on the neumann problem for pde s with a small parameter and the corresponding diffusion processes the diffusion process in a region g subset mathbb r 2 governed by the operator tilde l varepsilon frac 1 2 u xx frac1 2 varepsilon u zz inside the region and undergoing instantaneous co normal reflection at the boundary is considered we show that the slow component of this process converges to a diffusion process on a certain graph corresponding to the problem this allows to find the main term of the asymptotics for the solution of the corresponding neumann problem in g the operator tilde l varepsilon is up to the factor e 1 the result of small perturbation of the operator frac 1 2 u zz our approach works for other operators diffusion processes in any dimension if the process corresponding to the non perturbed operator has a first integral and the e process is non degenerate on non singular level sets of this first integral

order conditions for stochastic runge kutta methods preserving quadratic invariants of stratonovich sdes abstract in this paper we prove that for a stochastic runge kutta method the conditions for preserving quadratic invariants work as simplifying assumptions for such methods the method coefficients only have to satisfy one condition for each unrooted tree this is a generalization of the result obtained for deterministic runge kutta methods by sanz serna and abia in 1991

new solitons for the hirota equation and generalized higher order nonlinear schrodinger equation with variable coefficients in this paper multisoliton solutions of the hirota equation with variable coefficients are obtained by the darboux transformation based on the ablowitz kaup newell segur technology as an example we discuss the evolutional behaviour of a two soliton solution in a soliton control fibre system the results reveal that one may control the interaction between the pulses by choosing the third order dispersion parameters d4 and h appropriately meanwhile more generalized forms of bright soliton and dark soliton solutions of generalized higher order nonlinear schrodinger equations ghonlse with variable coefficients are obtained by the extended tanh function method moreover new bright and dark combined solitary wave kink solitary wave and m shaped solitary wave to ghonlse with variable coefficients are firstly reported in this paper especially the term proportional to 1 resulting from the group velocity decides the group velocity and the phase shift of these new solitary waves

regularity of solutions to quantum master equations a stochastic approach applying probabilistic techniques we study regularity properties of quantum master equations qmes in the lindblad form with unbounded coefficients a density operator is regular if roughly speaking it describes a quantum state with finite energy using the linear stochastic schrodinger equation we deduce that solutions of qmes preserve the regularity of the initial states under a general nonexplosion condition to this end we develop the probabilistic representation of qmes and we prove the uniqueness of solutions for adjoint quantum master equations by means of the nonlinear stochastic schrodinger equation we obtain the existence of regular stationary solutions for qmes under a lyapunov type condition

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

numerical solution of stochastic quantum master equations using stochastic interacting wave functions abstract we develop a new approach for solving stochastic quantum master equations with mixed initial states first we obtain that the solution of the jump diffusion stochastic master equation is represented by a mixture of pure states satisfying a system of stochastic differential equations of schrodinger type then we design three exponential schemes for these coupled stochastic schrodinger equations which are driven by brownian motions and jump processes hence we have constructed efficient numerical methods for the stochastic master equations based on quantum trajectories the good performance of the new numerical integrators is illustrated by simulations of two quantum measurement processes

stabilizing feedback controls for quantum systems no quantum measurement can give full information on the state of a quantum system hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory here we study the properties of controlled quantum filtering equations as classical stochastic differential equations we then develop methods using a combination of geometric control and classical probabilistic techniques for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator

estimates of low sobolev norms of solutions for some nonlinear evolution equations abstract in this work we obtain results on the estimates of low sobolev norms for solutions of some nonlinear evolution equations in particular we apply our method for the complex modified korteweg de vries type equation and benjamin ono equation

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

exact travelling wave solutions for a generalized nonlinear schrodinger equation a wide class of exact travelling wave solutions of a generalized nonlinear schrodinger equation gnls is obtained and analysed in detail this class of solutions incorporates bright and dark solitary waves periodic waves unbounded waves and other solitary waves as asymptotic limits of the periodic or unbounded modes the method of analysis adopted is based on reducing the gnls to an ordinary differential equation and studying the phase plane of the resulting dynamical system application of the obtained results to the problem of propagation of femtosecond duration pulses in nonlinear optical fibres is also discussed

the scaling and squaring method for the matrix exponential revisited the scaling and squaring method is the most widely used method for computing the matrix exponential not least because it is the method implemented in the matlab function expm the method scales the matrix by a power of 2 to reduce the norm to order 1 computes a pade approximant to the matrix exponential and then repeatedly squares to undo the effect of the scaling we give a new backward error analysis of the method in exact arithmetic that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency we also give a new rounding error analysis that shows the computed pade approximant of the scaled matrix to be highly accurate for ieee double precision arithmetic the best choice of degree of pade approximant turns out to be 13 rather than the 6 or 8 used by previous authors our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than the expm function in matlab 7 0 when the matrix norm exceeds 1 which can amount to a 37 saving in the number of multiplications and it is typically more accurate owing to the fewer required squarings we also investigate a different scaling and squaring algorithm proposed by najfeld and havel that employs a pade approximation to the function x coth x this method is found to be essentially a variation of the standard one with weaker supporting error analysis

heisenberg evolution of quantum observables represented by unbounded operators this paper deals with open quantum systems in particular we focus on the adjoint quantum master equations with initial conditions given by unbounded operators examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many body particle systems the article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the born markov approximations to this end we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of schrodinger type furthermore we examine quantum dynamical semigroup properties of the heisenberg evolutions of general classes of observables

symplectic runge kutta semidiscretization for stochastic schrodinger equation based on a variational principle with a stochastic forcing we indicate that the stochastic schrodinger equation in the stratonovich sense is an infinite dimensional stochastic hamiltonian system whose phase flow preserves symplecticity we propose a general class of stochastic symplectic runge kutta methods in the temporal direction to the stochastic schrodinger equation in the stratonovich sense and show that the methods preserve the charge conservation law we present a convergence theorem on the relationship between the mean square convergence order of a semi discrete method and its local accuracy order taking the stochastic midpoint scheme as an example of stochastic symplectic runge kutta methods in the temporal direction based on the theorem we show that the mean square convergence order of the semidiscrete scheme is 1 under appropriate assumptions

soliton like solutions of a generalized variable coefficient higher order nonlinear schrodinger equation from inhomogeneous optical fibers with symbolic computation for the long distance communication and manufacturing problems in optical fibers the propagation of subpicosecond or femtosecond optical pulses can be governed by the variable coefficient nonlinear schrodinger equation with higher order effects such as the third order dispersion self steepening and self frequency shift in this paper we firstly determine the general conditions for this equation to be integrable by employing the painleve analysis based on the obtained 3 3 lax pair we construct the darboux transformation for such a model under the corresponding constraints and then derive the nth iterated potential transformation formula by the iterative process of darboux transformation through the one and two soliton like solutions we graphically discuss the features of femtosecond solitons in inhomogeneous optical fibers

long time behavior of weakly coupled oscillators we consider small perturbations of a simple completely integrable system with many degrees of freedom a collection of independent one degree of freedom oscillators in the perturbed system the individual oscillators are no longer independent we show that the long time behavior of such a system even in the case of purely deterministic perturbations should in general be described as a stochastic process the limiting stochastic process is a markov process on an open book space corresponding to the collection of first integrals of the non perturbed system

averaging principle for the higher order nonlinear schrodinger equation with a random fast oscillation this work concerns the problem associated with averaging principle for a higher order nonlinear schrodinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction diffusion equation evolving with respect to the fast time this model can be translated into a multiscale stochastic partial differential equations stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time scales to be more precise under suitable conditions we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear schrodinger equation is an average with respect to the stationary measure of the fast varying process finally by using the khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation and as a consequence the system can be reduced to a single higher order nonlinear schrodinger equation with a modified coefficient

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

two kinds of new integrable decompositions of the gerdjikov ivanov equation the gerdjikov ivanov gi equation can be given from the first non trivial flow of the generalized kaup newell equations under a reduction condition in this paper we perform binary nonlinearization for the generalized kaup newell equations under two symmetry constraints by imposing the reduction condition on this nonlinearization we obtain two kinds of new integrable decompositions of the gi equation

stochastic symplectic and multi symplectic methods for nonlinear schrdinger equation with white noise dispersion we indicate that the nonlinear schrdinger equation with white noise dispersion possesses stochastic symplectic and multi symplectic structures based on these structures we propose the stochastic symplectic and multi symplectic methods which preserve the continuous and discrete charge conservation laws respectively moreover we show that the proposed methods are convergent with temporal order one in probability numerical experiments are presented to verify our theoretical results

high weak order methods for stochastic differential equations based on modified equations inspired by recent advances in the theory of modified differential equations we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations this approach is illustrated with the constructions of new methods of weak order two in particular semi implicit integrators well suited for stiff mean square stable stochastic problems and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system numerical examples confirm the theoretical results and show the versatility of our methodology

unique continuation property for a higher order nonlinear schrodinger equation abstract we prove that if a sufficiently smooth solution u to the initial value problem associated with the equation t u i x 2 u x 3 u i u 2 u u 2 x u e u 2 x u 0 x t r is supported in a half line at two different instants of time then u 0 to prove this result we derive a new carleman type estimate by extending the method introduced by kenig et al in ann inst h poincare anal non lineaire 19 2002 191 208

scattering solutions in networks of thin fibers small diameter asymptotics small diameter asymptotics is obtained for scattering solutions in a network of thin fibers the asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph we calculate the lagrangian gluing conditions at vertices v in gamma for the problems on the limiting graph if the frequency of the incident wave is above the bottom of the absolutely continuous spectrum the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

numerical simulation of stochastic evolution equations associated to quantum markov semigroups we address the problem of approximating numerically the solutions x t t 0 t of stochastic evolution equations on hilbert spaces h with respect to brownian motions arising in the unraveling of backward quantum master equations in particular we study the computation of mean values of x t ax t where a is a linear operator first we introduce estimates on the behavior of x t then we characterize the error induced by the substitution of x t with the solution x t n of a convenient stochastic ordinary differential equation it allows us to establish the rate of convergence of e to e where x t n denotes the explicit euler method finally we consider an extrapolation method based on the euler scheme an application to the quantum harmonic oscillator system is included

bright and dark soliton solutions and backlund transformation for the eckhaus kundu equation with the cubic quintic nonlinearity in this paper with symbolic computation the eckhaus kundu equation which appears in the quantum field theory weakly nonlinear dispersive water waves and nonlinear optics is studied via the hirota method by virtue of the dependent variable transformation the bilinear form is obtained bilinear backlund transformation is given with the help of exchange formulae and the corresponding one soliton solution is derived bright and dark n soliton solutions are obtained propagation and interaction of the bright and dark solitons are discussed analytically and graphically interactions of the two solitons are presented bound state of the two solitons can be suppressed via the choice of parameters

algebro geometric solutions for the gerdjikov ivanov hierarchy this paper is dedicated to provide theta function representations of algebro geometric solutions and related crucial quantities for the gerdjikov ivanov gi hierarchy our main tools include the polynomial recursive formalism the hyperelliptic curve with finite number of genus the baker akhiezer functions the meromorphic function the dubrovin type equations for auxiliary divisors and the associated trace formulas with the help of these tools the explicit representations of the baker akhiezer functions the meromorphic function and the algebro geometric solutions are obtained for the entire gi hierarchy

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

optical solitons in a monomode fiber we discuss the propagation of optical solitons in a monomode fiber as a model of long distance high bit rate transmission system we give several new results which did not appear in our previous papers on this subject such as 1 a derivation of the perturbed nonlinear schrodinger equation from the maxwell equation 2 on the integrability of the perturbed nonlinear schrodinger equation 3 a discussion of the soliton as a stable fixed point of certain infinite dimensional map generated by a transmission system with periodic excitations