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

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

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

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

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

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

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

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

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

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

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

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

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

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

method of lines for hyperbolic differential equations runge kutta methods are applied to the numerical solution of hyperbolic partial differential equations conditions that they are locally stable are derived

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

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

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

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

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

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

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

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

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

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

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