the riemann boundary value problem on closed riemann surfaces and integrable systems the riemann problem has different forms in several areas of physics scattering theory integrable systems and so on it is necessary to build an analytic function or a matrix with prescribed poles and zeros on a riemann surface another similar problem is to construct a section of a complex vector bundle over a riemann surface for example anti self dual yang mills equation as was established by a grothendieck and h rohrl this problem is equivalent to the riemann boundary value problem let m be a compact riemann surface a contour on m g p an n n matrix on determine an analytic matrix f p in m which is multiple of a given divisor and satisfies the boundary condition on f p g p f p this problem is well understood on the complex plane it is related to singular integral operators wiener hopf equations and banach algebras in the case of a riemann surface there are additional with the riemann roch and abel theorems jacobi varieties and riemann theta functions these relations are described here the riemann problem is equivalent to the so called problem u au recently this elliptic equation was used to study the inverse scattering problem finally we consider some physical problems finite gap potentials the landau lifschitz equation the painleve problems to illustrate the applications of the riemann problem on a riemann surface acquaintance with riemann surface theory and algebraic topology is not assumed

a numerical study of the 3 periodic wave solutions to kdv type equations abstract in this paper by using the direct method of calculating periodic wave solutions proposed by akira nakamura we present a numerical process to calculate the 3 periodic wave solutions to several kdv type equations the korteweg de vries equation the sawada koterra equation the boussinesq equation the ito equation the hietarinta equation and the 2 1 dimensional kadomtsev petviashvili equation some detailed numerical examples are given to show the existence of the three periodic wave solutions numerically

an integrable couplings of g wki equations hierarchy with self consistent sources we present a kind of integrable couplings of soliton equations hierarchy with self consistent sources by using the loop algebra sl 4 as an application a new integrable couplings of the generalized wadati konono ichikawa g wki equations hierarchy with self consistent sources is derived through the enlarged loop algebra sl 4

new stable numerical inversion of abel s integral equation abstract the 3 d image reconstruction from cone beam projections in computerized tomography leads naturally in the case of radial symmetry to the study of abel s type of integral equations a new method for the numerical solution of such equations when the experimental information is obtained through measured data on a discrete set of points is presented and rigorously analyzed

an adaptive huber method for non linear systems of weakly singular second kind volterra integral equations abstract numerical methods for systems of weakly singular volterra integral equations are rarely considered in the literature especially if the equations involve non linear dependencies between unknowns and their integrals in the present work an adaptive huber method for such systems is proposed by extending the method previously formulated for single weakly singular second kind volterra equations the method is tested on example systems of integral equations involving integrals with kernels k t t 1 2 k t exp t t 1 2 where 0 and k t 1 the magnitude of the errors and practical accuracy orders observed for ie systems are comparable to those for single ies in cases when the solution vector is not differentiable at t 0 the estimation of errors at t 0 is found somewhat less reliable for ie systems than it was for single ies the stability of the ie systems solved appears to be sufficient in practice for the numerical stability of the method

cyclically stationary brownian local time processes local time processes parameterized by a circle defined by the occupation density up to time t of brownian motion with constant drift on the circle are studied for various random times t while such processes are typically non markovian their laplace functionals are expressed by series formulae related to similar formulae for the markovian local time processes subject to the ray knight theorems for bm on the line and for squares of bessel processes and their bridges for t the time that bm on the circle first returns to its starting point after a complete loop around the circle the local time process is cyclically stationary with same two dimensional distributions but not the same three dimensional distributions as the sum of squares of two i i d cyclically stationary gaussian processes this local time process is the infinitely divisible sum of a poisson point process of local time processes derived from brownian excursions the corresponding intensity measure on path space and similar levy measures derived from squares of bessel processes are described in terms of a 4 dimensional bessel bridge by williams decomposition of ito s law of brownian excursions

quasiperiodic waves and asymptotic behavior for bogoyavlenskii s breaking soliton equation in 2 1 dimensions based on a multidimensional riemann theta function the hirota bilinear method is extended to explicitly construct multiperiodic quasiperiodic wave solutions for the 2 1 dimensional bogoyavlenskii breaking soliton equation among these periodic waves the one periodic waves are well known cnoidal waves their surface pattern is one dimensional and often they are used as one dimensional models of periodic waves in shallow water the two periodic biperiodic waves are a direct generalization of one periodic waves their surface pattern is two dimensional that is they have two independent spatial periods in two independent horizontal directions the two periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality a limiting procedure is presented to analyze asymptotic behavior of the one and two periodic waves in details the exact relations between the periodic wave solutions and the well known soliton solutions are established it is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit

existence of gauge field in any partially integrable systems for any integrable and more generally any partially integrable model the existence is shown of a yang mills gauge field with zero curvature whose gauge group consists of all general coordinate transformations it has been suggested that this fact may have some connection with existence of an affine connection in the space with zero riemann curvature tensor a brst like quadratically nilpotent operator is also constructed for any theory with poisson bracket structure whenever such a connection with zero riemann curvature exists

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

on the numerical study of nonlinear initial boundary value problems or initial value problems this paper deals with a numerical study for two problems first of which is a system of nonlinear initial boundary value problems for a certain version of boussinesque system of equations by using two quadratic forms the second one is to investigate and construct the numerical solution for a system of nonlinear initial value problems which is generated by a system of hyper elliptic type in some hilbert space

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

a riemann hilbert problem for the finite genus solutions of the kdv equation and its numerical solution abstract we derive a riemann hilbert problem satisfied by the baker akhiezer function for the finite gap solutions of the korteweg de vries kdv equation as usual for riemann hilbert problems associated with solutions of integrable equations this formulation has the benefit that the space and time dependence appears in an explicit linear and computable way we make use of recent advances in the numerical solution of riemann hilbert problems to produce an efficient and uniformly accurate numerical method for computing all periodic and quasi periodic finite genus solutions of the kdv equation

right inverses of levy processes and stationary stopped local times if x is a symmetric levy process on the line then there exists a non decreasing cadlag process h such that x h x x for all x 0 if and only if x is recurrent and has a non trivial gaussian component the minimal such h is a subordinator k the law of k is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of x when x is brownian motion k is just the usual ladder times process and this result extends the classical result of levy that the maximum process has the same law as the local time at 0 write g t for last point in the range of k prior to t in a parallel with classical fluctuation theory the process z x t x gt t 0 is markov with local time at 0 given by x gt t 0 the transition kernel and excursion measure of z are identified a similar programme is outlined for levy processes on the circle this leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle

algebro geometric solution of the 2 1 dimensional burgers equation with a discrete variable the quasiperiodic solution of the 2 1 dimensional burgers equation with a discrete variable is obtained through three steps a decomposition into a symplectic map plus two finite dimensional hamiltonian systems b straightening out of both the discrete and the continuous flows in the jacobian variety c inversion into the original variables inner relation with the modified kadomtsev petviashvili equation is presented the explicit theta function solutions for these two 2 1 integrable models are given

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

the constraints of potentials and the finite dimensional integrable systems restricting potential to the space spanned by the eigenvectors of the recursion operator leads to a natural constraint of potential and a finite dimensional integrable hamiltonian system the general method for proving the consistency of the two systems stemming from the lax pair and obtaining the constants of the motion for the hamiltonian system is illustrated by the classical boussinesq and akns hierarchies by using gauge transformation similar results for the jaulent miodek and kaup newell hierarchies are presented

a numerical scheme for periodic travelling wave simulations in some nonlinear dispersive wave models a numerical method for simulating periodic travelling wave solutions of some nonlinear dispersive wave equations is proposed the construction of the scheme is based on an efficient computation of the elements that characterize these solutions the initial profile and the velocity of the wave

some algebro geometric solutions for the coupled modified kadomtsev petviashvili equations arising from the neumann type systems a treatment is described for getting some algebro geometric solutions of the coupled modified kadomtsev petviashvili cmkp equations and a hierarchy of 1 1 dimensional integrable nonlinear evolution equations inlees by using the neumann type systems through three steps i according to the nonlinearization of lax pair the cmkp equations and the 1 1 dimensional inlees are reduced to a family of compatible neumann type systems on a symplectic submanifold whose involutive solutions give rise to the finite parametric solutions of the inlees in both 2 1 and 1 1 dimensions ii from the holomorphic differentials and the abel map on a hyperelliptic curve of riemman surface the abel jacobi variables are introduced to straighten out the neumann type flows the 1 1 and 2 1 dimensional flows giving the abel jacobi solutions iii based on the riemann theorem and the trace formulas the jacobi inversion is applied to the straightened flows for getting some new algebro geometric solutions of the cmkp equations and the 1 1 dimensional integrable hierarchy

high accuracy product integration methods for the abel equation and their application to a problem in scattering theory in this paper we present orsder three four and five backward product integration methods for the generalized abel equation we illustrate their use by silving the small angle deflection problem and show computationally that these methods are convergent for all 0 1 the coefficients of order three and four methods are given in applenidx i

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

on a periodic boundary value problem with a finite number of gaps in its spectrum co 4 9 n a 3 a t t wy 3 are called instability intervals all but the first interval in 3 are finite and may shrink to a point under special conditions erdelyi 3 discovered situations where all but a finite number of finite instability intervals vanish when q t is a suitable elliptic function lax 9 showed that a function q t which satisfies the nth order korteweg de vries equation requires 1 to have no more than n non vanishing finite instability intervals conversely if all finite instability intervals vanish then q t in 1 is necessarily a constant this fact was proved by borg 11 hochstadt 71 and ungar 121 when all but n finite instability intervals vanish q t satisfies a differential equation of the form

the symmetries of solitons in this article we will retrace one of the great mathematical adventures of this century the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries we will take an historical approach starting with a famous numerical experiment carried out by fermi pasta and ulam on one of the first electronic computers and with zabusky and kruskal s insightful explanation of the surprising results of that experiment and of a follow up experiment of their own in terms of a new concept they called solitons solitons however raised even more questions than they answered in particular the evolution equations that govern solitons were found to be hamiltonian and have infinitely many conserved quantities pointing to the existence of many non obvious symmetries we will cover next the elegant approach to solitons in terms of the inverse scattering transform and lax pairs and finally explain how those ideas led step by step to the discovery that loop groups acting by dressing transformations give a conceptually satisfying explanation of the secret soliton symmetries

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

stopped markov chains with stationary occupation times let e be a finite set equipped with a group g of bijective transformations and suppose that x is an irreducible markov chain on e that is equivariant under the action of g in particular if e g with the corresponding transformations being left or right multiplication then x is a random walk on g we show that when x is started at a fixed point there is a stopping time u such that the distribution of the random vector of pre u occupation times is invariant under the action of g when g acts transitively that is e is a homogeneous space any non zero finite expectation stopping time with this property can occur no earlier than the time s of the first return to the starting point after all states have been visited we obtain an expression for the joint laplace transform of the pre s occupation times for an arbitrary finite chain and show that even for random walk on the group of integers mod r the pre s occupation times do not generally have a group invariant distribution this appears to contrast with the brownian analog as there is considerable support for the conjecture that the field of local times for brownian motion on the circle prior to the counterpart of s is stationary under circular shifts