on a new class of completely integrable nonlinear wave equations i infinitely many conservation laws we point out a class of nonlinear wave equations which admit infinitely many conserved quantities these equations are characterized by a pair of exact one forms the implication that they are closed gives rise to equations the characteristics and riemann invariants of which are readily obtained the construction of the conservation laws requires the solution of a linear second order equation which can be reduced to canonical form using the riemann invariants the hodograph transformation results in a similar linear equation we discuss also the symplectic structure and backlund transformations associated with these equations

a class of nonlinear wave equations containing the continuous toda case we consider a nonlinear field equation which can be derived from a binomial lattice as a continuous limit this equation containing a perturbative friction like term and a free parameter reproduces the toda case in the absence of the friction like term and other equations of physical interest by choosing particular values of we apply the symmetry and the approximate symmetry approach and the prolongation technique our main aim is to check the limits of validity of different analytical methods in the study of nonlinear field equations we show that the equation under investigation with the friction like term is characterized by a finite dimensional lie algebra admitting a realization in terms of boson annhilation and creation operators in the absence of the friction like term the equation is linearized and connected with bessel type equations examples of exact solutions are displayed and the algebraic structure of the equation is discussed

a family of second lie algebra structures for symmetries of a dispersionless boussinesq system for the 3 component dispersionless boussinesq type system we construct two compatible nontrivial finite deformations for the lie algebra structure in the symmetry algebra

oscillator representations of the 2d conformal algebra and superalgebra for the virasoro algebra and for the neveu schwarz ramond nsr superalgebra oscillator representations are found by localising the miura and the super miura maps from the theory of the korteweg de vries kdv and the super kdv equations respectively

elliptic euler poisson darboux equation critical points and integrable systems the structure and properties of families of critical points for classes of functions obeying the elliptic euler poisson darboux equation e 1 2 1 2 are studied general variational and differential equations governing the dependence of critical points in variational deformation parameters are found explicit examples of the corresponding integrable quasi linear differential systems and hierarchies are presented there are the extended dispersionless toda nonlinear schrodinger hierarchies the inverse hierarchy and equations associated with the real analytic eisenstein series among them the specific bi hamiltonian structure of these equations is also discussed

the classical shallow water equations symplectic geometry the classical shallow water equations express the change with time of the height h and the velocity of a 1 dimensional fluid t x h x 0 h x h x 0 they possess an infinite number of integrals of motion due to benney 1973 and can be written in hamiltonian form relative to a symplectic structure introduced by manin 1978 the present paper deals with their complete integrability up to the advent of shocks this is proved in the small under an extra assumption satisfied by most height velocity pairs that hh only at isolated points

in any dimension a clamped plate with a uniform weight may change sign abstract positivity preserving properties have been conjectured for the bilaplace dirichlet problem in many versions in this note we show that in any dimension there exist bounded smooth domains such that even the solution of 2 u 1 in with the homogeneous dirichlet boundary conditions u u 0 on is sign changing in two dimensions this corresponds to the kirchhoff love model of a clamped plate with a uniform weight

an algebro geometric solution for a hamiltonian system with application to dispersive long wave equation by using an iterative algebraic method we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation it is shown that the hierarchy is integrable in liouville sense and possesses bi hamiltonian structure two commutators with zero curvature and lax representations for the hierarchy are constructed respectively by using two different systematic methods under a bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional hamiltonian system by introducing the abel jacobi coordinates an algebro geometric solution for the dispersive long wave equation is derived by resorting to the riemann theta function

extended symmetry analysis of an isothermal no slip drift flux model abstract we perform extended group analysis for a system of differential equations modeling an isothermal no slip drift flux the maximal lie invariance algebra of this system is proved to be infinite dimensional we also find the complete point symmetry group of this system including discrete symmetries using the megaideal based version of the algebraic method optimal lists of one and two dimensional subalgebras of the maximal lie invariance algebra in question are constructed and employed for obtaining reductions of the system under study since this system contains a subsystem of two equations that involves only two of three dependent variables we also perform group analysis of this subsystem the latter can be linearized by a composition of a fiber preserving point transformation with a two dimensional hodograph transformation to the klein gordon equation we also employ both the linearization and the generalized hodograph method for constructing the general solution of the entire system under study we find inter alia genuinely generalized symmetries for this system and present the connection between them and the lie symmetries of the subsystem we mentioned earlier hydrodynamic conservation laws and their generalizations are also constructed

on trust region methods for unconstrained minimization without derivatives we consider some algorithms for unconstrained minimization without derivatives that form linear or quadratic models by interpolation to values of the objective function then a new vector of variables is calculated by minimizing the current model within a trust region techniques are described for adjusting the trust region radius and for choosing positions of the interpolation points that maintain not only nonsingularity of the interpolation equations but also the adequacy of the model particular attention is given to quadratic models with diagonal second derivative matrices because numerical experiments show that they are often more efficient than full quadratic models for general objective functions finally some recent research on the updating of full quadratic models is described briefly using fewer interpolation equations than before the resultant freedom is taken up by minimizing the frobenius norm of the change to the second derivative matrix of the model a preliminary version of this method provides some very promising numerical results

reciprocal transformations of hamiltonian operators of hydrodynamic type nonlocal hamiltonian formalism for linearly degenerate systems reciprocal transformations of hamiltonian operators of hydrodynamic type are investigated the transformed operators are generally nonlocal possessing a number of remarkable algebraic and differential geometric properties we apply our results to linearly degenerate semi hamiltonian systems in riemann invariants a typical example being rti m 1nrm ri rxi i 1 2 n since all such systems are linearizable by appropriate generalized reciprocal transformations our formulas provide an infinity of mutually compatible nonlocal hamiltonian structures explicitly parametrized by n arbitrary functions of one variable

a hierarchy of nonlinear evolution equations its bi hamiltonian structure and finite dimensional integrable systems an isospectral problem and the associated hierarchy of nonlinear evolution equations is presented as a reduction a new generalized nonlinear schrodinger equation is obtained it is shown that the hierarchy possesses bi hamiltonian structure and is integrable in liouville sense moreover the eigenvalue problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenvalues

local symplectic operators and structures related to them matrices with entries being differential operators that endow the phase space of an evolution system with a pre symplectic structure are considered special types of such structures are explicitly described links with integrability geometry of loop spaces and backlund transformations are traces

symmetry transformations isovectors and conservation laws a system of second order partial differential equations is considered it is shown that a conservation law may be associated with any pair consisting of i a symmetry transformation ii a symmetry transformation if the system is self adjoint a solution to the adjoint of the equations of variation otherwise such conservation laws continue to hold when symmetry transformations are replaced by isovectors it is also proved that isovectors identify additional conservation laws by deformation of a given one whence it follows that there exists a natural conserved current associated with every isovector also an application is made to find conserved currents for the navier stokes equations

positivity for equations involving polyharmonic operators with dirichlet boundary conditions cranston fabes and zhao 26 5 established the uniform bound sup x y 2 x 6 y r g1 n x z g1 n z y dz g1 n x y m 1 1 where g1 n x y is the green function for the laplacian with dirichlet boundary conditions on a lipschitz domain rn with n 3 see 27 for n 2 this estimate was used in 23 and 18 to obtain positivity uniformly with respect to f 0 for noncooperative elliptic systems as

scaling symmetries conservation laws and action principles in one dimensional gas dynamics scaling symmetries of the planar one dimensional gas dynamic equations with adiabatic index are used to obtain lagrangian and eulerian conservation laws associated with the symmetries the known eulerian symmetry operators for the scaling symmetries are converted to the lagrangian form in which the eulerian spatial position of the fluid element is given in terms of the lagrangian fluid labels conditions for a linear combination of the three scaling symmetries to be a divergence or variational symmetry of the action are established the corresponding lagrangian and eulerian form of the conservation laws are determined by application of noether s theorem a nonlocal conservation law associated with the scaling symmetries is obtained by applying a nonlocal symmetry operator to the scaling symmetry conserved vector an action principle incorporating known conservation laws using lagrangian constraints is developed noether s theorem for the constrained action principle gives the same formulas for the conserved vector as the classical noether theorem except that the lie symmetry vector field now includes the effects of nonlocal potentials noether s theorem for the constrained action principle is used to obtain nonlocal conservation laws the scaling symmetry conservation laws only apply for special forms of the entropy of the gas

multi symplectic lagrangian one dimensional gas dynamics the equations of lagrangian ideal one dimensional compressible gas dynamics are written in a multi symplectic form using the lagrangian mass coordinate m and time t as independent variables and in which the eulerian position of the fluid element x x m t is one of the dependent variables this approach differs from the eulerian multi symplectic approach using clebsch variables lagrangian constraints are used to specify equations for xm xt and st consistent with the lagrangian map where s is the entropy of the gas we require st 0 corresponding to advection of the entropy s with the flow we show that the lagrangian hamiltonian equations are related to the de donder weyl multi momentum formulation the pullback conservation laws and the symplecticity conservation laws are discussed the pullback conservation laws correspond to invariance of the action with respect to translations in time energy conservation and translations in m in noether s theorem the conservation law due to m translatio

hamiltonian formulation of the kdv equation we consider the canonical formulation of whitham s variational principle for the kdv equation this lagrangian is degenerate and we have found it necessary to use dirac s theory of constrained systems in constructing the hamiltonian earlier discussions of the hamiltonian structure of the kdv equation were based on various different decompositions of the field which is avoided by this new approach

hamiltonian theory over noncommutative rings and integrability in multidimensions a generalization of the adler gel fand dikii scheme is used to generate bi hamiltonian structures in two spatial dimensions in order to implement this scheme a hamiltonian theory is built over a noncommutative ring namely the ring of formal pseudodifferential operators bi hamiltonian structures generated in this way can be used for the kadomtsev petviashvili equation as well as other integrable equations in 2 1

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

dirac s theory of constraints in field theory and the canonical form of hamiltonian differential operators a simple algorithm for constructing the canonical form of hamiltonian systems of evolution equations with constant coefficient hamiltonian differential operators is given the result of the construction is equivalent to the canonical system derived using dirac s theory of constraints from the corresponding degenerate lagrangian

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

on an integrable hierarchy derived from the isentropic gas dynamics in this paper we study a new hierarchy of equations derived from the system of isentropic gas dynamics equations where the pressure is a nonlocal function of the density we show that the hierarchy of equations is integrable we construct the two compatible hamiltonian structures and show that the first structure has three distinct casimirs while the second has one the existence of casimirs allows us to extend the flows to local ones we construct an infinite series of commuting local hamiltonians as well as three infinite series related to the three casimirs of nonlocal charges we discuss the zero curvature formulation of the system where we obtain a simple expression for the nonlocal conserved charges which also clarifies the existence of the three series from a lie algebraic point of view we point out that the nonlocal hierarchy of hunter zheng equations can be obtained from our nonlocal flows when the dynamical variables are properly constrained

the hamiltonian structure of the dispersionless toda hierarchy abstract the hamiltonian structure of the two dimensional dispersionless toda hierarchy is studied this being a particular example of a system of hydrodynamic type the polynomial conservation laws for the system turn out after a change of variable to be associated with the axially symmetric solutions of the 3 dimensional laplace equation and this enables a generating function for the hamiltonian densities to be derived in closed form

hamiltonian regularisation of shallow water equations with uneven bottom the regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations in a recent work the first two authors proposed a so called hamiltonian regularisation for nonlinear shallow water and isentropic euler equations the characteristic property of this method is that the regularisation of solutions is achieved without adding any artificial dissipation or dispersion the regularised system possesses a hamiltonian structure and thus formally preserves the corresponding energy functional in the present article we generalise this approach to shallow water waves over general possibly time dependent bottoms the proposed system is solved numerically with continuous galerkin method and its solutions are compared with the analogous solutions of the classical shallow water and dispersive serre green naghdi equations the numerical results confirm the absence of dispersive and dissipative effects in presence of bathymetry variations

canonical structures for dispersive waves in shallow water the canonical hamiltonian structure of the equations of fluid dynamics obtained in the boussinesq approximation are considered new variational formulations of these equations are proposed and it is found that as in the case of the kdv equation and the equations governing long waves in shallow water they are degenerate lagrangian systems therefore in order to cast these equations into canonical form it is again necessary to use dirac s theory of constraints it is found that there are primary and secondary constraints which are second class and it is possible to construct the hamiltonian in terms of canonical variables among the examples of boussinesq equations that are discussed are the equations of whitham broer kaup which kupershmidt has recently expressed in symmetric form and shown to admit tri hamiltonian structure

bi hamiltonian structures of d boussinesq and benney lax equations the dispersionless boussinesq and benney lax equations are equations of hydrodynamic type which can be obtained as reductions of the dispersionless kadomtsev petviashvili equation we find that for the three component reduction the dispersionless boussinesq and benney lax equations are the same up to a diffeomorphism this equivalence becomes manifest when the equations of motion are cast into the form of a triplet of conservation laws furthermore in this form we are able to recognize a non trivial scaling symmetry of these equations which plays an important role in the construction of their bi hamiltonian structure we exhibit a pair of compatible hamiltonian operators which belong to a restricted class of dubrovin and novikov operators appropriate to a system of conservation laws the recursion operator for this system generates three infinite sequences of conserved hamiltonians

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

canonical formulation of shallow water waves the author introduces a new potential and use it with the velocity potential to construct a new variational principle for long surface waves in shallow water applying dirac s theory of constraints the author casts it into canonical form and obtain an explicit expression for the exact hamiltonian

invariant solutions of hydrodynamic type equations conditions are established under which a system of hydrodynamic type equations with time dependent coefficients allows an infinite dimensional group of hydrodynamic type symmetries when this group exists it is used to linearize and solve the original system

an elementary proof that the biharmonic green function of an eccentric ellipse changes sign p r garabedian showed in 1951 that the green function for the biharmonic boundary value problem with vanishing dirichlet data changes sign in case the domain is a sufficiently eccentric ellipse this refuted a conjecture made by j hadamard in 1908 the proof of garabedian was based on kernel functions the present note gives an elementary proof

multi lagrangians for integrable systems we propose a general scheme to construct multiple lagrangians for completely integrable nonlinear evolution equations that admit multi hamiltonian structure the recursion operator plays a fundamental role in this construction we use a conserved quantity higher lower than the hamiltonian in the potential part of the new lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map this leads to some remarkable new developments we show that nonlinear evolutionary systems that admit n fold first order local hamiltonian structure can be cast into variational form with 2n 1 lagrangians which will be local functionals of clebsch potentials this number increases to 3n 2 when the miura transformation is invertible furthermore we construct a new lagrangian for polytropic gas dynamics in 1 1 dimensions which is a free local functional of the physical field variables namely density and velocity thus dispensing with the necessity of introducing clebsch potentials entirely this is a consequence of bi hamiltonian structure with a compatible pair of first and third order hamiltonian operators derived from sheftel s recursion operator

bi hamiltonian structure of n component kodama equations the author presents a simple way of constructing the second hamiltonian operators for n component kodama equations using dimensional analysis he is led to an ansatz for the hamiltonian operator as well as the conserved quantities in terms of ratios of polynomials the coefficients of these polynomials are determined from the jacobi identities the resulting bi hamiltonian structure consists of generalization of cavalcante and mckean s work 1982 for n 2 and his earlier results for n 3 4

linearization of universal field equations the universal field equations recently constructed as examples of higher dimensional dynamical systems which admit an infinity of inequivalent lagrangians are shown to be linearised by a legendre transformation this establishes the conjecture that these equations describe integrable systems while this construction is implicit in general there exists a large class of solutions for which an explicit form may be written comment 11pp dtp 92 47 ni 92 011

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

an example of a miura map the shallow water wave equations the linear hamiltonian structure of modified shallow water wave equations goes to the non linear and second hamiltonian structure of the bi hamiltonian shallow water wave equations a similar situation must exist for gas dynamics according to a result of dubrovin and novikov 1983

boundary conditions and the first order conserved density for gas dynamics boundary conditions for gas dynamics are chosen to make the first order conserved density finitely integrable the action of the corresponding third order noetherian symmetry on the boundary condition is discussed a generalisation to first order diagonalised quasilinear systems is considered

inverse variational problem and canonical structure of burgers equations it is demonstrated that burgers equations which are often believed to describe dissipative systems are non lagrangian following bateman s analysis of damped harmonic oscillator an action is defined to look for a lagrangian representation for equations in the burgers hierarchy the associated higher order lagrangian densities are found to be degenerate such that the hamiltonian structure could be studied by a repackaging of ostrogradski formalism and dirac s theory of constraints

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

compatible poisson brackets of hydrodynamic type some general properties of compatible poisson brackets of hydrodynamic type are discussed in particular a an invariant differential geometric criterion of the compatibility based on the nijenhuis tensor which is slightly different from those existing in the literature b the lax pair with a spectral parameter governing compatible poisson brackets in the diagonalizable case c the connection of this problem with the class of surfaces in euclidean space which possess non trivial deformations preserving the weingarten operator

topology optimization of a reinforced wing box for enhanced roll maneuvers in this paper the aileron reversal speed of a wing torsion box is maximized by reinforcing its upper skin we propose a topology optimization approach using the spectral level set methodology this is based on the level set methods which represent theinterface describing thereinforcements asthe zerolevel setofafunction accordingto the proposed methodology this function is discretized using its fourier coefficients these coefficients become the design variables of the optimization problem assigned to define the reinforcing layout results show that approximately the sameoptimal reinforcement layout isobtained from topologically different initialconfigurations the two underlying ideas of the present work are to propose the spectral level set methodology as a framework for topology optimization given its potential for reducing the design space dimension and considering the possibilities of new custom made materials to readdress the reinforcement perspective in aircraft structures without the weight penalty nomenclature ai bi problem domain boundaries corresponding to componenti ofx at t smoothness parameter av v smoothness parameter a0 lift curve slope bt minimum thickness c positive constant cl derivative of the rolling moment coefficient of the rigid airplane with respect to c y n twist aty produced by a unit moment at

quasiclassical limit of coupled kdv equations riemann invariants and multi hamiltonian structure abstract we consider the quasiclassical limit of the first nontrivial flow in coupled kdv hierarchy written down in riemann invariants it assumes the extremely simple form r i k 1 n r k 2r i r x i i 1 n up to certain natural equivalence this turns out to be the only hydrodynamic system with n 1 compatible purely local hamiltonian structures

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

monte carlo approach to dendritic growth the connection between diffusion limited aggregation and the equations of dendritic growth is critically examined a different type of monte carlo simulation is proposed and used to construct two dimensional dendrite like patterns the wavelength occurring for short times is in good agreement with the linear stability analysis the time dependence of the characteristic wavelengths is also determined

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

on a new class of completely integrable nonlinear wave equations ii multi hamiltonian structure the multi hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed infinitely many conservation laws had earlier been obtained for these equations starting from a primary hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi hamiltonian structure are obtained and it is shown that the second hamiltonian operator can be constructed solely through a knowledge of the first hamiltonian function the recursion operator which first appears at the level of bi hamiltonian structure gives rise to an infinite sequence of conserved hamiltonians it is found that in general there exist two different infinite sequences of conserved quantities for these equations the recursion relation defining higher hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the k 1 st hamiltonian operator which depends on the kth hamiltonian function the infinite sequence of conserved hamiltonians are common to all the higher hamiltonian structures the equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri hamiltonian structure with two distinct infinite sets of conserved quantities the isothermal case of 1 is an exceptional one that requires separate treatment this corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to poisson s equation in nonlinear acoustics

on positivity for the biharmonic operator under steklov boundary conditions the positivity preserving property for the inverse of the biharmonic operator under steklov boundary conditions is studied it is shown that this property is quite sensitive to the parameter involved in the boundary condition moreover positivity of the steklov boundary value problem is linked with positivity under boundary conditions of navier and dirichlet type

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

diagonalization and hamiltonian structures of hyperbolic systems a simple algorithm for casting two dimensional hyperbolic systems into the hamiltonian form is given first order conserved quantities are found and expressed in terms of eigenvalues it is shown that for some systems the dependence of the eigenvalues on the field variables can be used for classification

on symmetry analysis of differential systems on functional manifolds abstract in this short letter there is successively sketched a classical e noether based analytical scheme of constructing the complete set of global conservation laws to a smooth vector field on a functional manifold there is also described an analytical algorithm of constructing the complete set of differential conservation laws to a general smooth differential functional system on a functional manifold with values in an arbitrary vector bundle it is demonstrated that the algorithm reduces as a special case to that presented recently by c s anco

alternative bi hamiltonian structures for wdvv equations of associativity the wdvv equations of associativity in two dimensional topological field theory are completely integrable third order monge ampere equations which admit bi hamiltonian structure the time variable plays a distinguished role in the discussion of hamiltonian structure whereas in the theory of wdvv equations none of the independent variables merits such a distinction wdvv equations admit very different alternative hamiltonian structures under different possible choices of the time variable but all these various hamiltonian formulations can be brought together in the framework of the covariant theory of symplectic structure they can be identified as different components of the covariant witten zuckerman symplectic 2 form current density where a variational formulation of the wdvv equation that leads to the hamiltonian operator through the dirac bracket is available

multi hamiltonian structure of the born infeld equation the multi hamiltonian structure conservation laws and higher order symmetries for the born infeld equation are exhibited a new transformation of the born infeld equation to the equations of a chaplygin gas is presented and explored the born infeld equation is distinguished among two dimensional hyperbolic systems by its wealth of such multi hamiltonian structures

higher order symmetries of the compressible one dimensional isentropic fluid equations the equations of compressible one dimensional isentropic flow are shown to have infinitely many higher order symmetries by means of recursion operators furthermore a new first order conserved functional is found to be associated with a third order symmetry via noether s theorem

the shallow water equations explicit solutions and superposition principle by using the hodograph method the author finds almost all the solutions of the classical shallow water equations the author also constructs an auto backlund transformation superposition principle on the set of these hodograph solutions and shows that this transformation is canonical relative to a symplectic form introduced by manin 1978

multi hamiltonian structure of equations of hydrodynamic type the discussion of the hamiltonian structure of two component equations of hydrodynamic type is completed by presenting the hamiltonian operators for euler s equation governing the motion of plane sound waves of finite amplitude and another quasilinear second order wave equation there exists a doubly infinite family of conserved hamiltonians for the equations of gas dynamics that degenerate into one namely the benney sequence for shallow water waves infinite sequences of conserved quantities for these equations are also presented in the case of multicomponent equations of hydrodynamic type it is shown that kodama s generalization of the shallow water equations admits bi hamiltonian structure

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

dirac constraints in field theory lifts of hamiltonian systems to the cotangent bundle to every hamiltonian system is associated a degenerate lagrangian formulation following dirac s theory of constraints one finds a family of lifts of the system to the cotangent bundle of the underlying manifold the lifted system admits a reduction to the original equation such that the original hamiltonian formulation is the pullback of the canonical symplectic form invariants of the original equation can be lifted to invariants of the extended system this procedure is applied to integrable systems such as the korteweg de vries equation

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

non polynomial conservation law densities generated by the symmetry operators in some hydrodynamical models a new extra series of conserved densities for the polytropic gas model and nonlinear elasticity equation is obtained without any reference to the recursion operator or to the lax operator formalism our method is based on the utilization of the symmetry operators and allows us to obtain the densities of arbitrary homogeneity dimensions the non polynomial densities with logarithmic behaviour are presented as an example special attention is paid to the singular case 1 for which we have found new non homogeneous solutions expressed in terms of the elementary functions

introducing loading uncertainty in topology optimization uncertainty is an important consideration in structural design and optimization to produce robust and reliable solutions this paper introduces an efficient and accurate approach to robust structural topology optimization the objective is to minimize expected compliance with uncertainty in loading magnitude and applied direction where uncertainties are assumed normally distributed and statistically independent this new approach is analogous to a multiple load case problem where load cases and weights are derived analytically to accurately and efficiently compute expected compliance and sensitivities illustrative examples using a level set based topology optimization method are then used to demonstrate the proposed approach

hamiltonian structure of real monge ampere equations the variational principle for the real homogeneous monge ampere equation in two dimensions is shown to contain three arbitrary functions of four variables there exist two different specializations of this variational principle where the lagrangian is degenerate and furthermore contains an arbitrary function of two variables the hamiltonian formulation of these degenerate lagrangian systems requires the use of dirac s theory of constraints as in the case of most completely integrable systems the constraints are second class and dirac brackets directly yield the hamiltonian operators thus the real homogeneous monge ampere equation in two dimensions admits two classes of infinitely many hamiltonian operators namely a family of local as well as another family non local hamiltonian operators and symplectic 2 forms which depend on arbitrary functions of two variables the simplest non local hamiltonian operator corresponds to the kac moody algebra of vector fields and functions on the unit circle hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair in the case of real monge ampere equations with constant right hand side this compatible pair is the only pair of hamiltonian operators that survives then the complete integrability of all these real monge ampere equations follows by magri s theorem some of the remarkable properties we have obtained for the hamiltonian structure of the real homogeneous monge ampere equation in two dimensions turn out to be generic to the real homogeneous monge ampere equation and the geodesic flow for the complex homogeneous monge ampere equation in arbitrary number of dimensions hence among all integrable nonlinear evolution equations in one space and one time dimension the real homogeneous monge ampere equation is distinguished as one that retains its character as an integrable system in multiple dimensions

a family of completely integrable multi hamiltonian systems explicitly related to some celebrated equations by introducing a spectral problem with an arbitrary parameter we derive a kaup newell type hierarchy of nonlinear evolution equations which is explicitly related to many important equations such as the kundu equation the kaup newell kn equation the chen lee liu cll equation the gerdjikov ivanov gi equation the burgers equation the modified korteweg devries mkdv equation and the sharma tasso olver equation it is shown that the hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure under the bargann constraint between the potentials and the eigenfunctions the spectral problem is nonlinearized as a finite dimensional completely integrable hamiltonian system the involutive representation of the solutions for the kaup newell type hierarchy is also presented in addition an n fold darboux transformation of the kundu equation is constructed with the help of its lax pairs and a reduction technique according to the darboux transformation the solutions of the ku

conservation laws for shallow water waves on a sloping beach shallow water waves are governed by a pair of nonlinear partial differential equations we transfer the associated homogeneous and nonhomogeneous systems corresponding to constant and sloping depth respectively to the hodograph plane where we find all the nonsimple wave solutions and construct infinitely many polynomial conversation laws we also establish correspondence between conservation laws and hodograph solutions as well as backlund transformations by using the linear nature of the problems on the hodograph plane

structural boundary design via level set and immersed interface methods we develop and test an algorithmic approach to the boundary design of elastic structures the goal of our approach is two fold first to develop a method which allows one to rapidly solve the two dimensional lam equations in arbitrary domains and compute for example the stresses and second to develop a systematic way of modifying the design to optimize chosen properties at the core our approach relies on two distinct steps given a design we first apply an explicit jump immersed interface method to compute the stresses for a given design shape we then use a narrow band level set method to perturb this shape and progress towards an improved design the equations of 2d linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers this effectively reduces the dimensionality of the problem by one once the stresses are found the level set method which represents the design structure through an embedded implicit function is used in the second step to alter the shape with velocities depending on the stresses in the current design criteria are provided for advancing the shape in an appropriate direction and for correcting the evolving shape when given constraints are violated

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found

perfect fluid theory and its extensions we review the canonical theory for perfect fluids in euler s and lagrange s formulations the theory is related to a description of extended structures in higher dimensions internal symmetry and supersymmetry degrees of freedom are incorporated additional miscellaneous subjects that are covered include physical topics concerning quantization as well as mathematical issues of volume preserving diffeomorphisms and representations of chern simons terms vortex or magnetic helicity