We show that certain partial differential equations associated to nonisospectral scattering problems in 2+1 dimensions provide a key integrable hierarchies of both ordinary and equations. This is illustrated using (an extension of) known second-order two new third-order problems. These allow us derive equations, 1+1 dimensions, together with their underlying linear (isospectral nonisospectral); also again

We present a generalized non-isospectral dispersive water wave hierarchy in 2+1 dimensions. characterize our entire and its underlying linear problem using single equation together with corresponding scattering problem. This then allows straightforward construction of problems for the 2 + 1 hierarchy. Reductions this yield new integrable hierarchies 1+1 dimensions, also ordinary differential equations, all their problems. In particular, we obtain P_{IV} − P_{II} hierarchy; includes as...

We consider a non-isospectral scattering problem having as its spatial part an energy-dependent Schrödinger operator. This gives rise to new completely integrable multicomponent systems of equations in (2 + 1) dimensions. Their reductions (1 dimensions have isospectral problems and include extensions the AKNS equation also generalization Dym equation. An extension Fuchssteiner - Fokas Camassa Holm is presented.

The interesting result obtained in this paper involves using the generalized singular manifold method to determine Darboux transformations for equations. It allows us establish an iterative procedure obtain multisolitonic solutions. This is closely related Hirota -function method. In paper, we report how improve when equation has more than one Painlevé branch. such a way applied pair of equations 2 + 1 dimensions

The authors present a unified treatment of modified singular manifold expansion method as an improved variant the Painleve analysis for partial differential equations with two branches in expansion. They illustrate by fully applying it to Boussinesq classical system and Mikhailov-Shabat system.

A complete Painleve test (1900) is applied to the generalized Burgers-Huxley equation using version of analysis recently developed by Weiss, Tabor and Carnevale (1983) for partial nonlinear differential equations. In so doing, authors are able find a set new solutions as well recovering some previous particular already found ad hoc methods which have been published.

In this paper we discuss a new approach to the relationship between integrability and symmetries of nonlinear partial differential equation. The method is based heavily on ideas using both Painlevé property singular manifold analysis, which outstanding importance in understanding concept given our examples show that solutions possess Lie point correspond precisely so‐called nonclassical symmetries. We also out connection direct Clarkson Kruskal. Here function its reduced variable. Although...

We give a new nonisospectral generalization of the Volterra lattice equation to 2+1 dimensions. use this construct hierarchy in dimensions, along with its underlying linear problem. Reductions yield variety integrable hierarchies, including generalizations known discrete Painlevé all their corresponding problems. This represents an extension previously developed techniques case.

The relations between the different linear problems for Painlevé equations is an intriguing open problem. Here we consider our previously given second and fourth hierarchies [Publ. Res. Inst. Math. Sci. (Kyoto) 37, 327–347 (2001)], show that they could alternatively have been derived using of Jimbo Miwa. That is, give a gauge transformation these two which maps those themselves onto

We give a new 2+1 dimensional nonisospectral generalization of the Toda lattice hierarchy. Reductions yield variety integrable hierarchies along with their underlying linear problems, including 1+1 differential-delay (nonisospectral and isospectral), ordinary hierarchies, discrete Painlevé hierarchies. also show that reduction in components yields our previously obtained Volterra

We explain how to obtain closed-form analytic solutions from the set of equations that describe three-diode lumped-parameter equivalent circuit model proposed by Mazhari [1] portray undesirable S-shape often observed in I-V characteristics illuminated organic solar cells (OSCs), and occasionally seen other types cells. This allows quick extraction model's parameter values directly fitting resulting solution cell's measured data. Such mathematical simplification procedure facilitates...

We give a new non-isospectral extension to 2 + 1 dimensions of the Boussinesq hierarchy. Such third-order scattering problem xxx +U x +(V - ) = 0 has not been considered previously. This extends our previous results on one-component hierarchies in associated problems. characterize entire (2 1)-dimensional hierarchy and its linear using single partial differential equation corresponding problem. then allows an alternative approach construction problems for Reductions this yield integrable...

We give two new completely integrable sixth-order partial differential equations, together with their Lax pairs and Darboux transformations. From these last we are able to derive the corresponding Bäcklund A simple reduction of equations yields an fourth-order ordinary equation for which a linear problem was previously unknown. also this equation.

In a recent paper (Gordoa P R et al 1999 Nonlinearity 12 955-68) we presented new method of deriving Bäcklund transformations (BTs) for ordinary differential equations. The is based on consideration mappings that preserve natural subset movable poles, together with careful asymptotic analysis the transformed equation, near each type pole. our original applied this approach to second and fourth Painlevé equations, in short 2001 Glasgow Math. J. appear) gave preliminary results third fifth...

The truncation method is a collective name for techniques that arise from truncating Laurent series expansion (with leading term) of generic solutions nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties integrable PDEs, it has not been generally extended to ordinary (ODEs). Here we give new general provides such an extension show how apply the classical ODEs called Painleve equations. Our main idea consider...

In a recent paper we introduced new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy underlying linear problems.Here consider reductions this to hierarchies discrete equations, which obtain once again their problems.We generalized first Painlevé includes as special cases, after further summation, both standard and extended version thirty-fourth hierarchy.

This paper is an attempt to present and discuss at some length the Singular Manifold Method. Method based upon Painlev\'e Property systematically used as a tool for obtaining clear cut answers almost all questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, B\"acklund or Darboux Transformations well $\tau$-functions, in unified way. Besides basics of we exemplify this approach by applying it four equations $(1+1)$-dimensions. Two them are other two through Miura...