an algorithmic method for showing existence of nontrivial non classical symmetries of partial differential equations without solving determining equations abstract in this paper based on differential characteristic set theory and the associated algorithm also called wu s method an algorithmic method is presented to decide on the existence of a nontrivial non classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system the theory and algorithm give a partial answer for the open problem posed by p a clarkson and e l mansfield in 21 on non classical symmetries of partial differential equations as applications of our algorithm non classical symmetries and corresponding invariant solutions are found for several evolution equations

on critical phenomena in interacting growth systems part ii bounded growth this paper completes the classification of some infinite and finite growth systems which was started in part i components whose states are integer numbers interact in a local deterministic way in addition to which every component s state grows by a positive integerk with a probability ek 1 e at every moment of the discrete time proposition 1 says that in the infinite system which starts from the state all zeros percentages of elements whose states exceed a given valuek 0 never exceed ce k wherec const proposition 2 refers to finite systems it states that the same inequalities hold during a time which depends exponentially on the system size

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

on the local systems hamiltonian in the weakly non local poission brackets we study in this work the important class of non local poisson brackets pb which we call weakly non local they appeared recently in some investigations in soliton theory however there was no theory of such brackets except for very special first order case even in this case the theory was not developed enough in particular we introduce the physical forms and find casimirs momentum and canonical forms for the most important hydrodynamic type pb of that kind and their dependence on the boundary conditions

reciprocal transformations of two component hyperbolic systems and their invariants abstract criteria for the equivalence under reciprocal transformations of two component hyperbolic systems are established in terms of reciprocal invariants these results are applied to a 1 1 dimensional isentropic gasdynamics system to determine reciprocal equivalence to the canonical system with a polytropic gas law the four parameter model pressure density relations for which this reciprocal equivalence holds are parameterized in terms of the lagrangian signal speed

existence of an exact solution for a one phase stefan problem with nonlinear thermal coefficients from tirskii s method abstract the mathematical analysis of a one phase lame clapeyron stefan problem with nonlinear thermal coefficients following g a tirskii two exact solutions of stefan s nonlinear problem sov phys dokl 4 1959 288 292 is obtained two related cases are considered one of them has a temperature condition on the fixed face x 0 and the other one has a flux condition of the type q 0 t q 0 0 we obtain in both cases sufficient conditions for data in order to have the existence of an explicit solution of a similarity type which is given by using a double fixed point

on a family of symmetric polynomials abstract in this paper we provide two simple approaches to the explicit expression of a family of symmetric polynomials introduced and studied in milovanovic and cvetkovic j math anal appl 311 2005 191 thereby improving on their observations

symmetry reductions and exact solutions to the systems of carbon nanotubes conveying fluid abstract in this paper the systems of carbon nanotubes cnts conveying fluid are considered by the combination of lie symmetry analysis and the dynamical system method all of the geometric vector fields of the three systems are obtained then the symmetry reductions and exact solutions to the systems are investigated by the dynamical system and the power series method especially the convergence of the power series solution to the nonlinear system is shown under the condition which is easily satisfied generally

non homogeneous systems of hydrodynamic type related to quadratic hamiltonians with electromagnetic term abstract we consider a class of non homogeneous systems of hydrodynamic type q t i j i q q x j i q i 1 n which can be related to quadratic hamiltonians with electromagnetic terms whilst it is unlikely that our systems are generally integrable they do possess intriguing properties such as always having a higher conservation law and sometimes a 2 n parameter family of exact solutions in fact our systems coincide with those possessing a conservation law with the densitylbeing a quadratic expression in the first derivatives 1 2 i j 1 n g i j q x i q x j k 1 n a k q x k h we discuss several examples some of which have important applications

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

hydrodynamic chains and the classification of their poisson brackets infinite component poisson brackets of the dubrovin novikov type sov math dokl 27 665 669 1983 are considered the corresponding jacobi identity is significantly simplified in the liouville coordinates since the skew symmetry condition is automatically satisfied the concept of m poisson bracket connected with hydrodynamic chains is introduced then the jacobi identity is a nonlinear system of equations in partial derivatives which can be completely integrated in such a case a classification of infinite component poisson brackets of the dubrovin novikov type can be obtained two simplest examples m 0 and m 1 are considered also infinite component poisson brackets of the ferapontov am math soc transl 170 38 58 1995 type can be simplified in the liouville coordinates the jacobi identity for infinite component poisson brackets of the ferapontov mokhov type russ math surveys 45 218 219 1990 is presented in the liouville coordinates

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

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

conservation laws for some systems of nonlinear pdes via the symmetry adjoint symmetry pair method abstract this paper applies the symmetry adjoint symmetry pair method to find conservation laws for nonlinear partial differential equations as examples conservation laws are obtained for a combined potential system with nonlinear telegraph and nonlinear diffusion convection equations as special cases as well as for an extended nonlinear wave equation system in addition for both examples conservation laws are obtained for and nonlocally related equivalent systems the paper concludes with a posed open problem

on rediscovered methods for solving equations abstract in the recent papers of gerlach siam rev 36 1994 272 276 and ford and pennline siam rev 38 1996 658 659 a class of iterative methods for solving a single equation f x 0 with arbitrary rate of convergence has been presented in this paper we show that this class is equivalent to five other classes of iterative methods derived earlier in various ways and expressed in different forms the proofs of equivalence of all considered iteration formulas are given

equations admitting o 2 1 r t t 1 as a prolongation algebra derives several equations possessing o 2 1 r t t 1 as a prolongation algebra an example of a backlund transformation is given

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

application of reciprocal backlund transformations to a class of nonlinear boundary value problems it is shown that reciprocal backlund transformations may be used to reduce an important class of nonlinear boundary value problems for higher order nonlinear diffusion equations to linear canonical boundary value problems

binormal motion of curves of constant curvature and torsion generation of soliton surfaces the purely binormal motion of curves of constant curvature or torsion respectively is shown to lead to integrable extensions of the dym and classical sine gordon equations in the case of the extended dym equation a reciprocal invariance is used to establish the existence of novel dual soliton surfaces associated with a given soliton surface a cc ideal formulation is adduced to obtain a matrix darboux invariance for the extended dym and reciprocally linked m 2 kdv equations a backlund transformation with a classical constant length property is thereby constructed which allows the generation of associated soliton surfaces analogues of both backlund9s and bianchi9s classical transformations are derived for the extended sine gordon system

reciprocal transformations in 2 1 dimensions reciprocal backlund transformations are introduced for nonlinear integro differential equations in 2 1 dimensions invariance under such transformations is investigated

a comparison of conservation law construction approaches for the two dimensional incompressible mooney rivlin hyperelasticity model an extended kovalevskaya form is derived for the two dimensional incompressible mooney rivlin nonlinear hyperelasticity equations and is used to compute a complete set of local conservation laws of the model through the direct method conserved densities and fluxes of the conservation laws are derived and their physical interpretation is discussed since the model admits a variational formulation the equations are rewritten in the self adjoint form computation of local conservation laws through the direct method applied to the self adjoint form as well as a conservation law computation through the local symmetry analysis and the noether s first theorem is performed a correspondence between local variational symmetries and conservation law multipliers is illustrated it is argued that even though it leads to more complicated forms of multipliers the direct conservation law construction method applied to the kovalevskaya form of the equations is a preferred systematic way of conservation law computat

symmetries of the kdv equation and four hierarchies of the integrodifferential kdv equations using the inverse strong symmetry of the korteweg de vries kdv equation on the trivial symmetry and 0 symmetry one gets four new sets of symmetries of the kdv equation these symmetries are expressed explicitly by the multi integrations of the jost function of the kdv equation and constitute an infinite dimensional lie algebra together with two hierarchies of the known symmetries contrary to the general belief the time independent symmetry groups of the kdv and mkdv equations are non abelian and the infinite dimensional lie algebras of the kdv and mkdv equations are nonisomorphic though two equations are related by the miura transformation starting from these sets of symmetries four hierarchies of the integrodifferential kdv equations which can be solved by the schrodinger inverse scattering transformation method are obtained some of these hierarchies enjoy a common strong symmetry and or same local conserved densities

prolongation structures for harry dym type equations and backlund transformations of cc ideals the authors examine the prolongation structures for a certain class of differential equations the interrelation between the incomplete set of commutator relations and the defining relations for kac moody algebras is discussed constant coefficient ideals are introduced for which backlund transformations are found in a purely algebraic manner

comparing symmetries and conservation laws of nonlinear telegraph equations a comparison is made between the symmetries and conservation laws admitted by nonlinear telegraph nlt systems such systems are not variational unlike the situation for variational systems where all conservation laws arise from symmetries there are many nlt systems that admit more conservation laws than symmetries the results are summarized in a table which includes the numbers of symmetries and conservation laws for each nlt system it is also indicated when symmetries map conservation laws to other conservation laws

bi hamiltonian aspects of a matrix harry dym hierarchy we study the harry dym hierarchy of nonlinear evolution equations from the bi hamiltonian view point this is done by using the concept of an s hierarchy it allows us to define a matrix harry dym hierarchy of commuting hamiltonian flows in two fields that projects onto the scalar harry dym hierarchy we also show that the conserved densities of the matrix harry dym equation can be found by means of a riccati type equation

loop soliton interaction in an integrable nonlinear telegraphy model reciprocal and backlund transformations an integrable telegraphy equation linked in turn to the geometry of pseudospherical surfaces and a class of model nonlinear elastic stress strain laws descriptive of ideally hard material response is investigated a constant coefficient ideal formalism is introduced which leads to a reciprocal transformation between this nonlinear telegraphy equation and a system compatible with the classical sine gordon equation a backlund transformation is then constructed which is used to analyse two loop soliton interaction the latter exhibits a novel exchange particle phenomenon

miura type transformations and homogeneous spaces we relate miura type transformations mts over an evolution system to its zero curvature representations with values in lie algebras we prove that certain homogeneous spaces of produce mts and show how to distinguish these spaces for a scalar translation invariant evolution equation this allows us to classify all mts in terms of homogeneous spaces of the wahlquist estabrook algebra of the equation for other evolution systems this allows us to construct some mts as an example we study mts over the kdv equation a fifth order equation of harry dym type and the coupled kdv mkdv system of kersten and krasilshchik

reciprocal transformations and flat metrics on hurwitz spaces we consider hydrodynamic systems which possess a local hamiltonian structure to such a system there are also associated an infinite number of nonlocal hamiltonian structures we give the necessary and sufficient conditions so that after a nonlinear transformation of the independent variables the reciprocal system still possesses a local hamiltonian structure we show that under our hypotheses bi hamiltonicity is preserved by the reciprocal transformation finally we apply such results to the reciprocal systems of genus g whitham kdv modulation equations

backlund transformations for the un pumped maxwell bloch system and the fifth painleve equation a backlund transformation for the un pumped maxwell bloch system governing phenomena in nonlinear optics is derived the approach is based on the use of constant coefficient ideals cc ideals since the maxwell bloch system in a certain one dimensional reduction is connected to the fifth painleve equation via a contact transformation chains of solutions for the latter equation are obtained as illustration some rational solutions are explicitly given

jacobi structures of evolutionary partial differential equations abstract in this paper we introduce the notion of infinite dimensional jacobi structure to describe the geometrical structure of a class of nonlocal hamiltonian systems which appear naturally when applying reciprocal transformations to hamiltonian evolutionary pdes we prove that our class of infinite dimensional jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable the main technical tool is in a suitable generalization of the classical schouten nijenhuis bracket to the space of the so called quasi local multi vectors and a simple realization of this structure in the framework of supermanifolds these constructions are used to compute the lichnerowicz jacobi cohomologies and to prove a darboux theorem for jacobi structures with hydrodynamic leading terms we also introduce the notion of bi jacobi structures and consider the integrability of a system of evolutionary pdes that possesses a bi jacobi structure

group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations

conservation laws for nonlinear telegraph equations a complete conservation law classification is given for nonlinear telegraph nlt systems with respect to multipliers that are functions of independent and dependent variables it turns out that a very large class of nlt systems admits four nontrivial local conservation laws the results of this work are summarized in tables which display all multipliers fluxes and densities for the corresponding conservation laws a physical example is considered for possible applications

moving frames and prolongation algebras we consider differential ideals generated by sets of 2 forms which can be written with constant coefficients in a canonical basis of 1 forms by setting up a cartan ehresmann connection in a fiber bundle over a base space in which the 2 forms live one finds an incomplete lie algebra of vector fields in the fibers conversely given this algebra a prolongation algebra one can derive the differential ideal the two constructs are thus dual and analysis of either derives properties of both such systems arise in the classical differential geometry of moving frames examples of this are discussed together with examples arising more recently the korteweg de vries and harrison ernst systems

painleve property and pseudopotentials for nonlinear evolution equations the link between the singularity manifold equation and the pseudopotential equations is found for nonlinear evolution equations both in 1 1 and 2 1 dimensions many examples are given burgers korteweg de vries modified korteweg de vries harry dym a higher order korteweg de vries sawada kotera kaup kuperschmidt kadomtsev petviashvili boussinesq modified kadomtsev petviashvili and 2 1 dimensional sawada kotera equations

explicit solutions for a two phase unidimensional lame clapeyron stefan problem with source terms in both phases a two phase stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi infinite phase change material is studied we assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x 0 a constant temperature or a heat flux of the form q0 t q0 0 the internal heat source functions are given by gj x t lt j x2ajt j 1 solid phase j 2 liquid phase where j j are functions with appropriate regularity properties is the mass density l is the fusion latent heat by unit of mass aj2 is the diffusion coefficient x is the spatial variable and t is the temporal variable we obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x 0 moreover the equivalence of the two free boundary problems is also proved we generalize the solution obtained in j l menaldi d a tarzia generalized lame clapeyron solution for a one phase source stefan problem comput appl math 12 2 1993 123 142 for the one phase stefan problem finally a particular case where j j 1 2 are of exponential type given by j x exp x dj 2 with x and dj r is also studied in details for both boundary temperature conditions at x 0 this type of heat source terms is important through the use of microwave energy following e p scott an analytical solution and sensitivity study of sublimation dehydration within a porous medium with volumetric heating j heat transfer 116 1994 686 693 we obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x 0 is considered a similar result is obtained for a heat flux condition imposed on x 0 if an inequality for parameter q0 is satisfied

n loop solitons and their link with the complex harry dym equation n loop solitons are constructed by means of n cusp soliton solutions of the harry dym equation the proposed approach is based on a novel link between the loop soliton equation and the complex harry dym equation considered on a special curve

on a two phase free boundary problem a two phase free boundary problem associated with the burgers equation is considered the problem is reduced to a system of nonlinear integral equations which is analysed and shown to have a unique solution the system admits a two component shock solution which travels with the same velocity as that of the free boundary the stability analysis of such a solution shows the existence of stability and instability regions according to different values of the parameters characterizing the system

unification of ernst equation backlund transformations using a modified wahlquist estabrook technique the three known backlund transformations for the ernst equation are derived using a modification of the wahlquist estabrook prolongation procedure the modification requires that the equation to be studied be cast into a set of differential forms and their exterior derivatives such that all coefficients are constant a cc ideal analysis of the resulting equations produces 16 solutions composed of the three basic transformations combined with identity and other essentially trivial transformations the group structure of the transformations is discussed a backlund transformation already known for the ernst maxwell equations can be found by the same method promising generalizations are mentioned

a cauchy problem in nonlinear heat conduction a cauchy problem on the semiline for a nonlinear diffusion equation is considered with a boundary condition corresponding to a prescribed thermal conductivity at the origin the problem is mapped into a moving boundary problem for the linear heat equation with a robin type boundary condition such a problem is then reduced to a linear integral volterra equation of ii type which admits a unique solution

simple one dimensional interaction systems with superexponential relaxation times finite one dimensional random processes with local interaction are presented which keep some information of a topological nature about their initial conditions during time the logarithm of whose expectation grows asymptotically at least asm3 wherem is the size of the setr m of states of one component actuallyr m is a circle of lengthm at every moment of the discrete time every component turns into some kind of average of its neighbors after which it makes a random step along this circle all these steps are mutually independent and identically distributed in the present version the absolute values of the steps never exceed a constant the processes are uniform in space time and the set of states this estimation contributes to our awareness of what kind of stable behavior one can expect from one dimensional random processes with local interaction

algebraic construction of the darboux matrix revisited we present algebraic construction of darboux matrices for 1 1 dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case we discuss different approaches to the darboux backlund transformation based on different dependences of the darboux matrix polynomial sum of partial fractions or the transfer matrix form we derive symmetric n soliton formulae in the general case the matrix spectral parameter and dressing actions in loop groups are also discussed we describe reductions to twisted loop groups unitary reductions the matrix lax pair for the kdv equation and reductions of chiral models harmonic maps to su n and to grassmann spaces we show that in the kdv case the nilpotent darboux matrix generates the binary darboux transformation the paper is intended as a review of known results usually presented in a novel context but some new results are included as well e g general compact formulae for n soliton surfaces and linear and bilinear constraints on the nonisospectral lax pair matrices which are preserved by darboux transformations

negative powers of olver recursion operators olver s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator some negative order kdv equations are derived in this way in particular the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated schrodinger equation used in the inverse scattering solution of the kdv equation and this formulation is used to show that this new equation is the miura transform of a sinh gordon equation

conservation laws of 3 α dimensional time fractional diffusion equation abstract the concept of lie backlund symmetry plays a fundamental role in applied mathematics it is clear that in order to find conservation laws for a given partial differential equations pdes or fractional differential equations fdes by using lagrangian function firstly we need to obtain the symmetries of the considered equation fractional derivation is an efficient tool for interpretation of mathematical methods many applications of fractional calculus can be found in various fields of sciences as physics classic quantum mechanics and thermodynamics biology economics engineering and etc so in this paper we present some effective application of fractional derivatives such as fractional symmetries and fractional conservation laws by fractional calculations in the sequel we obtain our results in order to find conservation laws of the time fractional equation in some special cases

framework for nonlocally related partial differential equation systems and nonlocal symmetries extension simplification and examples any partial differential equation pde system can be effectively analyzed through consideration of its tree of nonlocally related systems if a given pde system has n local conservation laws then each conservation law yields potential equations and a corresponding nonlocally related potential system moreover from these n conservation laws one can directly construct 2n 1 independent nonlocally related systems by considering these potential systems individually n singlets in pairs n n 1 2couplets taken all together one n plet in turn any one of these 2n 1 systems could lead to the discovery of new nonlocal symmetries and or nonlocal conservation laws of the given pde system moreover such nonlocal conservation laws could yield further nonlocally related systems a theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems the planar gas dynamics equations and nonlinear telegraph equations are used as illustrative examples many new local

hodograph transformation and differential constraints for wave solutions to 2 2 quasilinear hyperbolic nonhomogeneous systems the differential constraint method is used to work out a reduction approach to determine solutions in a closed form to the highly nonlinear hodograph system arising from 2 2 hyperbolic nonhomogeneous models these solutions inherit all of the features of the standard wave solutions obtainable via the classical hodograph transformation and in the meantime incorporate the dissipative effects induced on wave processes by the source like term involved in the governing equations within such a theoretical framework the problem of integrating the standard linear hodograph system associated with 2 2 homogeneous models is also revisited and a number of results obtained elsewhere of relevant interest in wave problems are recovered as a particular case along the lines of the proposed reduction approach different examples of 2 2 governing models are analysed thoroughly in order to highlight the flexibility of the provided solutions to describe hyperbolic dissipative wave processes

infinite dimensional estabrook wahlquist prolongations for the sine gordon equation we are looking for the universal covering algebra for all symmetries of a given partial differential equation pde using the sine gordon equation as a typical example for a nonevolution equation for nonevolution equations estabrook wahlquist prolongation structures for nonlocal symmetries depend on the choice of a specific subideal of the contact module to define the pde for each inequivalent such choice we determine the most general solution of the prolongation equations as subalgebras of the infinite dimensional algebra of all vector fields over the space of nonlocal variables associated with the pde in the style of vinogradov covering spaces we show explicitly how previously known prolongation structures known to lie within the kac moody algebra a 1 1 are special cases of these general solutions although we are unable to identify the most general solutions with previously studied algebras we show the existence of gauge transformations between prolongation structures viewed as determinin

framework for potential systems and nonlocal symmetries algorithmic approach an algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations des each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree useful conservation laws play an essential role in the construction of an extended tree a useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system nonlocal symmetries for a given system of des can arise from any system in its extended tree we construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations and in both cases obtain new nonlocal symmetries more importantly due to the equivalence of solution sets any coordinate independent method of analysis qualitative numerical perturbation etc can be applied to any system within the tree and may yield simpler computations and or results that cannot be obtained when the method is directly applied to the given system

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

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

explicit solutions to the two phase stefan problem for storm type materials a reciprocal transformation is employed to reduce a two phase stefan problem in nonlinear heat conduction into a form which admits a class of exact solutions analogous to the classical neumann solution the problem is considered for materials of storm type rogers 1985 j phys a math gen 18 105 9 two related cases are considered one of them has a flux condition of the type q 0 t 1 2 q 0 0 and the existence and uniqueness of the solution is proved when q 0 satisfies a certain inequality which generalizes the work of tarzia 1981 q appl math 39 491 7 obtained for constant thermal coefficients the other one has a temperature condition on the fixed face and the existence and uniqueness is proved for all data

on critical phenomena in interacting growth systems part i general components which are placed in a finite or infinite space have integer numbers as possible states they interact in a discrete time in a local deterministic way in addition to which all the components states are incremented at every time step by independent identically distributed random variables we assume that the deterministic interaction function is translation invariant and monotonic and that its values are between the minimum and the maximum of its arguments theorems 1 and 2 based on propositions which we give in a separate part ii give sufficient conditions for a system to have an invariant distribution or a bounded mean other statements proved herein provide background for them by giving conditions when a system has no invariant distribution or the mean of its components states tends to infinity all our main results use one and the same geometrical criterion

soliton solutions for two nonlinear partial differential equations using a darboux transformation of the lax pairs two darboux transformations of the 1 1 dimensional wu zhang wz equation and the two component camassa holm 2ch system with the reciprocal transformation are obtained one loop and two loop soliton solutions and multisoliton like solutions of the 2ch system are obtained by using the darboux transformations and selecting different seed solutions of the corresponding equations the bidirectional soliton solutions of the 1 1 dimensional wz equation are also obtained the interactions of two soliton head on and overtaking collisions for the wz equation and the evolution of the two soliton like solutions for the 2ch system are studied

local and nonlocal symmetries for nonlinear telegraph equation in this paper local and nonlocal symmetry classifications are considered for four equivalent nonlinear telegraph equations a complete potential symmetry classification of a scalar nonlinear telegraph equation is given through the point symmetry classification of a related potential system six new classes of equations are shown to admit potential symmetries the relationships between local including contact and nonlocal potential symmetries of these equations are explored a physical example is considered for possible applications of the obtained potential symmetries

the modified camassa holm equation backlund transformation and nonlinear superposition formula this paper is concerned with the modified camassa holm equation and its backlund transformation with the aid of reciprocal transformation and the associated modified camassa holm equation a backlund transformation which involves both dependent and independent variables is constructed for the modified camassa holm equation the related nonlinear superposition formula is worked out and applications of both backlund transformation and nonlinear superposition formula are considered

preliminary group classification for the nonlinear wave equation utt f x u uxx g x u abstract a classification is given for the nonlinear wave equation u t t f x u u x x g x u admitting an extension by one dimension of the principal lie algebra of the equation under consideration the paper is one of the few applications of an algebraic approach to the problem of group classification the method of preliminary group classification a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained the result of the work is a wide class of equations summarized in table form

application of a reciprocal transformation to a two phase stefan problem a reciprocal transformation is employed to reduce a two phase stefan problem in nonlinear heat conduction to a form which admits a class of exact solutions analogous to the classical newmann solution

symmetry and singularity properties of a system of ordinary differential equations arising in the analysis of the nonlinear telegraph equations we investigate the symmetry and similarity properties of a system of equations arising in the analysis of the nonlinear telegraph equations the system of two equations can be decoupled and integrated although one of equations is linear in one of the dependent variables we are able to perform successfully a singularity analysis we are able to interpret the results of the singularity analysis in terms of the possibility of the existence of a subsidiary solution

a two phase free boundary problem for a nonlinear diffusion convection equation a two phase free boundary problem associated with a diffusion convection equation is considered the problem is reduced to a system of nonlinear integral equations which admits a unique solution for small times the system admits an explicit two component solution corresponding to a two component shock wave of the burgers equation the stability of such a solution is also discussed

reciprocal transformations and local hamiltonian structures of hydrodynamic type systems we start from a hyperbolic dubrovin and novikov dn hydrodynamic type system of dimension n which possesses riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that after such a transformation of the independent variables one of the metrics associated with the initial system is flat we prove the following statement let n 3 in the case of reciprocal transformations of a single independent variable or n 5 in the case of transformations of both the independent variables then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws i e the casimirs the momentum and the hamiltonian densities associated with the hamiltonian operator for the initial metric then we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat such characterization has an interesting geometric interpretation the hypersurfaces of two diagonalizable dn systems of dimension n 5 are lie equivalent if and only if the corresponding local hamiltonian structures are related by a canonical reciprocal transformation

on schroder s families of root finding methods schroder s methods of the first and second kind for solving a nonlinear equation f x 0 originally derived in 1870 are of great importance in the theory and practice of iteration processes they were rediscovered several times and expressed in different forms during the last 130 years it was proved in the paper of petkovic and herceg 1999 7 that even seven families of iteration methods for solving nonlinear equations are mutually equivalent in this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in schroder s generalized method of the second kind presented in 1870 in the continuation we consider smale s open problem from 1994 about possible link between schroder s methods of the first and second kind and state the link in a simple way

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

the dirichlet to neumann map for a nonlinear diffusion convection equation a method for constructing the dirichlet to neumann map for a nonlinear diffusion convection equation is presented the problem is reduced to the solution of a nonlinear integral equation in one independent variable existence and uniqueness of the solution may be proven for small times via a contraction mapping technique

nonlinear self adjointness and conservation laws the general concept of nonlinear self adjointness of differential equations is introduced it includes the linear self adjointness as a particular case moreover it embraces the strict self adjointness definition 1 and quasi self adjointness introduced earlier by the author it is shown that the equations possessing nonlinear self adjointness can be written equivalently in a strictly self adjoint form by using appropriate multipliers all linear equations possess the property of nonlinear self adjointness and hence can be rewritten in a nonlinear strictly self adjoint form for example the heat equation ut u 0 becomes strictly self adjoint after multiplying by u 1 conservation laws associated with symmetries are given in an explicit form for all nonlinearly self adjoint partial differential equations and systems