of the waves finite difference methods traditionally used for the solution of the kdv equation fails to preserve accurately these invariants for large times in this paper we show that this difficulty can be overcome by using a high accuracy finite difference hafd numerical method we present a strong stability preserving finite difference scheme and compare its performance using some relevant examples with those of two recently published numerical methods for solving this kind of equation a simpler second order finite difference method and a pseudospectral numerical scheme that enforces the conservation of energy the numerical comparison shows that the three methods have similar accuracy for short times but the simpler finite difference scheme is less accurate in preserving the three invariants affecting the numerical accuracy of the solution as time goes on on the contrary the hafd method presented here preserves the invariants with even better accuracy than the pseudospectral scheme but with a much lower computational cost in addition the numerical implementation of the hafd method is as easy as that of the simpler finite difference method being both much simpler than the intricate energy conservation pseudospectral scheme these advantages makes the hafd method presented here very appropriate for solving numerically this type of equations particularly for studying long time wave propagation

analytical and numerical studies of the variable coefficient gardner equation the gardner or extended korteweg de vries equation with the variable sign cubic non linear term is studied this equation occurred in the theory of interfacial waves in the stratified shear flow at some certain conditions on the medium stratification an asymptotic method is applied to study the solitary wave transformation when the coefficient of the cubic non linear term is varied slowly it is found the one branch of the solitary waves destroys in the vicinity of the bifurcation point where the exponential soliton transforms into the breather through algebraic soliton other solitary waves change their amplitude accordingly to the cubic non linear term variation the structure of the tail formed behind the solitary wave is determined direct numerical simulation of the gardner equation is performed to analyze the transformation and breaking of the solitary wave it confirms most of the theoretical results the calculated dynamics of the table top soliton due to the large values of the mass and momentum integrals differs from the asymptotic prediction because the characteristic time of the variation of the cubic non linear term has the same order as the duration of table top soliton

tdgl and mkdv equations for jamming transition in the lattice models of traffic abstract the lattice models of traffic are proposed to describe the jamming transition in traffic flow on a highway in terms of thermodynamic terminology of phase transitions and critical phenomena they are the lattice versions of the hydrodynamic model of traffic two lattice models are presented one is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other is the difference equation in which both time and space variables are discrete we apply the linear stability theory and the nonlinear analysis to the lattice models it is shown that the time dependent ginzburg landau tdgl equation is derived to describe the traffic flow near the critical point a thermodynamic theory is formulated for describing the phase transitions and critical phenomena it is also shown that the perturbed modified korteweg de vries mkdv equation is derived to describe the traffic jam

interaction of a solitary wave with an external force abstract using the forced korteweg de vries equation as a model we investigate the interaction of a solitary wave with an isolated moving external force of small amplitude we use both analysis and numerical calculations our analysis is based on a multi scale perturbation asymptotic analysis and is carried through to either the first or second order depending on whether the first or more accurate second order expression for the solitary wave speed is used the perturbation theory leads to a pair of autonomous equations for the solitary wave amplitude and position and these are analyzed in detail for the two extreme cases when the lengthscale of the external force is either much greater or much less than that of the solitary wave the theory predicts that the main regimes are passage when the solitary wave passes through the external force with some amplitude modulation repulsion when the solitary wave is reflected by the external force with a significant amplitude change and trapping when the solitary wave remains either partially or totally trapped at the location of the external force these theoretical predictions are confirmed in our numerical simulations of the forced korteweg de vries equation

boundary layers for the navier stokes equations of compressible fluids the global unique solvability is proved for the navier stokes equations of compressible fluids for the one dimensional spiral flows between two circular cylinders the zero shear viscosity limit 0 is justified the value o 0 1 2 is established for the boundary layer thickness

asymptotic behavior of solution to nonlinear evolution equations with damping in this paper we consider the global existence and the asymptotic behavior of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x xx t x 0 r t 1 x 2 x xx with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 4 1 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the parabolic system defined by 2 4 which are obtained by the convection diffusion equations 2 1 and solutions to cauchy problem e and i tend asymptotically to the convection diffusion system with exponential rates precisely speaking we derive the asymptotic profile of e by gauss kernel g t x as follows 2 2e 1 4 t xg y t 1 1 sin 2 y 0 cos 2 y 0 dy e 1 t 2e 1 4 t rg t y cos 2 y 1 sin 2 y sin 2 y cos 2 y u0 x y v0 x y dy lp rx e 1 4 to 1 t 1 1p the same problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 nishihara k nishihara asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity z angew math phys 57 4 2006 604 614 for the case of 0 0

interactions of solitons in nonintegrable systems direct perturbation method and applications abstract problems of approximate description of soliton interaction in nonintegrable systems are discussed it is shown that in an arbitrary lagrangian system in the first nontrivial order of the small parameter asymptotic scheme for a system of solitons with close velocities a lagrangian may be introduced analogous to that for classical particles with a pair interaction potential a classification of possible types of soliton interaction is given and criteria of the existence of bound states are formulated a possibility of generalization of approximated description of soliton interaction in the multidimensional case is noted

nonlinear stability for dissipative nonlinear evolution equations with ellipticity we study the cauchy problem for a set of nonlinear evolution equations which has been proposed for the investigation of nonlinear interaction between ellipticity and dissipation the zero equilibrium is linearly stable for coefficients within certain range we further establish the global existence nonlinear stability and optimal decay rate of the solution for coefficients within the same range though with suitable restriction to initial data

parametrically controlling solitary wave dynamics in the modified korteweg de vries equation we demonstrate the control of solitary wave dynamics of the modified korteweg de vries mkdv equation through the temporal variations of the distributed coefficients this is explicated through exact cnoidal wave and localized soliton solutions of the mkdv equation with variable coefficients the solitons can be accelerated and their propagation can be manipulated by suitable variations of the above parameters in sharp contrast with the nonlinear schrodinger equation the soliton amplitude and widths are time independent

kink soliton characterizing traffic congestion we study traffic congestion by analyzing a one dimensional traffic flow model developing an asymptotic method to investigate the long time behavior near a critical point we derive the modified korteweg de vries mkdv equation as the lowest order model there is an infinite number of kink solitons to the mkdv equation while it has been found by numerical simulations that the kink pattern arising in traffic congestion is uniquely determined irrespective of initial conditions in order to resolve this selection problem we consider higher order corrections to the mkdv equation and find that there is a kink soliton that can deform continuously with the perturbation represented by the addition of these corrections with numerical confirmation we show that this continuously deformable kink soliton characterizes traffic congestion we also discuss the relationship between traffic congestion and the slugging phenomenon in granular flow

breather lattice and its stabilization for the modified korteweg de vries equation we obtain an exact solution for the breather lattice solution of the modified korteweg de vries equation numerical simulation of the breather lattice demonstrates its instability due to the breather breather interaction however such multibreather structures can be stabilized through the concurrent application of ac driving and viscous damping terms

on barrier option pricing by erlangization in a regime switching model with jumps abstract we consider the risk neutral pricing of vanilla digital and down and out call options when the underlying asset price evolves like the exponential of a markov modulated brownian motion mmbm with two sided phase type jumps the price of such options is intimately related to the first passage properties of the mmbm to analyse these first passages we randomize the time horizon using erlang distributions with suitable parameters and apply matrix analytic methods this provides us with closed form approximations of the options prices with a very high precision as shown by several numerical illustrations in particular we consider an example in which the phase type jump distribution is constructed in such a way that it mimics fat tails

asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant abstract in this paper we consider an initial boundary value problem for some nonlinear evolution system with dissipation and ellipticity we establish the global existence and furthermore obtain the l p p 2 decay rates of solutions corresponding to diffusion waves the analysis is based on the energy method and pointwise estimates

option pricing under a double exponential jump diffusion model this paper aims to extend the analytical tractability of the black scholes model to alternative models with arbitrary jump size distributions more precisely we propose a jump diffusion model for asset prices whose jump sizes have a mixed exponential distribution which is a weighted average of exponential distributions but with possibly negative weights the new model extends existing models such as hyperexponential and double exponential jump diffusion models because the mixed exponential distribution can approximate any distribution as closely as possible including the normal distribution and various heavy tailed distributions the mixed exponential jump diffusion model can lead to analytical solutions for laplace transforms of prices and sensitivity parameters for path dependent options such as lookback and barrier options the laplace transforms can be inverted via the euler inversion algorithm numerical experiments indicate that the formulae are easy to implement and accurate the analytical solutions are made possible mainly because we solve a high order integro differential equation explicitly a calibration example for spy options shows that the model can provide a reasonable fit even for options with very short maturity such as one day this paper was accepted by michael fu stochastic models and simulation

propagation of a stationary shock layer in the presence of a boundary we are interested in the large time behavior of a stationary shock layer in the presence of a boundary effect it is shown that the stationary shock propagates away from the boundary with speed inversely proportional to the time t consequently the location of the wave front is of order log thinspace 1 t we show therefore that the zero viscosity and time asymptotic limits do not commute our analysis involves detailed point wise estimates which allow us to trace the shock

a note on boundary layer of a nonlinear evolution system with damping and diffusions abstract in this paper we consider an initial boundary value problem for a nonlinear evolution system with damping and diffusions on the strip our main purpose is to investigate zero diffusion limits which formally yield parabolic hyperbolic coupled equations resulting from parabolic parabolic coupled equations it is shown that the boundary layer thickness is of the order o with 0 3 4 as the diffusion parameter goes to zero

the modified kdv equation with variable coefficients exact uni bi variable travelling wave like solutions in this paper the modified korteweg de vries mkdv equation with variable coefficients vc mkdv equation is investigated via two kinds of approaches and symbolic computation on the one hand we firstly reduce the vc mkdv equation to a second order nonlinear nonhomogeneous ode using travelling wave like similarity transformation and then we obtain its many types of exact fractional solutions with one travelling wave like variable by applying some fractional transformations to the obtained nonlinear ode on the other hand we reduce the vc mkdv equation to two nonlinear pdes with variable coefficients using the anti tangent and anti hypertangent function transformations respectively and then we given its many types of exact solutions with two different travelling wave like variables by studying the obtained nonlinear pde with variable coefficients c 2008 elsevier inc all rights reserved

jamming transitions and the modified korteweg de vries equation in a two lane traffic flow abstract the two lattice models are presented to simulate the traffic flow on a two lane highway they are the lattice versions of the hydrodynamic model of traffic the one model a is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other model b is the difference equation in which both time and space variables are discrete the jamming transitions among the freely moving phase the coexisting phase and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation the modified korteweg de vries mkdv equations are derived from the lattice models near the critical point the traffic jam is described by a kink antikink solution obtained from the mkdv equation it is found that the critical point the coexisting curve and the neutral stability line decrease with increasing the rate of lane changing also the computer simulation is performed for the model b it is shown that the coexisting curves obtained from the mkdv equation are consistent with the simulation result

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

breaking of solitary and periodic nonlinear waves abstract a nontrivial behaviour of a nonlinear wave under influence of small disturbing factors like dissipation smooth inhomogeneity of medium parameters etc is considered by the example of sine gordon equation the stage of slow adiabatic variation of the parameters of quasi stationary wave is shown to change at some finite distance due to strong instability the wave form becomes essentially non stationary breaking of stationary wave structure the breaking condition is defined by the extremum of the wave adiabatic invariant the behaviour of a wave at the nonadiabatic stage is described using a galerkin procedure

zero viscosity limit of the linearized navier stokes equations for a compressible viscous fluid in the half plane the zero viscosity limit for an initial boundary value problem of the linearized navier stokes equations of a compressible viscous fluid in the half plane is studied by means of the asymptotic analysis with multiple scales we first construct an approximate solution of the linearized problem of the navier stokes equations as the combination of inner and boundary expansions next by carefully using the technique on energy methods we show the pointwise estimates of the error term of the approximate solution which readily yield the uniform stability result for the linearized navier stokes solution in the zero viscosity limit 1999 john wiley sons inc

the vanishing diffusivity limit for the 2 d boussinesq equations with boundary effect abstract in this paper we tackle the issue of the vanishing diffusivity limit of the 2 d incompressible boussinesq equations in the half plane our main purpose is to study the boundary layer effect and the convergence rates as the thermal diffusion parameter goes to zero under the homogeneous dirichlet boundary condition of velocity and the nonhomogeneous dirichlet boundary condition in the x direction for temperature we show that the boundary layer thickness is of the order o with 0 2 3 in contrast with jiang et al 2011 the bl thickness we got is thinner than that in jiang et al 2011 moreover we prove that as diffusivity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion equations we also obtain the convergence rates of the vanishing diffusivity limit

solitary wave transformation in a medium with sign variable quadratic nonlinearity and cubic nonlinearity abstract solitary wave transformation in a zone with a sign variable coefficient for the quadratic nonlinear term is studied in the framework of the variable coefficient extended korteweg de vries equation this situation can be realised for internal waves in a stratified ocean when the pycnocline lies midway between the sea bed and the sea surface the effects of the cases when the coefficient of the cubic nonlinear term can be either positive or negative are investigated for small amplitude solitary waves previous results when the cubic nonlinear term is ignored are confirmed the initial solitary wave is destroyed and solitary waves of the opposite polarity are generated after passage through the turning point i e the location where the coefficient of the quadratic nonlinear term is zero for large amplitude solitary waves the cubic nonlinear term and particularly its sign influence significantly both the polarity and amplitude of the resulting solitary waves if the cubic nonlinear term has a negative coefficient the amplitude of the large terminal wave is comparable with the initial value it is less than 50 when the cubic nonlinear term is ignored if the cubic nonlinear term has a positive coefficient the result depends on the initial wave amplitude large amplitude solitary waves pass through the transition zone keeping the solitary wave shape and its polarity moderate amplitude solitary wave are destroyed and transformed to a strongly pulsating wave packet breather

asymptotics of dissipative nonlinear evolution equations with ellipticity different end states in this paper we consider the global existence and asymptotic behaviors of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x xx t 1 x 2 x xx with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 4 1 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the diffusion waves defined by 2 5 which are obtained by the diffusion equations 2 1 solutions to cauchy problem e and i tend asymptotically to those diffusion waves with exponential rates the analysis is based on the energy method the same problem was studied by tang and zhao j math anal appl 233 1999 336 358 for the case of 0 0

nonautonomous mixed mkdv sinh gordon hierarchy the construction of a nonautonomous mixed mkdv sine gordon model is proposed by employing an infinite dimensional affine lie algebraic structure within the zero curvature representation a systematic construction of soliton solutions is provided by an adaptation of the dressing method which takes into account arbitrary time dependent functions a particular choice of those arbitrary functions provides an interesting solution describing the transition of a pure mkdv system into a pure sine gordon soliton

boundary layer effects for the 2 d boussinesq equations with vanishing diffusivity limit in the half plane nsfc 40890150 10801111 10971171 national basic research program 2005cb321700 natural science foundation of fujian province of china 2010j05011 fundamental research funds for the central universities 2010121006

boundary layer stability in real vanishing viscosity limit in the previous paper 20 an evans function machinery for the study of boundary layer stability was developed there the analysis was restricted to strongly parabolic perturbations that is to an approximation of the form ut f u x b u ux x 1 with an elliptic matrix b however real models like the navier stokes approximation of the euler equations for a gas flow involve incompletely parabolic perturbations b is not invertible in general

efficient hedging currency options in fractional brownian motion model with jumps abstract this paper deals with the problem of efficient hedging for european currency option and exchange option in case that exchange rate is driven by a fractional brownian motion with jumps in a complete market as long as a seller receives a price of the option equal to the expectation under the risk neutral probability measure of the discounted payoff the seller can hedge this payoff perfectly as the price is often quite expensive the investors need partial hedges which require less capital and reduce the risk the efficient hedge is one of partial hedges the efficient hedging results for options are obtained under fractional brownian motion model with jumps using the equivalent martingale measure simulation results and empirical studies confirm the theoretical findings and show that the fractional brownian motion model with jumps is reasonable

on partial differential equations related to lorenz system partial differential equations are constructed such that a truncation scheme as adopted by lorenz j atmos sci 20 130 1963 will lead to lorenz equations the partial differential equations are much simpler than those of the rayleigh bernard problem and are essentially of the mixed type various aspects of the partial differential equations are explored it is suggested that the switching back and forth between the elliptic and hyperbolic regimes represents the chaotic behavior of the system in the context of partial differential equations

on pricing barrier options with regime switching we consider the valuation of both european style and american style barrier options in a markovian regime switching black scholes merton economy where the price process of an underlying risky asset is governed by a markovian regime switching geometric brownian motion both the probabilistic and partial differential equation pde approaches are used to price the barrier options for the probabilistic approach to value a european style barrier option we employ the fundamental matrix solution and the fourier transform space to derive a semi analytical solution the pde approach is employed to value an american barrier option where we obtain a system of free boundary coupled pdes and an analytical quadratic approximation to the price by solving the free boundary problem

hyperelliptic solutions of modified korteweg de vries equation of genus g essentials of the miura transformation explicit hyperelliptic solutions of the modified korteweg de vries equations without any ambiguous parameters were constructed in terms of only the hyperelliptic al functions over the non degenerate hyperelliptic curve y2 f x of arbitrary genus g in the derivation any functions or baker akhiezer functions were not essentially used then the miura transformation naturally appears as the connection between the hyperelliptic functions and hyperelliptic al functions