pseudounitary symmetry and the gaussian pseudounitary ensemble of random matrices employing the currently discussed notion of pseudo hermiticity we define a pseudounitary group further we develop a random matrix theory that is invariant under such a group and call this ensemble of pseudo hermitian random matrices the pseudounitary ensemble we obtain exact results for the nearest neighbor level spacing distribution for 2 x 2 pt invariant hamiltonian matrices that have forms approximately sln 1 s near zero spacing for three independent elements and approximately s for four independent elements this shows a level repulsion in a marked distinction with an algebraic form s beta in the wigner surmise we believe that this paves the way for a description of varied phenomena in two dimensional statistical mechanics quantum chromodynamics and so on

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

stability analysis and synthesis of nonlinear time delay systems via linear takagi sugeno fuzzy models abstract this paper concerns with the stability analysis and synthesis of nonlinear retarded systems via linear takagi sugeno ts fuzzy model approach first the ts fuzzy models with time delay are present and the stability conditions are derived using lyapunov razumikhin functional approach then a stabilization approach for nonlinear retarded systems through fuzzy state feedback and fuzzy observer based controller is proposed it shows that the analysis results provide an efficient technique for the design of fuzzy controllers sufficient conditions for the existence of fuzzy state feedback gain and fuzzy observer gain of the time delay systems are derived through the numerical solution of a set of coupled linear matrix inequalities

multi soliton and double wronskian solutions of a 2 1 dimensional modified heisenberg ferromagnetic system a 2 1 dimensional modified heisenberg ferromagnetic system which arises in the motion of magnetization vector of the isotropic ferromagnet and biological pattern formation is investigated via the hirota bilinear method multi soliton solutions of such a system are derived it is proved that the system possesses the n soliton solutions expressed in terms of the double wronskian determinant head on and overtaking elastic interactions are exhibited elastic interaction behavior between the two solitons has been interpreted through the asymptotic analysis namely amplitude and velocity of each soliton remain unchanged except for the phase shift after the interaction inelastic interactions including the soliton fusion and fission between two solitons are shown during the soliton propagation for the product of two fields the soliton with the smaller amplitude can travel faster than with the larger while for the third field the soliton with the larger amplitude can travel faster than with the smaller on the other hand the soliton for the third field may exhibit the solitoff like property with respect to the three solitons head on elastic interaction can be found

the boussinesq equation revisited the continuous spectrum and soliton solutions for the boussinesq equation are investigated using the dressing method solitons demonstrate quite extraordinary behavior they may decay or form a singularity in a finite time formation of singularity collapse of solitons for the boussinesq equation was discovered several years ago systematic study of the solitonic sector is presented

an extended noncommutative kp hierarchy introducing squared eigenfunctions in the moyal deformed lax equations generates an extended noncommutative kp nckp hierarchy the compatibility equations between the nckp flows and the extended nckp flows and the compatibility equations among the extended nckp flows themselves are constructed the n reductions and the k constraints are analyzed and thus the nckp hierarchy with self consistent sources the constrained nckp hierarchy and the constrained nckp hierarchy with self consistent sources are presented various illustrative examples are worked out together with their lax representations involving pseudodifferential lax operators

a multistage reduction technique for feedback stabilizing distributed time lag systems abstract a direct pole relocation theory is advanced for linear time invariant systems with distributed delays in both state and control variables the principal tools of the theory include i the finite cardinality of the unstable spectrum ii a set of matrices each of which is a left zero of the system characteristic quasi polynomial matrix and iii a linear transformation which reduces the delay system to a sufficiently high order delay free system whose spectrum contains the delay system unstable spectrum it is shown that if the delay system is spectrally stabilizable then it shares a common feedback stabilizing control law with its delay free counterpart this point of contact with a delay free system permits the determination of the control law using well established ordinary system methods the workability of the approach hinges on the ability to partition the unstable spectrum augmented with additional poles from the stable spectrum if necessary into n symmetric sets when this partition is impossible a spectral controllability invariance theorem facilitates resolution of the problem

positon solutions of the sine gordon equation the concept of positons is introduced for the sine gordon equation exploiting its covariance under darboux transformations positon solutions represent potentials in the associated linear system corresponding to a vanishing reflection coefficient and a transmission coefficient identically one the mutual interaction of positons and the interaction of positons with solitons is considered

two new wronskian conditions for the 3 1 dimensional jimbo miwa equation abstract in this paper we obtain two kinds of sufficient conditions consisting of systems of linear partial differential equations which guarantee that the corresponding wroskian determinant solves the 3 1 dimensional jimbo miwa equation in the hirota bilinear form our results suggest that more general conditions could be derived by further study

hirota bilinear equations with linear subspaces of solutions we explore when hirota bilinear equations possess linear subspaces of solutions first we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to hirota bilinear equations second we show that multivariate polynomials whose zeros form a vector space can generate the desired hirota bilinear equations with given linear subspaces of solutions and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero third applying an algorithm using weights we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired hirota bilinear equations

extended gram type determinant wave and rational solutions to two 3 1 dimensional nonlinear evolution equations new exact grammian determinant solutions to two 3 1 dimensional nonlinear evolution equations are derived extended set of sufficient conditions consisting of linear partial differential equations with variable coefficients is presented moreover a systematic analysis of linear partial differential equations is used for solving the representative linear systems the bilinear backlund transformations are also constructed for the equations also as an application of the bilinear backlund transforms a new class of wave and rational solutions to the equations are explicitly computed

spacing distributions for rhombus billiards we show that the spacing distributions of rational rhombus billiards fall in a family of universality classes distinctly different from the wigner dyson family of random matrix theory and the poisson distribution some of the distributions find explanation in a recent work of bogomolny gerland and schmit for the irrational billiards despite ergodicity we get the same distribution for the examples considered once again distinct from the wigner dyson distributions all the results are obtained numerically by a method that allows us to reach very high energies

abundant lump type solutions of the jimbo miwa equation in 3 1 dimensions abstract based on the hirota bilinear form of the 3 1 dimensional jimbo miwa equation ten classes of its lump type solutions are generated via maple symbolic computations whose analyticity can be easily achieved by taking special choices of the involved parameters those solutions supplement the existing lump type solutions presented previously in the literature

generalized wigner surmise for 2 2 random matrices we present new analytical results concerning the spectral distributions for 2 2 random real symmetric matrices which generalize the wigner surmise

quantum chaos random matrix theory statistical mechanics in two dimensions and the second law a case study we present a theory where the statistical mechanics for dilute ideal gases can be derived from the random matrix approach we show the connection of this approach with the srednicki approach which connects berry conjecture with statistical mechanics we further establish a link between berry conjecture and random matrix theory in the course of arguing for these connections we also observe sum rules associated with the outstanding counting problem in the theory of braid groups we believe that these arguments developed for a special example connecting the properties of eigenfunctions and random matrices to the second law of thermodynamics will eventually prove to be more general

models of intermediate spectral statistics based on numerical results it is conjectured that the spectral statistics of certain pseudointegrable billiards have a special form similar to that of the anderson model at the transition point a simple theoretical model where such statistics can be obtained analytically is briefly discussed a few models with similar behavior are considered in particular we analytically found the eigenvalue statistics of a poisson distributed matrix perturbed by a rank one matrix which is a good model for spectral statistics of a singular billiard

finite spectrum assignment problem for systems with multiple commensurate delays in state variables abstract this paper is concerned with the finite spectrum assignment problem for systems with multiple commensurate delays in state variables the necessary and sufficient condition for finite spectrum assignability is clarified a systematic design procedure which can straightforwardly yield a control law is presented

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

negaton and positon solutions of the kdv and mkdv hierarchy we give a systematic classification and a detailed discussion of the structure motion and scattering of the recently discovered negaton and positon solutions of the korteweg de vries hierarchy there are two distinct types of negaton solutions which we label and where n 1 is the order of the wronskian used in the derivation for negatons the number of singularities and zeros is finite and they show very interesting time dependence the general motion is in the positive x direction except for certain negatons which exhibit one oscillation around the origin in contrast there is just one type of positon solution which we label for positons one gets a finite number of singularities for n odd but an infinite number for even values of n the general motion of positons is in the negative x direction with periodic oscillations negatons and positons retain their identities in a scattering process and their phase shifts are discussed we obtain a simple explanation of all phase shifts by generalizing the notions of mass and centre of mass to singular solutions finally it is shown that negaton and positon solutions of the kdv hierarchy can be used to obtain corresponding new solutions of the modified kdv hierarchy

construction of exact solutions to the jimbo miwa equation through backlund transformation and symbolic computation based on a simple transformation and with the aid of symbolic computation a backlund transformation relating the jimbo miwa equation and a system of linear partial differential equations is obtained which enables us to construct exact solutions of the jimbo miwa equation through the wronskian determinants of independent solutions of the linear system particularly explicit wronskian form n soliton solutions for the jimbo miwa equation are presented moreover the introduced transformation also helps to construct bi soliton like solutions of the jimbo miwa equation due to the arbitrary functions they contain the bi soliton like solutions can represent various waves such as classical cross line bi solitons curved bi solitons and bi soliton like breathers

random matrix ensembles for pt symmetric systems recently much effort has been made towards the introduction of non hermitian random matrix models respecting pt symmetry here we show that there is a one to one correspondence between complex pt symmetric matrices and split complex and split quaternionic versions of hermitian matrices we introduce two new random matrix ensembles of a gaussian split complex hermitian and b gaussian split quaternionic hermitian matrices of arbitrary sizes we conjecture that these ensembles represent universality classes for pt symmetric matrices for the case of 2 2 matrices we derive analytic expressions for the joint probability distributions of the eigenvalues the one level densities and the level spacings in the case of real eigenvalues

grammian type determinant solutions to generalized kp and bkp equations abstract grammian type determinant identities of the bilinear kp hierarchy and a useful property on the derivatives of grammians are proposed based on these grammian type determinant identities three grammian structures are furnished to a large class of generalized nonlinear kp and bkp type equations all the generating functions for matrix entries satisfy a system of combined linear partial differential equations in the first case of the class of generalized kp and bkp equations general rational solutions are given making use of its grammian formulation

multisoliton multipositon multinegaton and multiperiodic solutions of a coupled volterra lattice system and their continuous limits this paper aims to find new explicit solutions including multisoliton multipositon multinegaton and multiperiodic for a coupled volterra lattice system this coupled lattice system is an integrable discrete version of the coupled korteweg devries kdv equation which has many physical applications the dynamical properties of these new solutions are discussed in detail we also prove that the theory of the coupled volterra lattice system including the lax pair the darboux transformation and explicit solutions yield the corresponding theory of the coupled kdv equation in the continuous limit

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

direct search for exact solutions to the nonlinear schrodinger equation a five dimensional symmetry algebra consisting of lie point symmetries is firstly computed for the nonlinear schrodinger equation which together with a reflection invariance generates two five parameter solution groups three ansatze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear schrodinger equation various examples of exact solutions with constant trigonometric function type exponential function type and rational function amplitude are given upon careful analysis a bifurcation phenomenon in the nonlinear schrodinger equation is clearly exhibited during the solution process

wronskian and grammian solutions to a 3 1 dimensional generalized kp equation abstract wronskian and grammian formulations are established for a 3 1 dimensional generalized kp equation based on the plucker relation and the jacobi identity for determinants generating functions for matrix entries satisfy a linear system of partial differential equations involving a free parameter examples of wronskian and grammian solutions are computed and a few particular solutions are plotted

the periodic wave solutions for a 2 1 dimensional akns equation in this paper a 2 1 dimensional akns equation is considered bilinear backlund transformation is presented to explicitly construct periodic wave solutions based on a multidimensional riemann theta function then the one and two periodic wave solutions are presented and it is also shown that the soliton solutions can be reduced from the periodic wave solutions

on invariant 2 2 β ensembles of random matrices abstract we introduce and solve exactly a family of invariant 2 2 random matrices depending on one parameter and we show that rotational invariance and real dyson index are not incompatible properties the probability density for the entries contains a weight function and a multiple trace trace interaction term which corresponds to the representation of the vandermonde squared coupling on the basis of power sums as a result the effective dyson index eff of the ensemble can take any real value in an interval two weight functions gaussian and non gaussian are explored in detail and the connections with ensembles of dumitriu edelman and the so called poisson wigner crossover for the level spacing are respectively highlighted a curious spectral twinning between ensembles of different symmetry classes is unveiled as a consequence the identification between symmetry group orthogonal unitary or symplectic and the exponent of the vandermonde 1 2 4 is shown to be potentially deceptive the proposed technical tool more generically allows for designing actual matrix models which i are rotationally invariant ii have a real dyson index eff iii have a pre assigned confining potential or alternatively level spacing profile the analytical results have been checked through numerical simulations with an excellent agreement eventually we discuss possible generalizations and further directions of research

rational solutions to a kdv like equation two classes of rational solutions to a kdv like nonlinear differential equation are constructed the basic object is a generalized bilinear differential equation based on a prime number p 3 a conjecture is made that the two presented classes of rational solutions contain all rations solutions to the considered kdv like equation which are generated from polynomial solutions to the corresponding generalized bilinear equation

modified method of simplest equation and its applications to the bogoyavlenskii equation in this paper we construct the exact traveling wave solutions of the bogoyavlenskii equation using modified method of simplest equation the simplest equation herewith is a special riccati equation in the meantime the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields

non universal spectral rigidity of quantum pseudo integrable billiards we obtain the dyson mehta statistic for pseudo integrable billiards and show that it is non universal with a universal trend also that this trend is similar to the one for integrable billiards we present a formula based on exact semiclassical calculations and the proliferation law of periodic orbits which gives rigidity for the entire range of l to consolidate our theory we discuss several examples finding complete agreement with the numerical results and also the underlying fundamental reasons for the non universality

soliton solutions for a negative order non isospectral akns equation abstract in this paper the bilinear form of a negative order non isospectral akns equation is given the soliton solutions are obtained through hirota s direct method and the wronskian technique respectively

soliton solution to bkp equation in wronskian form we find out a group of identities named wronskian identities of bilinear kp hierarchy here as well as two useful properties of d operator this makes it possible to easily search for some new wronskian solutions for pde which owns bilinear form and to simplify the process of the proof as an application of this new method we propose the first wronskian condition for the bkp i equation and for the bkp ii equation respectively in addition we also discuss several generalized bkp equations and obtain their wronskian conditions

gaussian ensemble of 2 2 pseudo hermitian random matrices we present a random matrix theory for systems invariant under the joint action of parity and time reversal and more generally for pseudo hermitian systems this brings out the appearance of the metric in a systematic way so that consistency with the postulates of quantum mechanics is maintained here we specialize only to 2 2 matrices and we construct a pseudo unitary group with explicit examples nearest neighbour level spacing distributions for various classes of ensembles are found to exhibit a degree of level repulsion different from those hitherto known this work is not only relevant to quantum chaos but also to two dimensional statistical mechanics and consistent non local relativistic theories

recursive predictor design for state and output feedback controllers for linear time delay systems this paper presents a recursive method to design state and output feedback controllers for mimo block feedforward linear systems with delays in the inputs outputs and interconnections between the blocks the resulting controller is of predictor type which means that it contains finite integrals over past state and input values the method is a generalization of the well known model reduction approach for systems with input delay a recursive procedure replaces delay terms with non delay ones step by step from the top of the cascade structure down controller gains are computed for the proxy system without delays while the construction guarantees the same closed loop poles for the delay system and the proxy one the observer is designed by applying the duality argument and the separation principle is also shown to apply

wronskian determinant solutions of the 3 1 dimensional jimbo miwa equation abstract a set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the wronskian determinant solves the 3 1 dimensional jimbo miwa equation in the bilinear form upon solving the linear conditions the resulting wronskian formulations bring solution formulas which can yield rational solutions solitons negatons positons and interaction solutions

hirota bilinear equations with linear subspaces of hyperbolic and trigonometric function solutions linear superposition principles of hyperbolic and trigonometric function solutions are analyzed for hirota bilinear equations with an aim to construct a specific sub class of n soliton solutions formulated by linear combinations of hyperbolic and trigonometric functions an algorithm using weights is discussed and a few illustrative application examples are presented

resonant soliton and complexiton solutions for 3 1 dimensional boiti leon manna pempinelli equation abstract in this letter the linear superposition principle is used to discuss the 3 1 dimensional boiti leon manna pempinelli equation with bilinear derivatives as a result we obtain new resonant soliton and complexiton solutions by discussing two different cases involved the parameters these solutions are a class of n wave solutions of linear combinations of exponential traveling waves

mixed rational soliton solutions of two differential difference equations in casorati determinant form by reconsidering soliton solutions of the toda lattice and differential difference kdv equation in the casorati determinant form with new entries we obtain rational and mixed rational soliton solutions in the casorati determinant form all these solutions are verified by direct substitutions into bilinear equations the method used is general and can apply to other discrete systems

generalized wronskian solutions for the 3 1 dimensional jimbo miwa equation abstract more general wronskian conditions are constructed for the 3 1 dimensional jimbo miwa equation as a result the generalized wronskian solutions including rational solutions positons negatons solitons and interaction solutions are obtained moreover some new solutions together with some known solutions given by other solving methods are derived from these wronskian solutions

feedback stabilization of linear autonomous time lag systems a stabilization theory which employs well established finite dimensional control system tools is developed for the stabilization of linear autonomous time lag systems the main ideas include 1 a set whose elements are matrices each of which is a left zero of the system characteristic quasi polynomial matrix and 2 a linear transformation which reduces the delay system to a delay free system whose state matrix is a direct sum of n elements of the matrix set where n is some positive integer from the definition of this matrix set it is shown that each of its elements inherits its spectrum from that of the delay system so that by design the system unstable poles may be embedded in the spectrum of the delay free system under the assumption of spectral stabilizability it is then shown how to obtain a stabilizing feedback control law on the basis of the delay free system numerical examples are presented to confirm the theory

rectangular billiard in the presence of a flux line we analyze the quantum spectrum of a rectangular billiard coupled to flux line the nearest neighbor spacing distribution shows a transition from poisson to almost wigner distribution as the flux parameter is varied even though classically the system remains pseudointegrable and nonchaotic in fact the level statistics do not display any generic behavior the calculations presented here are simple and avoid the complexity involved in modeling the singular quantum billiard problem

solitonic solutions for a variable coefficient variant boussinesq system in the long gravity waves variable coefficient variant boussinesq vcvb system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth in this paper with symbolic computation a lax pair associated with the vcvb system under some constraints for variable coefficients is derived and based on the lax pair two sorts of basic darboux transformations are presented by applying the darboux transformations some solitonic solutions are obtained with the relevant constraints given in the text in addition the vcvb system is transformed to a variable coefficient broer kaup system solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics

soliton solutions backlund transformation and wronskian solutions for the extended 2 1 dimensional konopelchenko dubrovsky equations in fluid mechanics abstract this paper is to investigate the extended 2 1 dimensional konopelchenko dubrovsky equations which can be applied to describing some phenomena in the stratified shear flow the internal and shallow water waves and plasmas bilinear form equations are transformed from the original equations and n soliton solutions are derived via symbolic computation bilinear form backlund transformation and single soliton solution are obtained and illustrated wronskian solutions are constructed from the backlund transformation and single soliton solution

rational solutions to two new kp like equations in this paper we will present a new 3 1 dimensional kp like nonlinear partial differential equation and construct the rational solutions to this new 3 1 dimensional kp like equation the proposed method in this work is based on a generalized bilinear differential equation which implies that we can build the rational solutions to the presented kp like equation from polynomial solutions to the aforementioned generalized bilinear equation moreover for y z this new model reduces to the new 2 1 dimensional kp like nonlinear partial differential equation which will be studied as well herein

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

complexiton solutions to integrable equations abstract complexiton solutions or complexitons for short are exact solutions newly introduced to integrable equations are newly introduced starting with the solution classification for a linear differential equation the korteweg de vries equation and the toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations the crucial step in the solution process is to apply the wronskian and casoratian techniques for hirota s bilinear equations correspondence between complexitons of the korteweg de vries equation and complexitons of the toda lattice equation is provided

the structure of the rational solutions to the boussinesq equation rational solutions of the one dimensional boussinesq equation both with zero and nonzero boundary conditions at the infinity are obtained by reducing the known solutions of the kadomtsev petviashvili equation the structure of the found solutions generalizes a family of rational solutions to the korteweg de vries equation to the case of two wave processes

negatons positons rational like solutions and conservation laws of the korteweg de vries equation with loss and non uniformity terms solitons negatons positons rational like solutions and mixed solutions of a non isospectral equation the korteweg de vries equation with loss and non uniformity terms are obtained through the wronskian technique the non isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using mathematica we also derive an infinite number of conservation laws of the equation

n fold darboux transformation and solitonic interactions of a variable coefficient generalized boussinesq system in shallow water abstract under consideration in this paper is a variable coefficient generalized boussinesq system for the long weakly nonlinear and weakly dispersive surface waves in shallow water with the aid of symbolic computation n fold darboux transformation n dt is constructed for that system analytic solutions of the system are obtained via the n dt elastic interactions of three bell shaped and periodic bell shaped solitons are obtained fusion interactions and periodic fusion fission interactions of the solitary waves are graphically analyzed which are inelastic

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

space of state vectors in symmetric quantum mechanics the space of states of symmetric quantum mechanics is examined the requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric self consistent expressions for the probability amplitude and the average value of operator are suggested further specification of the space of state vectors yields a superselection rule redefining the notion of the superposition principle an expression for the probability current density satisfying the equation of continuity and vanishing for the bound state is proposed

traveling wave solutions and conservation laws for nonlinear evolution equation in this work the riccati bernoulli sub ordinary differential equation and modified tanh coth methods are used to reach soliton solutions of the nonlinear evolution equation we acquire new types of traveling wave solutions for the governing equation we show that the equation is nonlinear self adjoint by obtaining suitable substitution therefore we construct conservation laws for the equation using new conservation theorem the obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium the constraint condition for the existence of solitons is stated some three dimensional figures for some of the acquired results are illustrated

linear systems with delayed controls a reduction linear systems with delayed control action are transformed into systems without delays under an absolute continuity condition the new system is an ordinary differential control equation in the general case the new system is a measure differential control system it is shown how the controllability stabilization and various optimization problems can be analyzed via the reduced systems

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

positons for the toda lattice and related spectral problems the concept of positons is introduced for the toda lattice equation it is shown that these multiparametric oscillating and slowly decaying solutions when inserted as potentials in the finite difference schrodinger equation of the corresponding lax pair lead to a trivial s matrix the resulting eigenvalues are embedded in the continuous spectrum of this infinite jacobi matrix the singularities connected with the one positon solution are discussed and compared with those of the positons of continuous integrable models the special features of the soliton positon interaction are analysed

feedback stabilization of linear systems with distributed delays in state and control variables solving a generalized characteristic matrix equation gcme of a linear retarded system is the crux for stabilizing the system using a reduction technique this note gives the solution of gcme under the general condition that the eigenvalues occur in the spectrum of the system with algebraic and geometric multiplicities being greater than or equal to one the main idea is to transform the gcme into a group of linear algebraic equations a sufficient condition for the existence of the solution to the linear algebraic equations is established a key lemma for stabilizing the systems under the above general condition is also proved

asymptotic solutions for singularly perturbed boussinesq equations abstract we consider a family of singularly perturbed boussinesq equations we obtain a rational weak solution to the classical boussinesq equation and demonstrate that this solution can be used to construct perturbation solutions for singularly perturbed high order boussinesq equations these solutions take the form of an algebraic function which behaves similarly to a peakon and which decays as time becomes large we show that approximate solutions obtained via perturbation for the singularly perturbed models are asymptotic to the true solutions as the residual errors rapidly decay away from the origin

output feedback stabilization of delay systems via generalization of the transformation method the left characteristic matrix equation l c m e of a delay system has been employed to develop a state feedback stabilization theory for delay systems analogously the right characteristic matrix equation r c m e has led to the development of an observer theory for this class of systems the characteristic matrix equation approach however only permits the relocation of integral multiples of n poles where n represents the system order this restriction is removed in this work through a generalization of the l c m e and r c m e in a manner which permits the stabilization of an arbitrary but finite number of unstable modes the resulting state feedback controller and state estimator permit the output feedback stabilization of the most general linear autonomous delay system under the minimal assumptions of spectral stabilizability and detectability

pseudo reality and pseudo adjointness of hamiltonians we define pseudo reality and pseudo adjointness of a hamiltonian h as h 1 h and h 1 h respectively we prove that the former yields the necessary condition for a spectrum to be real whereas the latter helps in fixing a definition for the inner product of the eigenstates here we separate out the adjointness of an operator from its hermitian adjointness it turns out that a hamiltonian possessing a real spectrum is first pseudo real further it could be hermitian pt symmetric or pseudo hermitian

pfaffian solutions to a 3 1 dimensional generalized b type kadomtsev petviashvili equation and its modified counterpart abstract a class of exact pfaffian solutions to a 3 1 dimensional generalized b type kadomtsev petviashvili equation is obtained a set of sufficient conditions consisting of systems of linear partial differential equations involving free parameters is generated to guarantee that the pfaffian solves the equation a backlund transformation of the equation is presented the equation is transformed into a set of bilinear equations and a few classes of traveling wave solutions rational solutions and pfaffian solutions to the extended bilinear equations are furnished examples of the pfaffian solutions are explicitly computed and a few solutions are plotted

weak pseudo hermiticity and antilinear commutant we inquire into some properties of diagonalizable pseudo hermitian operators showing that their definition can be relaxed and that the pseudo hermiticity property is strictly connected with the existence of an antilinear symmetry this result is then illustrated by considering the particular case of the complex morse potential

a symbolic computation approach to constructing rogue waves with a controllable center in the nonlinear systems abstract a symbolic computation approach to constructing higher order rogue waves with a controllable center of the nonlinear systems is presented making use of their hirota bilinear forms as some examples it turns out that some higher order rogue wave solutions of the kadomtsev petviashvili kp type equations in 3 1 and 2 1 dimensions are obtained some features of controllable center of rogue waves are graphically discussed

new analytical positon negaton and complexiton solutions of a coupled kdv mkdv system new positon negaton and complexiton solutions of a coupled kdv mkdv system are constructed directly from zero seed solution by means of darboux transformation these new exact solutions are all analytical with different spectral parameters and constants selection

linear superposition principle of hyperbolic and trigonometric function solutions to generalized bilinear equations linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations are analyzed necessary and sufficient conditions are presented to apply the linear superposition principles applications of an algorithm using weights are made together with a few concrete examples

a pseudo hermitian β hermite family of matrices abstract a family of pseudo hermitian random matrices is constructed whose eigenvalues are the same of the hermite ensemble of random matrix theory it is shown that matrices of this family are unstable in such a way that small perturbations cause the eigenvalues to move into the complex plane in conjugate pairs