extensions of frechet ϵ subdifferential calculus and applications abstract in this paper we establish some calculus rules for the limiting frechet subdifferentials of marginal functions and composite functions necessary conditions for approximate solutions of a constrained optimization problem are derived

linear integral equations and nonlinear difference difference equations abstract in this paper we present a systematic method to obtain various integrable nonlinear difference difference equations and the associated linear integral equations from which their solutions can be inferred it is argued that these difference difference equations can be regarded as arising from bianchi identities expressing the commutativity of backlund transformations applying an appropriate continuum limit we first obtain integrable nonlinear differential difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations as special cases we treat the difference difference versions and the differential difference versions of the korteweg de vries equation the modified korteweg de vries equation the nonlinear schrodinger equation the isotropic classical heisenberg spin chain and the complex and real sine gordon equation

the spectral wronskian tool and the investigation of the kdv hierarchy the recently introduced spatial transform st method for providing solutions to nonlinear evolution equations is developed when the basic equation is the spectral schrodinger equation s a fundamental tool is the spectral wronskian which allows one to take advantage of the structure of a two dimensional module for some set of solutions of s this leads easily to the kdv hierarchy contrary to the usual spectral transform or inverse scattering transform method there is no a priori assumption on the long distance behavior of the solutions a recursion operator is exhibited local conservation laws and backlund relations are also derived

geometrical integration of landau lifshitz gilbert equation based on the mid point rule landau lifshitz gilbert llg equation is the fundamental equation to describe magnetization vector field dynamics in microscale and nanoscale magnetic systems this equation is highly nonlinear in nature and for this reason it is generally solved by using numerical techniques in this paper the mid point rule time stepping technique is applied to the numerical time integration of llg equation and the relevant properties of the numerical scheme are discussed the mid point rule is an unconditionally stable and second order accurate scheme which preserves the fundamental geometrical properties of llg dynamics first it exactly preserves the llg property of conserving the magnetization magnitude at each spatial location second for constant in time applied fields it preserves the llg lyapunov structure namely the fact that the free energy is a decreasing function of time in addition in the case of zero damping the mid point rule preserves the conservation of the system free energy the above preservation properties are unconditionally valid i e they are fulfilled for any value of the time step finally the llg hamiltonian structure in the case of zero damping is preserved up to the third order terms with respect to the time step the main difficulty related to this scheme is the necessity of solving a large system of globally coupled nonlinear equations this problem has been circumvented by using special and reasonably fast quasi newton iterative technique the proposed numerical scheme is then tested on the standard micromagnetic problem no 4 in the numerical computations the spatial discretization is obtained by finite difference technique and the magnetostatic field is computed through the fast fourier transform method

nonlinear evolution equations associated with a riemann hilbert scattering problem in an earlier paper nonlinear evolution equations associated with a riemann hilbert scattering problem which reduces in an appropriate limit to the zakharov shabat akns scattering problem were considered here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations moreover the direct linearization of the nonlinear evolution equations and an algorithm to construct ann soliton solution are given

baklund transformations and singular integral equations abstract a systematic method for deriving backlund transformations for singular linear integral equations is presented the method leads in a natural way to the backlund transformations for the corresponding integrable nonlinear partial differential equations by repeated use of the method a hierarchy of singular integral equations and backlund transformations can be derived corresponding to so called multi modified partial differential equations specific examples include the korteweg de vries equation and the first second and third modified korteweg de vries equation the nonlinear schrodinger equation and the anisotropic heisenberg spin chain the sine gordon equation and the modified sine gordon equation

an inverse problem for multidimensional first order systems an inverse problem associated with n first order equations in n 1 dimensions n 1 is considered given appropriate inverse data t reconstruct the potential q x0 x where q is an n n off diagonal matrix although q depends on n 1 variables it turns out that t depends on 3n 1 variables this necessitates imposing certain constraints on t i e t must be suitably characterized the characterization problem for t is solved explicitly furthermore the problem of reconstructing q is reduced to one for reconstructing a 2 2 matrix potential in two dimensions the inverse data needed for the reduced problem are obtained in closed form from t a method for solving two dimensional inverse problems has recently appeared in the literature

string equations for the kdv hierarchy and the grassmannian the grassmannian model for the korteweg de vries hierarchy is used to describe the translational galilean and scaling self similar solutions of this hierarchy these solutions are characterized by the string equations appearing in two dimensional quantum gravity in particular the subsets of the sato grassmannian corresponding to solutions of the string equations are found the well known adler moser rational solutions are obtained as well as a three parameter family of solutions associated with the painleve ii equation

the ekeland variational principle for set valued maps involving coderivatives in this paper we use the frechet clarke and mordukhovich coderivatives to obtain variants of the ekeland variational principle for a set valued map f and establish optimality conditions for set valued optimization problems our technique is based on scalarization with the help of a marginal function associated with f and estimates of subdifferentials of this function in terms of coderivatives of f

davey stewartson type equations in 4 2 and 3 1 possessing soliton solutions an integrable generalisation of a davey stewartson type system of two equations involving two scalar functions in 4 2 i e in four spatial and two temporal dimensions has been recently derived by one of the authors here we first show that there exists a reduction of this system to a single equation involving a scalar function in 4 2 we will refer to this equation as the 4 2 davey stewartson equation we then show that it is possible to reduce this equation to an equation in 3 1 which we will refer to as the 3 1 davey stewartson equation furthermore by employing the so called direct linearising method we compute 1 and 2 soliton solutions for both the 4 2 and the 3 1 davey stewartson equations

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

a unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization in this paper we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now we establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations with the help of quasi efficient solutions a generalized subdifferential of a vector mapping is introduced which generates a number of approximate subdifferentials frequently used in optimization in a unifying way the generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization an application of generalized subdifferential to express optimality conditions for quasi efficient solutions is also given

approximate mean value theorem for upper subderivatives the purpose of this paper is to give a mean value theorem for a lower semicontinuous function f on a banach space using the upper subderivatives defined by rockafellar l as will be shown by example section 4 it may happen that the subdifferential af x is empty for all x in the closed line segment a 61 where a and b belong to x therefore we cannot formulate the mean value theorem in such a form as that obtained in 2 and 3 for other generalized derivatives we prove among other factors that there exists a sequence xk converging to some point x e a 61 x b such that the sets df xj are nonempty and

on some linear integral equations generating solutions of nonlinear partial differential equations abstract two types of linear inhomogeneous integral equations which yield solutions of a broad class of nonlinear evolution equations are investigated one type is characterized by a two fold integration with an arbitrary measure and contour over a complex variable k and thier complex conjugates whereas the other one has a two fold integration over one and the same contour the inhomogeneous term which may contain an arbitrary function of k makes it possible to define a matrix structure on the solutions of the integral equations the elements of these matrices are shown to obey a system of partial differential equations the special form of which depends on the choice of the dispersion relation occurring in the integral equations for special elements of the matrices closed partial differential equations can be derived such as e g the nonlinear schrodinger equation and the real and complex modified korteweg de vries and sine gordon equations the relations between the matrix elements are shown to lead to miura transformations between the various partial differential equations

lattice equations hierarchies and hamiltonian structures ii kp type of hierarchies on 2d lattices abstract we investigate the continuum limit properties of an integrable lattice version of the kadomtsev petviashvili kp equation by applying continuum limits with vertex operators involving an infinite number of continous variables to the lattice kp we obtain hierarchies of integrable equations for fields depending on one continous variable at the sites of a two dimensional lattice the direct linearization of the hierarchies is obtained applying the same limit to the free wave function in the integral equation for lattice kp on the basis of the relation between the generator of the hierarchy and the monodromy factor for the scattering problem on a line which is not involved in the continuum limit we establish the hamiltonian structure of the equations of the hierarchy

a novel formulation for the numerical computation of magnetization modes in complex micromagnetic systems the small oscillation modes in complex micromagnetic systems around an equilibrium are numerically evaluated in the frequency domain by using a novel formulation which naturally preserves the main physical properties of the problem the landau lifshitz gilbert llg equation which describes magnetization dynamics is linearized around a stable equilibrium configuration and the stability of micromagnetic equilibria is discussed special attention is paid to take into account the property of conservation of magnetization magnitude in the continuum as well as discrete model the linear equation is recast in the frequency domain as a generalized eigenvalue problem for suitable self adjoint operators connected to the micromagnetic effective field this allows one to determine the normal oscillation modes and natural frequencies circumventing the difficulties arising in time domain analysis the generalized eigenvalue problem may be conveniently discretized by finite difference or finite element methods depending on the geometry of the magnetic system the spectral properties of the eigenvalue problem are derived in the lossless limit perturbation analysis is developed in order to compute the changes in the natural frequencies and oscillation modes arising from the dissipative effects it is shown that the discrete approximation of the eigenvalue problem obtained either by finite difference or finite element methods has a structure which preserves relevant properties of the continuum formulation finally the generalized eigenvalue problem is solved for a rectangular magnetic thin film by using the finite differences and for a linear chain of magnetic nanospheres by using the finite elements the natural frequencies and the spatial distribution of the natural modes are numerically computed

variational analysis in nonsmooth optimization and discrete optimal control this paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite dimensional spaces the main focus is on the study of problems with nonsmooth structures which require the usage of advanced tools of generalized differentiation in this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete time inclusions in infinite dimensions the principal difference between finite dimensional and infinite dimensional frameworks of optimization and control consists of the lack of compactness in infinite dimensions which leads to imposing certain normal compactness properties and developing their comprehensive calculus together with appropriate calculus rules of generalized differentiation on the other hand one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems in particular we fully avoid the requirements of this type imposed on target endpoint sets in infinite dimensional optimal control for discrete time inclusions

linearization of the nonlinear schrodinger equation and the isotropic heisenberg spin chain abstract a new description in terms of one and the same linear inhomogeneous integral equation is proposed for the nonlinear schrodinger equation nls as well as for the equation of motion for the classical isotropic heisenberg spin chain in the continuum limit ihsc from the integral equation which contains a two fold integration over an arbitrary contour in the complex plane with an arbitrary measure one can obtain the various solutions of the nls as well as of the ihsc in a direct way without going through the details of the inverse scattering formalism well known properties such as the miura transformation the gel fand levitan equation and the lax representations for nls and ihsc can be derived as a corollary from the integral equation the treatment leads also to a few more general integrable partial differential equations which contain the nls and the ihsc as special cases

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

a fuzzy necessary optimality condition for non lipschitz optimization in asplund spaces in this paper we consider a general optimization problem in the broad class of asplund spaces we derive a new necessary optimality condition in the so called lagrangian fuzzy form without standard lipschitz conditions we also give a chain rule for frechet subdifferentials and subdifferential representations of frechet normals to constrained sets

the direct linearization of a class of nonlinear evolution equations the paper deals with the direct linearization an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scattering transform linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1 1 and 2 1 dimensions

lattice equations hierarchies and hamiltonian structures by now many integrable systems with soliton solutions are known in two dimensions for fields u n1 n2 defined at the sites nln2 of a two dimensional 2d lattice for time dependent fields u n t defined at the sites n of a one dimensional chain and for fields u x t depending on two continuous variables x and t the relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method 1 2 which is based on a linear integral equation with arbitrary measure and contour 3 4 in the treatment use is made of backlund transformations bt s which are generated by scalar multiplications of the free wave function and or measure in the integral equation 5 the integrable lattice equations are obtained in the form of bianchi identities expressing the commutativity of bt s the integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time

on the initial value problem of the second painleve transcendent the initial value problem associated with the second painleve transcendent is linearized via a matrix discontinuous homogeneous riemann hilbert rh problem defined on a complicated contour six rays intersecting at the origin this problem is mapped through a series of transformations to three different simple riemann hilbert problems each of which can be solved via a system of two fredholm integral equations the connection of these results with the inverse scattering transform in one and two dimensions is also pointed out

the anisotropic heisenberg spin chain and the nonlinear schrodinger equation abstract it is shown that the solutions of the continuous anisotropic heisenberg spin chain ahsc can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the isotropic heisenberg spin chain ihsc and the nonlinear schrodinger equation nls an explicit expression is obtained for the miura transformation which maps the solutions of the ahsc on solutions of the nls in the second part of the paper we investigate the similarity solutions of these partial differential equations which leads to ordinary differential equations of painleve type as an application we discuss some new solutions of painleve iv

on the numerical integration of the linear shape functions times the 3 d green s function or its gradient on a plane triangle formulas for the static and dynamic potentials due to linearly varying source distributions defined on a planar triangle are presented the general linear variation of the source is represented in terms of three linear basis functions each one associated with a different vertex of the triangular domain the singular kernels considered are given by the 3 d green s function and its gradient both for the static and dynamic case in the static case the evaluation of potentials is performed analytically and compact form results are given in the dynamic case the results are given in a form suited for numerical integration

geometric integration on spheres and some interesting applications geometric integration theory can be employed when numerically solving odes or pdes with constraints in this paper we present several one step algorithms of various orders for odes on a collection of spheres to demonstrate the versatility of these algorithms we present representative calculations for reduced free rigid body motion a conservative ode and a discretization of micromagnetics a dissipative pde we emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions

on local inverse transforms the direct linearization of a nonlinear partial differential equation when it is possible is a purely algebraic process which should not imply strong conditions on the solutions at infinity thus it should be possible to manage it in such a way that the relevant inversion equation is not of the marchenko form but others e g the true gelfand levitan form which preserves the local character of the approach this is demonstrated on the korteweg de vries case

approximations and metric regularity in mathematical programming in banach space this paper establishes verifiable conditions ensuring the important notion of metric regularity for general nondifferentiable programming problems in banach spaces these conditions are used to obtain lagrange kuhn tucker multipliers for minimization problems with infinitely many inequality and equality constraints

direct linearization approach to discrete integrable systems associated with ℤ graded lax pairs fordy and xenitidis j phys a math theor 50 2017 165205 doi 10 1088 1751 8121 aa639a recently proposed a large class of discrete integrable systems which include a number of novel integr

iteration of some discretizations of the nonlinear schrodinger equation we consider several discretizations of the nonlinear schrodinger equation which lead naturally to the study of some symmetric difference equations of the form n 1 n 1 f n we find a variety of interesting and exotic behaviour from simple closed orbits to intricate patterns of orbits and loops in the n 1 n phase plane some analytical results for a special case are also presented