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

generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

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

wave solutions of evolution equations and hamiltonian flows on nonlinear subvarieties of generalized jacobians the algebraic geometric approach is extended to study evolution equations associated with the energy dependent schr dinger operators having potentials with poles in the spectral parameter in connection with hamiltonian flows on nonlinear subvarieties of jacobi varieties the general approach is demonstrated by using new parametrizations for constructing quasi periodic solutions of the shallow water and dym type equations in terms of theta functions a qualitative description of real valued solutions is provided

periodic functional determinants examines and evaluates the functional determinants associated with a class of periodic operators which arise in the quantum theory of solitons the functional determinants are expressed in terms of a finite subset of the spectrum of the operator a subset which corresponds to the edges of the valency or conduction bands of the schrodinger equation

lump mixed lump stripe and rogue wave stripe solutions of a 3 1 dimensional nonlinear wave equation for a liquid with gas bubbles abstract liquids with gas bubbles are very common in science engineering physics nature and life under investigation in this paper is a 3 1 dimensional nonlinear wave equation for a liquid with gas bubbles with respect to the velocity of the liquid gas bubble mixture we obtain the lump rogue wave mixed lump stripe soliton and mixed rogue wave stripe soliton solutions via the symbolic computation based on the mixed lump stripe soliton solutions we construct the rogue wave and investigate the fusion and fission phenomena between a lump and the one stripe soliton we graphically study the mixed lump stripe soliton under the influence of the parameters and which represent the dispersion perturbed effect disturbed wave velocities along the y and z i e the two transverse directions respectively with the decreasing value of to 1 the graph from a lump and one stripe soliton shows a soliton with the increasing value of to 3 location of the lump moves along the negative x axis with the value of increasing to 0 5 location of the lump moves along the positive x axis with the increasing value of to 3 location and range of the lump soliton keep unchanged with respect to the velocity of the mixture we obtain the interaction between a rogue wave and a pair of stripe solitons according to mixed rogue wave stripe soliton solutions a lump is provided

on the uniqueness of solution for all wavenumbers in acoustic radiation the integral relations equivalent to helmholtz s equation with closed boundary conditions possess nonunique exterior solutions at eigenwavenumbers of the corresponding interior problems if the green s function for helmholtz s equation can be expanded into a series of orthogonal functions then an infinite set of integral equations at the boundary results these null field equations are shown to possess unique solutions for all wavenumbers furthermore these solutions can be used to represent the solutions in the exterior domain as algebraic series

duals and propagators a canonical formalism for nonlinear equations a new formalism for solving general nonlinear equations is proposed given a nonlinear operator equation a dual operator is constructed canonically similar to the adjoint operator in linear theory by using this dual operator advanced and retarded propagators analogs of the green s functions in linear theory are generated these propagators satisfy the customary reciprocity and semigroup properties and yield the formal solution of the original nonlinear problem it is further shown that these propagators can be obtained either by solving a linear equation that still contains an implicit dependence on the problem s solution or by solving a closed form nonlinear integral equation from which the problem s solution itself is completely eliminated the formalism is canonical in the sense that its applicability is not affected by the particularities of the nonlinear operator boundary conditions and underlying phase space all aspects of applying this formalism to nonlinear problems are illustrated analy

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

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

the constraint on potential and decomposition for 2 1 dimensional integrable systems besides a sufficient condition a necessary condition is given to determine some kind of consistent constraint on the potential of a 2 1 dimensional integrable system which cannot be obtained from the sufficient condition under this kind of constraint on the potential a 2 1 dimensional integrable system can be decomposed into two commuting 1 1 dimensional integrable systems and can be further decomposed into three commuting finite dimensional integrable hamiltonian systems the bkp hierarchy is taken as an illustrative example

classical liouville completely integrable systems associated with the solutions of boussinesq burgers hierarchy two new finite dimensional completely integrable systems in the liouville sense are obtained the solutions of boussinesq burgers hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems

zonal schrodinger operators on the n sphere inverse spectral problem and rigidity we study the direct and inverse spectral problems for a class of schrodinger operatorsh v onsn withzonal axisymmetric potentials spectrum ofh is known to consist of clusters of eigenvalues km k k n 1 km m k the main result of the work is to derive asymptotic expansion of spectral shifts km in powers ofk 1 and to link coefficients of the expansion to certain transforms ofv as a corollary we solve the inverse problem get explicit formulae for the weinsteinband invariants of cluster distribution measures and establishlocal spectral rigidity for zonal potential the latter provides a partial answer to a long standing spectral rigidity hypothesis of v guillemin

mass corrections to periodic solitons the quantum mass corrections to static periodic solitons are considered and expressed in terms of a functional quantity the discriminant of the hill s operator appearing several general expansions for this and hence the mass corrections are given in particular the connection with integrable systems and inverse scattering is clearly seen this enables one to obtain simple expressions for these corrections

the positive and negative camassa holm γ hierarchies zero curvature representations bi hamiltonian structures and algebro geometric solutions in this paper we extend the camassa holm ch equation to two different hierarchies of nonlinear evolution equations by resorting to two lenard recursion sequences one is of positive order ch hierarchy including a harry dym type equation while the other is of negative order ch hierarchy of integrodifferential equations with nonzero integration constants including the ch equation we provide the zero curvature representations for both hierarchies by solving a key matrix equation moreover the lenard sequences are used to construct a lax matrix that satisfies a stationary zero curvature equation which enables us to establish bi hamiltonian structures for both hierarchies by applying trace identity in addition the lenard sequence and lax pair are used to systematically derive the its matveev trace formula and the dubrovin type equations associated with the ch equation the hyperelliptic curve and abel jacobi coordinates are then introduced to linearize the associated flow from which the algebro geometric solutions to the ch equation are constructed by using standard jacobi inversion technique in this paper we extend the camassa holm ch equation to two different hierarchies of nonlinear evolution equations by resorting to two lenard recursion sequences one is of positive order ch hierarchy including a harry dym type equation while the other is of negative order ch hierarchy of integrodifferential equations with nonzero integration constants including the ch equation we provide the zero curvature representations for both hierarchies by solving a key matrix equation moreover the lenard sequences are used to construct a lax matrix that satisfies a stationary zero curvature equation which enables us to establish bi hamiltonian structures for both hierarchies by applying trace identity in addition the lenard sequence and lax pair are used to systematically derive the its matveev trace formula and the dubrovin type equations associated with the ch equation the hyperelliptic curve and abel jacobi coordinates are then introduced to linearize the associated flow from which the a

conservation laws of evolution equations generic non existence evolution type partial differential equations in one space variable are formulated in terms of exterior differential systems the space of conservation laws is discussed in this geometric context and a familiar classical condition for conservation laws is derived it is shown that the generic even order evolution equation with one space variable possesses no conservation laws of order greater than the order of the equation

quantum mass correction for the twisted kink we present an analytic result for the 1 loop quantum mass correction in semiclassical quantization for the twisted 4 kink on s1 without explicit knowledge of the fluctuation spectrum for this purpose we use the contour integral representation of the spectral zeta function by solving the bethe ansatz equations for the n 2 lam equation we obtain an analytic expression for the corresponding spectral discriminant we discuss the renormalization issues of this model an energetically preferred size for the compact space is finally obtained

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

from stackel systems to integrable hierarchies of pde s benenti class of separation relations we propose a general scheme of constructing of soliton hierarchies from finite dimensional stackel systems and related separation relations in particular we concentrate on the simplest class of separation relations called benenti class i e certain stackel systems with quadratic in momenta integrals of motion

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

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

classical and quantum algebras of non local charges in σ models we investigate the algebras of the non local charges and their generating functionals the monodromy matrices in classical and quantum non linear models in the case of the classical chiral models it turns out that there exists no definition of the poisson bracket of two monodromy matrices satisfying antisymmetry and the jacobi identity thus the classical non local charges do not generate a lie algebra in the case of the quantum o n non linear model we explicitly determine the conserved quantum monodromy operator from a factorization principle together withp t and o n invariance we give closed expressions for its matrix elements between asymptotic states in terms of the known two particles matrix the quantumr matrix of the model is found the quantum non local charges obey a quadratic lie algebra governed by a yang baxter equation

the expansions of spectral function and the corresponding finite dimensional integrable systems abstract in this paper taking the lax pairs of kdv equation as an illustrative example with the help of a transformation of the spectral problem another form of spectral function is used to give finite dimensional integrable systems these integrable systems are generated by two different kinds of polynomial expansion of spectral function on spectral parameter further the root variables of the spectral function are introduced to study the canonical equations of hamilton finally based on the hamilton jacobi theory the action angle variables are built and the soliton solutions of kdv equation are obtained by inversion

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

semiclassical limit of the focusing nls whitham equations and the riemann hilbert problem approach abstract the main goal of this paper is to put together a the whitham theory applicable to slowly modulated n phase nonlinear wave solutions to the focusing nonlinear schrodinger fnls equation and b the riemann hilbert problem approach to particular solutions of the fnls in the semiclassical small dispersion limit that develop slowly modulated n phase nonlinear wave in the process of evolution both approaches have their own merits and limitations understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fnls

hamiltonian formulation of energy conservative variational equations by wavelet expansion abstract hamiltonian formulation of various energy conservative evolution equations is given by means of wavelet expansion of solutions on the whole real axis r the kdv equation wave equations and schrodinger equations are treated in a unified similar manner a matrix representation of operators with respect to a nice wavelet base plays an important role in the formulation since the procedure is very concrete our results can be used to efficiently compute numerical solutions of partial differential equations described in the text in fact we may also use symplectic schemes to solve derived hamiltonian systems

sound radiation from a finite fluid filled submerged cylindrical shell with porous material sandwich a theoretical model is developed to predict the far field sound radiation from a finite fluid filled submerged cylindrical thin shell with porous material sandwich a combination of the wave number domain approach and the transfer matrix method is presented which is convenient to analyze the vibratory responses in terms of wave number expressions for the spectral radial velocity of the outer surface of the shell are also formulated a prediction model for sound radiation at far field is given by using the boundary integral equation extensive numerical results are also presented to illustrate the general characteristics of the far field sound pressure as a function of frequency an experimental verification is performed and a good correlation between the theoretical results and the experimental results shows that the theoretical study work in this paper is correct

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

invariant tori for the cubic szego equation we continue the study of the following hamiltonian equation on the hardy space of the circle i partial tu pi u 2u where denotes the szego projector this equation can be seen as a toy model for totally non dispersive evolution equations in a previous work we proved that this equation admits a lax pair and that it is completely integrable in this paper we construct the action angle variables which reduces the explicit resolution of the equation to a diagonalisation problem as a consequence we solve an inverse spectral problem for hankel operators moreover we establish the stability of the corresponding invariant tori furthermore from the explicit formulae we deduce the classification of orbitally stable and unstable traveling waves

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

finite gap integration of the derivative nonlinear schrodinger equation a riemann hilbert method abstract in this paper we retrieve finite gap solutions of the gerdjikov ivanov type derivative nonlinear schrodinger equation by using the algebro geometric method and the riemann hilbert method we show that the baker akhiezer function of the derivative nonlinear schrodinger equation can be described in terms of two solvable matrix riemann hilbert problems on with 2 and 3 symmetry conditions based on the technique developed in the study of long time asymptotics our main tools include matrix baker akhiezer function asymptotic analysis algebraic curve and riemann theta function matrix riemann hilbert problem and associated deformation procedures

negative order korteweg de vries equations in this paper based on the regular korteweg de vries kdv system we study negative order kdv nkdv equations particularly their hamiltonian structures lax pairs conservation laws and explicit multisoliton and multikink wave solutions thorough bilinear backlund transformations the nkdv equations studied in our paper are differential and actually derived from the first member in the negative order kdv hierarchy the nkdv equations are not only gauge equivalent to the camassa holm equation through reciprocal transformations but also closely related to the ermakov pinney systems and the kupershmidt deformation the bi hamiltonian structures and a darboux transformation of the nkdv equations are constructed with the aid of trace identity and their lax pairs respectively the single and double kink wave and bell soliton solutions are given in an explicit formula through the darboux transformation the one kink wave solution is expressed in the form of tanh while the one bell soliton is in the form of sech and both forms are very standard the collisions of two kink wave and two bell soliton solutions are analyzed in detail and this singular interaction differs from the regular kdv equation multidimensional binary bell polynomials are employed to find bilinear formulation and backlund transformations which produce n soliton solutions a direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the nkdv equations furthermore the relations between quasiperiodic wave solutions and soliton solutions are clearly described finally we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions

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

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

a simple lagrangian for integrable systems a simple lagrangian for integrable models that can describe the entire hierachy of equations associated with the system is proposed in addition the lax equation the recursion relation between the conserved quantities as well as the vanishing of the nijenhuis torsion tensor can also be derived from the system of equations following from this lagrangian

the kp equation with quasiperiodic initial data abstract the kadomtsev petviashvili kp equation is known to admit exact quasiperiodic solutions that can be written in terms of riemann theta functions with a finite number of phases in each solution in this paper we propose a method to solve the initial value problem for the kp equation for initial data taken from this class of quasiperiodic functions

minimal conditions for implications of gronwall bellman type abstract gronwall bellman type inequalities entail the following implication if a sufficiently integrable function satisfies a certain homogeneous linear integral inequality then it is nonpositive we present a minimal necessary and sufficient condition on the borel measure underlying the integrals for this implication to hold the condition is also a necessary prerequisite for any nontrivial bound on solutions to inhomogeneous linear integral inequalities of gronwall bellman type

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

quasi periodic solutions of 3 1 generalized bkp equation by using riemann theta functions this paper is focused on quasi periodic wave solutions of 3 1 generalized bkp equation because of some difficulties in calculations of n 3 periodic solutions hardly ever has there been a study on these solutions by using riemann theta function in this study we obtain one and two periodic wave solutions as well as three periodic wave solutions for 3 1 generalized bkp equation moreover we analyze the asymptotic behavior of the periodic wave solutions tend to the known soliton solutions under a small amplitude limit

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

numerical solutions of the korteweg di vries equation using the periodic scattering transform m representation abstract we develop a numerical procedure based upon the periodic scattering transform for determining the spatial and temporal evolution of solutions to the korteweg de vries kdv equation with periodic boundary conditions in this approach solutions to kdv are found in terms of the representation i e the wave amplitude is given by a trace formula consisting of a linear superposition of hyperelliptic functions j x t 1 j n which are governed by n nonlinearly coupled ordinary differential equations our approach consists of the numerical integration of these equations using a second order taylor series expansion with variable integration step size it is well known that the j are generalizations of elliptic functions to the hyperelliptic class nevertheless we find that the j x t bear little or no resemblance to elliptic functions except in the small amplitude linear limit or for the case of a single soliton we study in particular the most difficult numerical case which occurs for the n soliton wave train solution of the periodic kdv equation

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

on the instability tongues of the hill equation coupled with a conservative nonlinear oscillator abstract we study the asymptotics for the lengths l n q of the instability tongues of hill equations that arise as iso energetic linearization of two coupled oscillators around a single mode periodic orbit we show that for small energies i e q 0 the instability tongues have the same behavior that occurs in the case of the mathieu equation l n q o q n the result follows from a theorem which fully characterizes the class of hill equations with the same asymptotic behavior in addition in some significant cases we characterize the shape of the instability tongues for small energies motivation of the paper stems from recent mathematical works on the theory of suspension bridges

on a class of eigenfunction identities for hill operators two very general identities are proved involving the eigenfunctions of two hill operators and the spectrum of a third various specializations lead to a number of identities involving squares and fourth powers of the periodic eigenfunctions

an integrable couplings of dirac soliton hierarchy with self consistent sources abstract a kind of the integrable couplings of soliton equations hierarchy with self consistent sources associated with sl 4 is presented as an application example we construct a new integrable couplings of the dirac soliton hierarchy with self consistent sources by using of loop algebra sl 4

algebro geometric finite gap solutions to the korteweg de vries equation as primitive solutions abstract in this paper we show that all algebro geometric finite gap solutions to the korteweg de vries equation can be realized as a limit of n soliton solutions as n diverges to infinity see remark 1 for the precise meaning of this statement this is done using the primitive solution framework initiated by dyachenko et al 2016 and zakharov et al 2016 25 26 one implication of this result is that the n soliton solutions can approximate any bounded periodic solution to the korteweg de vries equation arbitrarily well in the limit as n diverges to infinity we also study primitive solutions numerically that have the same spectral properties as the algebro geometric finite gap solutions but are not algebro geometric solutions

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

finite gap solutions of 2 1 dimensional integrable nonlinear evolution equations generated by the neumann systems each soliton equation in the korteweg de vries kdv hierarchy the 2 1 dimensional breaking soliton equation and the 2 1 dimensional caudrey dodd gibbon kotera sawada cdgks equation are reduced to two or three neumann systems on the tangent bundle tsn 1 of the unit sphere sn 1 the lax moser matrix for the neumann systems of degree n 1 is deduced in view of the mckean trubowitz identity and a bilinear generating function whose favorite characteristic accounts for the problem of the genus of riemann surface matching to the number of elliptic variables from the lax moser matrix the constrained hamiltonians in the sense of dirac poisson bracket for all the neumann systems are written down in a uniform recursively determined by integrals of motion the involution of integrals of motion and constrained hamiltonians is completed on tsn 1 by using a lax equation and their functional independence is displayed over a dense open subset of tsn 1 by a direct calculation which contribute to the liouville integ

resonant vibrations of a submerged beam abstract forced vibration of a simply supported submerged beam of circular cross section is investigated by the use of two mathematical methods in the first approach the problem formulation is reduced to a singular integro differential equation for the transverse deflection in the second approach the method of matched asymptotic expansions is employed the integro differential equation is solved numerically to yield an exact solution for the frequency response subsequent use of a representation integral yields the radiated far field acoustic pressure the exact results for the beam deflection are compared with approximate results that are available in the literature next a matched asymptotic expansion is worked out by constructing inner and outer expansions for frequencies near and not near resonance frequencies respectively the two expansions are matched in an appropriate manner to yield a uniformly valid solution the leading term of the matched asymptotic solution is compared with exact numerical results

integrability condition and finite periodic toda lattice a further generalization to construct integrable dynamical systems with hierarchy has been developed based upon a generalization of the zero nijenhuis tensor condition in a symplectic manifold with double symplectic structures as an example the periodic toda lattice hamiltonian has been shown to satisfy these conditions the case of the three dimensional coulomb potential problem has also been analyzed in a similar fashion

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

new integrable hierarchy its parametric solutions cuspons one peak solitons and m w shape peak solitons in this paper we propose a new completely integrable hierarchy particularly in the hierarchy we draw two new soliton equations 1 mt 12 1 m2 xxx 12 1 m2 x 2 mt mx u2 ux2 2m2ux 0 m u uxx the first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy both equations can be derived from the two dimensional euler equation by using the approximation procedure all equations in the hierarchy are proven to have bi hamiltonian operators and lax pairs through solving a crucial matrix equation moreover we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints one is for all negative members of the hierarchy on a symplectic submanifold and the other is for all positive members in the standard symplectic space the most interesting things are both equations possess new type of peaked solitons continuous and piecewise smooth w m shape peak soliton solutions in addition we find new cusp solit

singularities of the complex korteweg de vries flows on etudie le probleme isospectral de l equation de hill complexe etant donne 2g 1 valeurs propres simples dans le plan complexe trouver la classe spectrale correspondante q on considere la caracterisation de la classe isospectrale et ses sections singulieres on construit q a l aide des flots non lineaires de korteweg de vries dans c 1 les fonctions regulieres de periode 1

two component description of dynamical systems that can be approximated by solitons the case of the ion acoustic wave equations of plasma physics a new approach to the perturbative analysis of dynamical systems which can be described approximately by soliton solutions of integrable non linear wave equations is employed in the case of small amplitude solutions of the ion acoustic wave equations of plasma physics instead of pursuing the traditional derivation of a perturbed kdv equation the ion velocity is written as a sum of two components elastic and inelastic in the single soliton case the elastic component is the full solution in the multiple soliton case it is complemented by the inelastic component the original system is transformed into two evolution equations an asymptotically integrable normal form for ordinary kdv solitons and an equation for the inelastic component the zero order term of the elastic component is a single soliton or multiple soliton solution of the normal form the inelastic component asymptotes into a linear combination of single soliton solutions of the normal form with amplitudes determined by soliton interactions plus a second order decaying dispersive wave satisfaction of a conservation law by the inelastic component and of mass conservation by the disturbance to the ion density is determined solely by the initial data and or boundary conditions imposed on the inelastic component the electrostatic potential is a first order quantity it is affected by the inelastic component only in second order the charge density displays a triple layer structure the analysis is carried out through the third order

small dispersion limit of the korteweg de vries equation with periodic initial conditions and analytical description of the zabusky kruskal experiment abstract we study the small dispersion limit of the korteweg de vries kdv equation with periodic boundary conditions and we apply the results to the zabusky kruskal experiment in particular we employ a wkb approximation for the solution of the scattering problem for the kdv equation i e the time independent schrodinger equation to obtain an asymptotic expression for the trace of the monodromy matrix and thereby of the spectrum of the problem we then perform a detailed analysis of the structure of said spectrum i e band widths gap widths and relative band widths as a function of the dispersion smallness parameter we then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of finally by performing an appropriate rescaling we compare our results to those in the famous zabusky and kruskal s paper showing very good agreement with the numerical results

on the integrability of stationary and restricted flows of the kdv hierarchy a bi hamiltonian formulation for stationary flows of the kdv hierarchy is derived in an extended phase space a map between stationary flows and restricted flows is constructed in one case it connects an integrable henon heiles system and the garnier system moreover a new integrability scheme for hamiltonian systems is proposed that holds in the standard phase space

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

integral inequalities in measure spaces the inequality 2 is usually known as a gronwall bellman type integral inequality special cases of 1 and 2 both continuous and discrete have been studied by many authors for example see 1 wx 4 6 7 9 and they have important applications first we give an explicit upper bound for the solutions of 2 then by using this result we obtain estimates for the solutions of 1 at the same time we deduce a method to get various explicit upper bounds for thesolutions of 2 by applying this method we can obtain much better estimates than the first one finally we consider the integral equationcorresponding to 1 we prove the existence and the uniqueness of the solution and we show that this solution is the best possible estimate forthe solutions of 1 we give some examples to illustrate the results