the quasiclassical limit of the modified kp hierarchy in the quasiclassical limit the mkp hierarchy and its lax representation turn into a modified benney hierarchy and its poisson representation a miura map is constructed and shown to be canonical from the modified benney hierarchy into the unmodified one the modified hierarchy is given both hydrodynamical and kinetic representations and the miura map is given a kinetic form explicit combinatorial formulae are proved for the infinite number of conserved densities of the modified benney hierarchy

on twistor solutions of the dkp equation the factorization problem for the group of canonical transformations close to the identity and the corresponding twistor equations for an ample family of canonical variables are considered a method to deal with these reductions is developed for the construction of classes of nontrivial solutions of the dkp equation

on dispersionless lax representations for two primary models the equivalence between bi hamiltonian formulation and lax representation of dispersionless integrable hierarchies associated with two primary models of frobenius manifolds is explicitly verified

coisotropic deformations of associative algebras and dispersionless integrable hierarchies the paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies like the dispersionless kp and modified kp hierarchies and the universal whitham hierarchy of genus zero it stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras it discusses the link between the structure constants and hirota s tau function and shows that the dispersionless hirota bilinear equations are within this approach a way of writing the associativity conditions for the structure constants in terms of the tau function it also suggests a simple interpretation of the algebro geometric construction of the universal whitham equations of genus zero due to krichever

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

the chebyshev collocation path following method for solving sixth order sturm liouville problems in this paper we implement a chebyshev collocation method to approximate the eigenvalues of nonsingular sixth order sturm liouville problem this method transforms the sturm liouville problem to a sparse singular linear system which is solved by the path following technique numerical results demonstrate the accuracy and efficiency of the present algorithm

dispersionless scalar integrable hierarchies whitham hierarchy and the quasiclassical dressing method the quasiclassical limit of the scalar nonlocal problem is derived and a quasiclassical version of the dressing method is presented dispersionless kadomtsev petviashvili kp modified kp and dispersionless two dimensional toda lattice 2dtl hierarchies are discussed as illustrative examples it is shown that the universal whitham hierarchy is nothing but the ring of symmetries for the quasiclassical problem the reduction problem is discussed and in particular the d2dtl equation of b type is derived

differential geometric approach to the integrability of hydrodynamic chains the haantjes tensor the integrability of an m component system of hydrodynamic type u t v u u x by the generalized hodograph method requires the diagonalizability of the m m matrix v u this condition is known to be equivalent to the vanishing of the corresponding haantjes tensor we generalize this approach to hydrodynamic chains infinite component systems of hydrodynamic type for which the matrix v u is sufficiently sparse for such systems the haantjes tensor is well defined and the calculation of its components involves finite summations only we illustrate our approach by classifying broad classes of conservative and hamiltonian hydrodynamic chains with the zero haantjes tensor we prove that the vanishing of the haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi hamiltonian hydrodynamic reductions thus providing an easy to verify necessary condition for the integrability

on dispersionless hirota equations of the dispersionless dym hierarchy we show that under twistor construction the only finite dimensional reduction of the dispersionless dym hierarchy which can be characterized by a free energy is the two reduction model with lax operator of the form up u0 p 1 the primary variables u and u0 can be expressed in terms of second derivatives of the associated free energy and the hierarchy flows can be written as a set of dispersionless hirota equations

dispersionless limit of the noncommutative potential kp hierarchy and solutions of the pseudodual chiral model in 2 1 dimensions the usual dispersionless limit of the kp hierarchy does not work in the case where the dependent variable has values in a noncommutative e g matrix algebra passing over to the potential kp hierarchy there is a corresponding scaling limit in the noncommutative case which turns out to be the hierarchy of a pseudodual chiral model in 2 1 dimensions pseudodual to a hierarchy extending ward s modified integrable chiral model applying the scaling procedure to a method generating exact solutions of a matrix potential kp hierarchy from solutions of a matrix linear heat hierarchy leads to a corresponding method that generates exact solutions of the matrix dispersionless potential kp hierarchy i e the pseudodual chiral model hierarchy we use this result to construct classes of exact solutions of the su m pseudodual chiral model in 2 1 dimensions including various multiple lump configurations

s functions reductions and hodograph solutions of the rth dispersionless modified kp and dym hierarchies we introduce an s function formulation for the recently found rth dispersionless modified kp and rth dispersionless dym hierarchies giving also a connection of these s functions with the orlov functions of the hierarchies then we discuss a reduction scheme for the hierarchies that together with the s function formulation leads to hodograph systems for the associated solutions we consider also the connection of these reductions with those of the dispersionless kp hierarchy and with hydrodynamic type systems in particular for the one component and two component reduction we derive for both hierarchies ample sets of examples of explicit solutions

the initial value problem for the dkp hierarchy the solution corresponding to each initial condition for the dispersionless kp hierarchy can be found from the integration of a hamilton jacobi equation by means of a transformation of coordinates the solution is explicitly determined in a parametric form as well as a power series in a deformation parameter in addition a twistor formulation of the solution is given

dressing operator approach to the toroidal model of higher dimensional dispersionless kp hierarchy based upon the dressing operator approach we investigate the twistor theoretical construction additional symmetries hodograph solutions and miura transformation for the toroidal model of higher dimensional dispersionless kp hierarchy introduced by takasaki

additional symmetries and solutions of the dispersionless kp hierarchy the dispersionless kp hierarchy is considered from the point of view of the twistor formalism a set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied a method for dealing with the twistor equations by taking advantage of hodograph type equations is proposed this method is applied for determining the orbits of solutions satisfying reduction constraints of gelfand dikii type under the action of additional symmetries

the dispersionless 2d toda equation dressing cauchy problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse spectral problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation arising as commutation of vector fields in this paper we make use of the above theory i to construct the nonlinear riemann hilbert dressing for the so called two dimensional dispersionless toda equation elucidating the spectral mechanism responsible for wave breaking ii we present the formal solution of the cauchy problem for the wave form of it exp tt xx yy iii we obtain the longtime behaviour of the solutions of such a cauchy problem showing that it is essentially described by the longtime breaking formulae of the dkp solutions confirming the expected universal character of the dkp equation as prototype model in the description of the gradient catastrophe of two dimensional waves iv we finally characterize a class of spectral data allowing one to linearize the rh problem corresponding to a class of implicit solutions of the pde

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

a note on the dispersionless bkp hierarchy we discuss the miura map as well as the poisson algebras associated with the dispersionless dym hierarchy particularly we study explicitly the bi hamiltonian structure of a truncated dym system with two variables in which a new hierarchy flow generated by logarithmic hamiltonians appears we then show that this new hierarchy emerges naturally from the topological recursion relation in the landau ginzburg formulation

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

the approach to the dispersionless 2 1 harry dym hierarchy a approach is adopted to study the dispersionless harry dym dhd hierarchy moreover this formulism is applied to construct some explicit solutions of the dhd hierarchy

on the miura map between the dispersionless kp and dispersionless modified kp hierarchies the miura map between the dispersionless kp and dispersionless modified kp hierarchies is investigated it is shown that the miura map is canonical with respect to their bi hamiltonian structures moreover inspired by the works of takasaki and takebe the twistor construction of solution structure for the dispersionless modified kp hierarchy is given

on the rth dispersionless toda hierarchy factorization problem additional symmetries and some solutions for a family of poisson algebras parametrized by and an associated lie algebraic splitting we consider the factorization of given canonical transformations in this context we rederive the recently found rth dispersionless modified kp hierarchies and rth dispersionless dym hierarchies giving a new miura map among them we also found a new integrable hierarchy which we call the rth dispersionless toda hierarchy moreover additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the rth dispersionless kp rth dispersionless dym and rth dispersionless toda equations some solutions are derived by examining the imposition of a time invariance to the potential rth dispersionless dym equation for which a complete integral is presented and therefore an appropriate envelope leads to a general solution symmetries and miura maps are applied to get new solutions and solutions of the rth dispersionless modified kp equation