einstein maxwell solitons the application of the inverse scattering method to the einstein maxwell equations for stationary axisymmetric exterior fields leads the authors to an explicit formula for the ernst and electromagnetic potentials of new exact solutions generated from an arbitrary seed solution

exact solutions to higher order nonlinear equations through gauge transformation abstract a highly nonlinear hydrodynamic equation proposed by johnson is solved exactly by gauge transforming gt it to the mixed nonlinear schrodinger equation the gt is generalized further to generate a hierarchy of integrable higher order nonlinear systems including some known equations along with their lax pairs infinite sets of conserved quantities and exact solutions

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

global existence and nonexistence in a system of petrovsky in this paper we consider the nonlinearly damped semilinear petrovsky equationutt 2u aut ut m 2 bu u p 2in a bounded domain where a b 0 we prove the existence of a local weak solution and show that this solution blows up in finite time if p m and the energy is negative we also show that the solution is global if m p

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

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

an optimal result for global existence and boundedness in a three dimensional keller segel stokes system with nonlinear diffusion abstract this paper investigates the following keller segel stokes system with nonlinear diffusion k s f n t u n n m n c x t 0 c t u c c c n x t 0 u t p u n x t 0 u 0 x t 0 under homogeneous boundary conditions of neumann type for n and c and of dirichlet type for u in a three dimensional bounded domains r 3 with smooth boundary where w 2 m 0 it is proved that if m 4 3 then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system k s f which in view of the known results for the fluid free system mentioned below see introduction is an optimal restriction on m

finite band solutions of the classical boussinesq burgers equations the classical boussinesq burgers hierarchy is decomposed into two systems of solvable ordinary differential equations with the help of the lax representations of stationary evolution equations from which the finite band solutions of the higher order classical boussinesq burgers equations are obtained

a bargmann system and the involutive representation of solutions of the levi hierarchy the author considers a linear spectral problem associated with the levi hierarchy of equations a bargmann system under the so called bargmann constraint between the eigenfunctions and potentials is proposed and is proved to be a completely integrable hamiltonian system in the liouville sense moreover by the involutive solution of the compatible systems under the bargmann constraint he presents the involutive representation of solutions of the levi hierarchy especially he obtains the involutive representation of the solution of the well known burgers equation ut uxx 2uux

limit behavior of the solution to nonlinear viscoelastic marguerre von karman shallow shell system abstract this paper is concerned with the nonlinear full marguerre von karman shallow shell system with a dissipative mechanism of memory type the model depends on one small parameter the main purpose of this paper is to show that as the parameter approaches zero the limiting system is the well known full von karman model with memory for thin plates

blow up of positive initial energy solutions of a nonlinear viscoelastic hyperbolic equation abstract in this paper we consider the nonlinear viscoelastic equation u t t u 0 t g t u d u t u t m 2 u u p 2 with initial conditions and dirichlet boundary conditions for nonincreasing positive functions g and for p m we prove that there are solutions with positive initial energy that blow up in finite time

blow up of solutions for some non linear wave equations of kirchhoff type with some dissipation abstract the initial boundary value problem for non linear wave equations of kirchhoff type with dissipation in a bounded domain is considered we prove the blow up of solutions for the strong dissipative term u t and the linear dissipative term u t by the energy method and give some estimates for the life span of solutions we also show the nonexistence of global solutions with positive initial energy for non linear dissipative term by vitillaro s argument

finite dimensional discrete systems and integrable systems through nonlinearization of the discrete eigenvalue problem a method is developed that extends the nonlinearization procedure of eigenvalue problems to the discrete case from which the finite dimensional discrete systems and finite dimensional completely integrable systems associated with the toda hierarchy are obtained a new model for solving discrete soliton equations is proposed which enables one to get the solutions of the toda lattice equations by solving the compatible systems of discrete equations and nonlinear ordinary differential equations

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

global nonexistence for a semilinear petrovsky equation abstract in this paper we consider a semilinear petrovsky equation with damping and source terms it is proved that the solution blows up in finite time if the positive initial energy satisfies a suitable condition moreover for the linear damping case we show that the solution blows up in finite time even for vanishing initial energy this is an important breakthrough since it is only well known that the solution blows up in finite time if the initial energy is negative from all the previous literature

lax representation r matrix method and separation of variables for the neumann type restricted flow the r matrix method and the method of separation of variables for restricted flows with neumann constraints are considered restricted jaulent miodek flows are taken as illustrative examples lax representations are obtained from the adjoint representation of the corresponding soliton hierarchy the r matrix formula based on this lax representation is derived and the separation of variables for the hamilton jacobi equation are performed

c neumann and bargmann systems associated with the coupled kdv soliton hierarchy under two different constraints between the potentials and the eigenfunctions the eigenvalue problem associated with the coupled kdv hierarchy is nonlinearised to be a completely integrable c neumann system on the tangent bundle of sphere tsn 1 with the hamiltonian h 1 2 lambda q p 1 2 q q p p 1 2 q p 2 1 2 q p lambda p p and a completely integrable bargmann system r2n dpdq h with the hamiltonian h 1 2 lambda q p 1 2 p p q p 1 2 q q respectively the involutive solutions of the coupled kdv equation associated with the two systems are given

blow up of solution for a nonlinear petrovsky type equation with memory in this paper we consider the nonlinear petrovsky type equation u t t 2 u 0 t g t s 2 u t s d s u t m 2 u t u p 2 u with initial conditions and dirichlet boundary conditions under suitable conditions of the initial data and the relaxation function we prove that the solution with upper bounded initial energy blows up in finite time moreover for the linear damping case we show that the solution blows up in finite time by different method for nonpositive initial energy

an analytical study of bifurcation problems for equations involving fredholm mappings on considere des equations de la forme t v l v m v v d 0 ou est un sous ensemble ouvert d un espace norme pour un fixe t l et m sont des applications de la fermeture d 0 d un voisinage d 0 de l origine dans un espace de banach x dans un autre espace de banach y avec t 0 l 0 m 0 0 on etudie l existence de points de bifurcation

decay estimates for second order evolution equations with memory abstract this paper develops a unified method to derive decay estimates for general second order integro differential evolution equations with semilinear source terms depending on the properties of convolution kernels at infinity we show that the energy of a mild solution decays exponentially or polynomially as t our approach is based on integral inequalities and multiplier techniques these decay results can be applied to various partial differential equations we discuss three examples a semilinear viscoelastic wave equation a linear anisotropic elasticity model and a petrovsky type system

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

uniform decay rates for nonlinear viscoelastic marguerre von karman equations abstract in this paper we consider a model of nonlinear viscoelastic shallow shell that is referred to as the full marguerre von karman under the presence of long time memory we show that the energy functional associated with the system decays exponentially to zero as time goes to infinity

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation