a second wronskian formulation of the boussinesq equation a wronskian formulation leading to rational solutions is presented for the boussinesq equation it involves third order linear partial differential equations whose representative systems are systematically solved the resulting solutions formulas provide a direct but powerful approach for constructing rational solutions positon solutions and complexiton solutions to the boussinesq equation various examples of exact solutions of those three kinds are computed the newly presented wronskian formulation is different from the one previously presented by li et al which does not yield rational solutions

quasi periodic solutions for some 2 1 dimensional discrete models abstract some new 2 1 dimensional discrete models are proposed with the help of the 1 1 dimensional nonlinear network equations describing a volterra system the nonlinearization of the lax pairs associated with the 1 1 dimensional nonlinear network equations leads to a new integrable symplectic map and a class of finite dimensional integrable hamiltonian systems these 2 1 dimensional discrete models are decomposed into two hamiltonian systems of ordinary differential equations plus the discrete flow generated by the symplectic map the evolution of various flows is explicitly given through the abel jacobi coordinates quasi periodic solutions for these 2 1 dimensional discrete models are obtained resorting to the riemann theta functions

a note on controlled diffusions on line with time averaged cost existence of stable optimal markov controls is established for a class of cost functions for controlled one dimensional diffusions with time averaged cost

inverse problems for lorentzian manifolds and non linear hyperbolic equations we study two inverse problems on a globally hyperbolic lorentzian manifold m g the problems are 1 passive observations in spacetime consider observations in an open set v subset m the light observation set corresponding to a point source at q in m is the intersection of v and the light cone emanating from the point q let w subset m be an unknown open relatively compact set we show that under natural causality conditions the family of light observation sets corresponding to point sources at points q in w determine uniquely the conformal type of w 2 active measurements in spacetime we develop a new method for inverse problems for non linear hyperbolic equations that utilizes the non linearity as a tool this enables us to solve inverse problems for non linear equations for which the corresponding problems for linear equations are still unsolved to illustrate this method we solve an inverse problem for semilinear wave equations with quadratic non linearities we assume that we are given the neighborhood v of the time like path mu and the source to solution operator that maps the source supported on v to the restriction of the solution of the wave equation to v when m is 4 dimensional we show that these data determine the topological differentiable and conformal structures of the spacetime in the maximal set where waves can propagate from mu and return back to mu

quasi periodic waves and an asymptotic property for the asymmetrical nizhnik novikov veselov equation based on a multi dimensional riemann theta function the hirota bilinear method is extended to explicitly construct multi periodic quasi periodic wave solutions for the asymmetrical nizhnik novikov veselov equation among these periodic waves two periodic waves are a direct generalization of well known cnoidal waves their surface pattern is two dimensional the main physical result is the description of the behavior of nonlinear waves in shallow water a limiting procedure is presented to analyze asymptotic properties of the two periodic waves in details 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

a vector potential kdv equation and vector ito equation soliton solutions bilinear backlund transformations and lax pairs a vector potential kdv equation and vector ito equation are proposed based on their bilinear forms soliton solutions expressed by pfaffians are obtained bilinear backlund transformations and the corresponding lax pairs for the vector potential kdv equation and the vector ito equation are derived

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

on smooth cauchy hypersurfaces and geroch s splitting theorem given a globally hyperbolic spacetime m we show the existence of a smooth spacelike cauchy hypersurface s and thus a global diffeomorphism between m and r s

recovering singularities of a potential from singularities of scattering data in this paper we show that the leading singularities of certain potentials can be determined from the leading singularities of the backscattering as well as other determined sets of scattering data the potentials in question are conormal with respect to smooth surfaces of arbitrary dimension the restrictions on their orders allow for unbounded potentials in all dimension greater than or equal to three

an integrable symmetric 2 1 dimensional lotka volterra equation and a family of its solutions a symmetric 2 1 dimensional lotka volterra equation is proposed by means of a dependent variable transformation the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables a bilinear backlund transformation is found and then the corresponding lax pair is derived explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric 2 1 dimensional lotka volterra equation are given as a special case of the pfaffian solutions we obtain soliton solutions and dromions

long term optimal investment strategies in the presence of adjustment costs we consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment in particular we consider an investment project that yields payoff at a rate that depends on its installed capacity level and on a random economic indicator such as the price of the project s output commodity we model this economic indicator by means of a general one dimensional ergodic diffusion at any time the project s capacity level can be increased or decreased at given proportional costs the aim is to maximize an ergodic performance criterion that reflects the long term average payoff resulting from the project s management we solve this genuinely two dimensional stochastic control problem by constructing an explicit solution to an appropriate hamilton jacobi bellman equation and by fully characterizing an optimal investment strategy

soliton solutions of kd system using similarity transformations method abstract present article deals with some exact solutions of 2 1 dimensional system of coupled konopelchenko dubrovsky equations similarity transformations method is proposed to seek the solution of the system using lie group theory the lie group theory is a very strong tool by which complicated nonlinear partial differential equations under the group transformations remains invariant a brief review of lie symmetries of a system of partial differential equations has been described during the process method reduces the number of independent variables by one hence the system of partial equations reduces into a new system of ordinary differential equations in addition the infinitesimals after first reductions are more general than the previous established results by us kumar et al 2016 consequently solutions so derived are more general than previously known results we have obtained nine solutions in the explicit form some of them are more general and some are new for the best knowledge of us ultimately single solitons multi solitons and shocks behaviour is represented graphically through the numerical simulation for physical validation of the results

the hirota s direct method for multiple soliton solutions for three model equations of shallow water waves abstract multiple soliton solutions for three model equations for shallow water waves are determined the three models are completely integrable the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these equations the tanh coth method is used to obtain single soliton solutions and other solutions for these three models the three models have different linear dispersion relations but possess the same coefficients for the polynomials of exponentials

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

ergodic control of multidimensional diffusions 1 the existence results the existence of optimal stable markov relaxed controls for the ergodic control of multidimensional diffusions is established by direct probabilistic methods based on a characterization of a s limit sets of empirical measures the optimality of the above is established in the strong i e almost sure sense among all admissible controls under very general conditions

multiple soliton solutions for the lax kadomtsev petviashvili lax kp equation a new completely integrable dispersive equation is derived the new equation is obtained by extending the lax fifth order equation using the sense of the kadomtsev petviashvili kp equation in extending the kdv equation the newly derived lax kadomtsev petviashvili lax kp equation is investigated by using the tanh coth method and the hirota bilinear method to derive single soliton solution and n soliton solutions respectively the study highlights the multiple soliton solutions of the derived completely integrable equation

local propagation of impulsive gravitationalwaves in this paper we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves we construct such spacetimes as solutions to the characteristic initial value problem of the einstein vacuum equations with a data curvature delta singularity we show that in the resulting spacetime the delta singularity propagates along a characteristic hypersurface while away from that hypersurface the spacetime remains smooth unlike the known explicit examples of impulsive gravitational spacetimes this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions the arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data 2015 wiley periodicals inc

on minimum cost per unit time control of markov chains the minimum cost per unit time control problem is studied for a class of markov chains that though important in applications does not fit the conventional framework for this problem existence of optimal stationary strategies and necessary and sufficient conditions for optimality are established

a class of exact periodic solutions of nonlinear envelope equations a class of periodic solutions of nonlinear envelope equations e g the nonlinear schrodinger equation nls is expressed in terms of rational functions of elliptic functions the hirota bilinear transformation and theta functions are used to extend and generalize this class of solutions first reported for nls earlier in the literature in particular a higher order nls and the davey stewartson ds equations are treated doubly periodic standing waves solutions are obtained for both the dsi and dsii equations a symbolic manipulation software is used to confirm the validity of the solutions independently

a criterion for invariant measures of markov processes using the martingale formulation for markov processes introduced by stroock and varadhan we develop a criterion for checking if a measure happens to be invariant

complexiton solutions of the toda lattice equation a set of coupled conditions consisting of differential difference equations is presented for casorati determinants to solve the toda lattice equation one class of the resulting conditions leads to an approach for constructing complexiton solutions to the toda lattice equation through the casoratian formulation an analysis is made for solving the resulting system of differential difference equations thereby providing the general solution yielding eigenfunctions required for forming complexitons moreover a feasible way is presented to compute the required eigenfunctions along with examples of real complexitons of lower order

multiple soliton solutions of two extended model equations for shallow water waves abstract multiple soliton solutions for two extensions of model equations for shallow water waves presented in m j ablowitz d j kaup a c newell h segur the inverse scattering transform fourier analysis for nonlinear problems stud appl math 53 1974 249 315 r hirota j satsuma n soliton solutions of model equations for shallow water waves j phys soc jpn 40 2 1976 611 612 are obtained the two extensions possess identical dispersion relations but with different structures for the n soliton solutions for n 1 the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these extended equations the tanh coth method is used to obtain single soliton solutions

quasi periodic wave solutions for the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation abstract the hirota bilinear method is implemented to directly construct quasi periodic wave solutions in terms of theta functions for a hirota bilinear equation the result is applied to the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation thereby yielding their quasi periodic wave solutions under the backlund transformation u y 3 l n f x y t x

ergodic control problem for one dimensional diffusions with near monotone cost the existence and uniqueness of the solution to the bellman equation for ergodic control of one dimensional diffusions is established under a near monotonicity condition on the cost necessary and sufficient conditions for the optimality of a stable markov control are given in terms of the same