the master integrable two dimensional system with a quartic second integral in this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities this system is quite general and involves 21 parameters we also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters the results are applied to problems of particle and rigid body dynamics new integrable cases are obtained as special versions of the new system these cases include motions in a plane lobachevsky plane sphere and surfaces of variable curvature they also include generalizations of the classical cases of kovalevskaya chaplygin and goriatchev with the addition of certain singular terms to the potential

theory of group representations and integration of nonlinear systems xa zz exp kx a abstract a constructive proof of complete integrability of the system xa zf z exp kx a and its explicit solution are given for k coinciding with the cartan matrix of simple lie algebras the technique for dealing with this system of essentially nonlinear two dimensional partial differential equations involves lax type representation and representation theory of the corresponding group

generating function associated with the rational solutions of the painleve ii equation we consider the hankel determinant representation for the rational solutions of the painleve ii equation we give an explicit formula for the generating function of the entries in terms of logarithmic derivative of the airy function

a new integrable toda related three particle system a study of a three particle system that possesses in addition to the energy function two nonlinear constants of motion is presented herein the system is obtained by modification of the lagrangian of the toda lattice the explicit expression of the two integrals is obtained and it is proven that this new three particle lagrangian is integrable finally the possible generalization to the case of n particles is discussed

lax formalism for a family of integrable toda related n particle systems this article can be divided in two parts first a lax representation is obtained for a family of integrable three particle toda related systems the second part contains the generalization to the case of n particles

integrable three particle systems hidden symmetries and deformations of the calogero moser system the properties of some three particle systems are studied using as an approach the theory of generalized or hidden symmetries it is proven that the existence of a family that possesses in addition to the energy function two nonlinear constants of motion in involution this family of integrable systems is obtained by considering deformations of the lagrangian of the calogero moser system finally the generalization of the results to the general case of n particles is discussed

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

integrability of toda lattice by generalized variational symmetry approach a systematic method of finding nontrivial generalized variational symmetries are presented herein and the corresponding first integrals for nonlinear dynamical system with two degrees of freedom are derived then a method is applied to a toda lattice given by the potential v ex y c1e2y c2ey c3e 2x c4 e x where c1 c2 c3 and c4 are parameters the symmetries and constants of motion are presented explicitly

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

nonlinear equations with superposition formulas and the exceptional group g2 i complex and real forms of g2 and their maximal subalgebras in order to study nonlinear ordinary differential equations with superposition principles related to the exceptional simple lie group g2 the complex and real forms of its lie algebra are examined and their maximal subalgebras are summarized in particular the parabolic subalgebras of the noncompact real form gnc2 r are determined explicit matrix realizations of the fundamental representation d 1 0 are used and studied in connection with invariant subspaces in a seven dimensional complex or real vector space the results are collected in three tables of specific interest for the study of nonlinear differential equations which will be developed in paper ii of this series

on the non integrability of some generalized toda lattices abstract we consider a one parameter family of two degrees of freedom generalized toda hamiltonians using ziglin s theorem with some extensions we prove the non existence of an additional single valued analytic integral except for special values of the parameter

an integrable three particle system a study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized dynamical or hidden symmetries two lagrangians are considered both obtained by modifying the toda lagrangian first a two particle system is studied and then the results are generalized to a three particle system it is shown that in both cases the lagrangians possess nonlinear constants of motion in involution and thus they are integrable

integrals of motion for toda systems with unequal masses we present new integrals of motion for the toda lattice chain of particles in one dimension with exponential interaction for two special cases of boundary conditions the free end lattice with three non equal mass particles and the fixed end lattice for two particles in both cases we use two distinct approaches in order to identify the integrable cases direct search of the integral of motion and group theoretical methods our results are in agreement with the predictions of painleve analysis

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

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

integrability of the toda chain with free boundaries the hamilton function for the toda chain with free boundaries is studied for different interaction and mass parameters for certain systems the connection to simple lie algebras is shown and the integrability is proved by constructing the corresponding lax representations for the four particle system a painleve analysis is presented

superroughening by linear growth equations with spatiotemporally correlated noise we give an extensive study on a class of interfacial superroughening processes with finite lateral system size in 1 1 dimensions described by linear growth equations with spatiotemporally power law decaying correlated noise since some of these processes have an extremely long relaxation time we first develop a very efficient method capable of simulating the interface morphology of these growth processes even in very late time we numerically observe that this class of superrough growth processes indeed gradually develops macroscopic structures with the lateral size comparable to the lateral system size through the rigorous analytical study of the equal time height difference correlation function the different time height difference correlation function and the local width we explicitly evaluate not only the leading anomalous dynamic scaling term but also all the subleading anomalous dynamic scaling terms which dominate over the ordinary dynamic scaling term moreover the relation between the macroscopic structure formation and anomalous interfacial roughening of the superrough growth processes is analytically investigated in detail

how to construct integrable fokker planck and electromagnetic hamiltonians from ordinary integrable hamiltonians integrable hamiltonians with velocity dependent potentials including those of fokker planck type h 1 2 p2x p2y kxpx kypy are constructed from integrable hamiltonians of type h 1 2 p2x p2y v x y using certain canonical and noncanonical transformations some of the hamiltonians obtained this way are integrable only for zero energy candidates for the potential which is of interest for fokker planck models are constructed in several cases

a remark on the hankel determinant formula for solutions of the toda equation we consider the hankel determinant formula of the functions of the toda equation we present a relationship between the determinant formula and the auxiliary linear problem which is characterized by a compact formula for the functions in the framework of the kp theory similar phenomena that have been observed for the painleve ii and iv equations are recovered the case of finite lattice is also discussed

schlesinger backlund transformations for the n component kp the n component kadomtsev petviashvili kp hierarchy was introduced by sato as the set of deformation equations i e lax equations of n 1n n matrix pseudodifferential operators on such a set of operators we define a schlesinger backlund transformation which produces from a set of solutions of the kp hierarchy a new set of solutions applying such transformations recursively we obtain a whole family of solutions from which we can then derive the kp tau function

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases