Pilar garcia estevez

University of Salamanca · Department of Fundamental Physics

We present a generalized study and characterization of the integrability properties of the derivative non-linear Schrödinger equation in 1+1 dimensions. A Lax pair is derived for this equation by means of a Miura transformation and the singular manifold method. This procedure, together with the Darboux transformations, allow us to construct a wide...

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are strai...

A review of the mathematical and physical aspects of the Ermakov systems is presented. The main properties of Lie algebra invariants are extensively used. The two and tridimensional Ermakov systems are fully analyzed and the correspondent invariants found. Then, we go over quantization with special emphasis in the two dimensional case. An applicati...

We present a generalized study and characterization of the integrability properties of the derivative non-linear Schr\"odinger equation in 1+1 dimensions. A Lax pair is derived for this equation by means of a Miura transformation and the singular manifold method. This procedure, together with the Darboux transformations, allow us to construct a wid...

An integrable two-component nonlinear Schrödinger equation in 2+1 dimensions is presented. The singular manifold method is applied in order to obtain a three-component Lax pair. The Lie point symmetries of this Lax pair are calculated in terms of nine arbitrary functions and one arbitrary constant that yield a non-trivial infinite-dimensional Lie a...

Reciprocal transformations mix the role of the dependent and independent variables of (nonlinear partial) differential equations to achieve simpler versions or even linearized versions of them. These transformations help in the identification of a plethora of partial differential equations that are spread out in the physics and mathematics literatu...

An integrable two-component nonlinear Schr\"odinger equation in $2+1$ dimensions is presented. The singular manifold method is applied in order to obtain a three-component Lax pair. The Lie point symmetries of this Lax pair are calculated in terms of nine arbitrary functions and one arbitrary constant that yield a non-trivial infinite-dimensional L...

We study an effective integrable nonlinear model describing an electron moving along the axis of a deformable helical molecule. The helical conformation of dipoles in the molecular backbone induces an unconventional Rashba-like interaction that couples the electron spin with its linear momentum. In addition, a focusing nonlinearity arises from the...

A non-isospectral linear problem for an integrable 2+1 generalization of the non linear Schrödinger equation, which includes dispersive terms of third and fourth order, is presented. The classical symmetries of the Lax pair and the related reductions are carefully studied. We obtain several reductions of the Lax pair that yield in some cases non-is...

We investigate the generalized (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document} Nizhnik–Novikov–Veselov equation and construct its linear eige...

In this paper an effective integrable non-linear model describing the electron spin dynamics in a deformable helical molecule with weak spin-orbit coupling is presented. Non-linearity arises from the electron-lattice interaction and it enables the formation of a variety of stable solitons such as bright solitons, breathers and rogue waves, all of t...

It is widely admitted that the helical conformation of certain chiral molecules may induce a sizable spin selectivity observed in experiments. Spin selectivity arises as a result of the interplay between a helicity-induced spin-orbit coupling and electric dipole fields in the molecule. From the theoretical point of view, different phenomena might a...

We show that the nonlinear Schrödinger equation in 2 + 1 dimensions possesses a class of regular and rationally decaying solutions associated to interacting solitons. The interesting dynamics of the associated pulses is studied in detail and related to homothetic Lagrange configurations of certain N- body problems. These solutions correspond to the...

We present the iterative classical point symmetry analysis of a shallow water wave equation in \(2+1\) dimensions and that of its corresponding nonisospectral, two-component Lax pair. A few reductions arise and are identified with celebrate equations in the Physics and Mathematics literature of nonlinear waves. We pay particular attention to the is...

We propose and examine an integrable system of nonlinear equations that generalizes the nonlinear Schrödinger equation to 2+1 dimensions. This integrable system of equations is a promising starting point to elaborate more accurate models in nonlinear optics and molecular systems within the continuum limit. The Lax pair for the system is derived aft...

A Lie system is a nonautonomous system of first-order differential equations admitting a superposition rule, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. Using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called Vessi...

We consider a natural integrable generalization of nonlinear Schrödinger equation to 2+1 dimensions. By studying the associated spectral operator we discover a rich discrete spectrum associated with regular rationally decaying solutions, the lumps, which display interesting nontrivial dynamics and scattering. Particular interest is placed in the dy...

We present the iterative classical point symmetry analysis of a shallow water wave equation in $2+1$ dimensions and that of its corresponding nonisospectral, two component Lax pair. A few reductions arise and are identified with celebrate equations in the Physics and Mathematics literature of nonlinear waves. We pay particular attention to the isos...

This work presents a classical Lie point symmetry analysis of a two-component, non-isospectral Lax pair of a hierarchy of partial differential equations in $2+1$ dimensions, which can be considered as a modified version of the Camassa-Holm hierarchy in $2+1$ dimensions. A classification of reductions for this spectral problem is performed. Non-isos...

This work presents a classical Lie point symmetry analysis of a two-component, non-isospectral Lax pair of a hierarchy of partial differential equations in 2+1 dimensions, which can be considered as a modified version of the Camassa-Holm hierarchy in 2 + 1 dimensions. A classification of reductions for this spectral problem is performed. Non-isospe...

We present two hierarchies of partial differential equations in $2+1$ dimensions. Since there exist reciprocal transformations that connect these hierarchies to the Calogero-Bogoyavlenski-Schiff equation and its modified version, we can prove that one of the hierarchies can be considered as a modified version of the other. The connection between th...

We develop a method based upon the Singular Manifold Method that yields an iterative and analytic procedure to construct solutions for a Bogoyavlenskii-Kadomtsev-Petviashvili equation. This method allows us to construct a rich collection of lump solutions with a nontrivial evolution behavior

We here present two different hierarchies of PDEs in 1+1 dimensions whose first and second member are the shallow water wave Camassa-Holm and Qiao equations, correspondingly. These two hierarchies can be transformed by reciprocal methods into the Calogero-Bogoyanlevski-Schiff equation (CBS) and its modified version (mCBS), respectively. Considering...

We here present two different hierarchies of PDEs in 1 + 1 dimensions, whose first and second member are the Camassa-Holm and Qiao equations, correspond-ingly. These hierarchies can be transformed into the Calogero-Bogoyanlevski-Schiff equation and its modified version, respectively, by reciprocal methods. Bearing in mind that there exists a Miura...

The solutions of a class of nonlinear second-order differential equations with a cubic term in the dependent variable being related to Duffing oscillators are obtained by means of the factorization technique. The Lagrangian, the Hamiltonian and the constant of motion are also found through a correspondence with an autonomous system. A physical exam...

The non-isospectral problem (Lax pair) associated with a hierarchy in 2+1 dimensions that generalizes the well known Camassa-Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries can be written in terms of five arbitrary consta...

The classical Lie method is applied to a nonisospectral problem associated with a system of partial differential equations in 2+1 dimensions (Maccari A, J. Math. Phys. 39, (1998), 6547-6551). Identification of the classical Lie symmetries provides a set of reductions that give rise to different nontrivial spectral problems in 1+1 dimensions. The fo...

A new integrable 2+1 hierarchy that generalizes the 1+1 Qiao hierarchy is presented. A reciprocal transformation is defined such that the independent x-variable is considered as a dependent field. The hierarchy transforms into n copies of an equation that has a two component non-isospectral Lax pair. Coming back to the original variables, by invert...

This paper deals with the spectral problem of the Manakov Santini system. The point Lie symmetries of the Lax pair have been identified. Several similarity reductions arise from these symmetries. An important benefit of our procedure is that the study of the Lax pair instead of the partial differential equations yields the reductions of the eigenfu...

We present two reciprocal transformations for a spectral problem in 2+1 dimensions. Reductions of the transformed equations to 1+1 dimensions include the Degasperis-Procesi and Vakhnenko-Parkes equations.

We derive a class of localized solutions of a 2+1 nonlinear Schrödinger (NLS) equation and study their dynamical properties. The ensuing dynamics of these configurations is a superposition of a uniform, “center of mass” motion and a slower, individual motion; as a result, nontrivial scattering between humps may occur. Spectrally, these solutions co...

In this paper we apply truncated Painleve expansions to the Lax pair of a PDE to derive gauge- Backlund transformations of this equation. It allows us to construct an algorithmic method to derive solutions by starting from the simplest one. Actuall y, we use this method to obtain an infinite set of lump solutions that can be classified by means o f...

A class of particular travelling wave solutions of the generalized Benjamin–Bona–Mahony equation is studied systematically using the factorization technique. Then, the general travelling wave solutions of Benjamin–Bona–Mahony equation, and of its modified version, are also recovered.

A general type of almost linear second-order differential equations, which are directly related to several interesting physical problems, is characterized. The solutions of these equations are obtained using the factorization technique, and their non-autonomous invariants are also found by means of scale transformations.

The singular manifold method is used to generate lump solutions of a generalized integrable nonlinear Schrödinger equation in 2 + 1 dimensions. We present several essentially different types of lump solutions. The connection between this method and the Ablowitz–Villarroel scheme is also analysed.

We use the singular manifold method to generate lump solutions of a Schrdinger equation in 2+1 dimensions and present three different types of such solutions.

The travelling wave solutions of the two-dimensional Korteweg-de Vries-Burgers and Kadomtsev-Petviashvili equations are studied from two complementary points of view. The first one is an adaptation of the factorization technique that provides particular as well as general solutions. The second one applies the Painlevé analysis to both equations, th...

In this Letter we present the reductions arising from the classical Lie symmetries of a Lax pair in 2+1 dimensions. We obtain several interesting reductions and prove that, by analyzing not only a PDE but also its associated linear problem, it is possible to obtain the reduction of the PDE together with the reduced Lax pair. Specially relevant is t...

We consider solutions of a generalization of the Camassa-Holm hierarchy to 2+1 dimensions that include, in particular, the well-known multipeakon solutions of the celebrated Camassa-Holm equation.

he Singular Manifold Method is presented as an excellent tool to study a 2+1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1+1 reductions of the same equation. Nevertheless these problems are solved when the number of dimensions of the equation is increased.

A generalization of the negative Camassa-Holm hierarchy to 2+1 dimensions is presented under the name CHH(2+1). Several hodograph transformations are applied in order to transform the hierarchy into a system of coupled CBS (Calogero-Bogoyavlenskii-Schiff) equations in 2+1 dimensions that pass the Painleve test. A non-isospectral Lax pair for CHH(2+...

For the 1+1-dimensional nonlinear diffusion equations with x-dependent convection and source terms ut=(D(u)ux)x+Q(x,u)ux+P(x,u), we obtain conditions under which the equations admit the second-order generalized conditional symmetries η(x,u)=uxx+H(u)ux2+G(x,u)ux+F(x,u) and the first-order sign-invariants J(x,u)=ut−A(u)ux2−B(x,u)ux−C(x,u) on the solu...

The singular manifold method (SMM) is applied to an equation in 2+1 dimensions 0=η x +u 2 ,0=u xy +2uη y +4ω,0=u t -ω x that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and therefore, it can be considered as the modified version of an equation with just one br...

In a previous work7, we have obtained the Lax pair and Darboux transformations for an integrable equation in (2 + 1) dimensions. In the present paper we derive an iterative method of obtaining solutions in a rather easy way. Several solutions are presented for real and complex version of the equation.

A (1+1)-dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator V[(S)\tilde]0\tilde S_0 that depends on two constants and n1. When =0, it reduces to the invariant set S 0 introduced by Galaktionov. We also introduce a generalization of both the scaling and rotation groups,...

This paper considers a general form of the porous medium equation with nonlinear source term: u t =D (u) u x n x +F(u),n≠1, for a single function u of two variables t and x· The functions D(u) and F(u) are, respectively, the diffusion and the source term. The functional separation of variables of this equation is studied by using the generalized co...

We develop a generalized conditional symmetry approach for the functional separation of variables in a nonlinear wave equation with a nonlinear wave speed. We use it to obtain a number of new (1+1)-dimensional nonlinear wave equations with variable wave speeds admitting a functionally separable solution. As a consequence, we obtain exact solutions...

In this paper we study an equation in (2+1) dimensions. The singular manifold method is used to derive its nonisospectral Lax pair and Darboux transformation.

The real version of a (2 + 1) dimensional integrable generalization of the nonlinear Schrödinger equation is studied from the point of view of Painlevé analysis. In this way we find the Lax pair, Darboux transformations and Hirota's functions as well as solitonic and dromionic solutions from an iterative procedure.

In this paper the singular manifold method allows us to obtain a non-isospectral Lax pair, Darboux transformations and Miura transformations for an equation in (2 + 1) dimensions and its modified version. In this way we can iteratively build different kinds of solutions with solitonic behaviour.

In this paper the Singular Manifold Method has allowed us to obtain the Lax pair, Darboux transformations and tau functions for a non-linear Schr\"odiger equation in 2+1 dimensions. In this way we can iteratively build different kind of solutions with solitonic behavior.

The Painleve expansion for the second Painleve equation (PII) and fourth Painleve equation (PIV) have two branches. The singular manifold method therefore requires two singular manifolds. The double singular manifold method is used to derive Miura transformations from PII and PIV to modified Painleve type equations for which auto-Backlund transform...

A recently found soliton sector of the classical Boussinesq system is analysed from the point of view of both Hirota's method (1985) and the singular manifold method. Multisoliton solutions involving processes of soliton fusion and fission are characterized. Backlund transformations relating these solitons to those previously found by Kaup (1975) a...

This paper discusses two equations with the conditional Painlevé property. The usefulness of the singular manifold method as a tool for determining the non-classical symmetries that reduce the equations to ordinary differential equations with the Painlevé property is confirmed once more. The examples considered in this paper are particularly intere...

The interesting result obtained in this paper involves using the generalized singular manifold method to determine the Darboux transformations for the equations. It allows us to establish an iterative procedure to obtain multisolitonic solutions. This procedure is closely related to the Hirota -function method. In this paper, we report how to impro...

The nonlinear ordinary differential equation of the form u"+f1u'+f2u+f3u3=O representing a generalized damped cubic equation is fully analysed. The author finds the general expression for f2 in terms of f1 and f3 that renders the equation integrable. Also he shows that if f2 satisfies the above mentioned condition, the former nonlinear differential...

In this paper a generalization of the direct method of Clarkson and Kruskal (1989) for finding similarity reductions of partial differential equations is found and discussed for the Burgers and Burgers-Huxley equations. The generalization incorporates the singular manifold method largely based upon the Painleve property. This singular manifold can...

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

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

In this paper the nonlinear equation mty=(myxx+mxmy)x is thoroughly analysed. The Painleve test is performed yielding a positive result. The Backlund transformations are found and the Darboux-Moutard-Matveev formalism arises in the context of this analysis. The singular manifold method, based upon the Painleve analysis, is proved to be a useful too...

Painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2+1 dimensions. This procedure has allowed us to obtain the Lax pair, Darboux transformation and tau functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

A Dispersive Wave Equation in 2+1 dimensions (2LDW) widely discussed by different authors is shown to be nothing but the modified version of the Generalized Dispersive Wave Equation (GLDW). Using Singularity Analysis and techniques based upon the Painleve Property leading to the Double Singular Manifold Expansion we shall find the Miura Transformat...

This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlev\'e Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, B\"acklund or Darboux Transformations as wel...

In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painlevé property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial d...

The nonlinear equation m ty = (m yxx +m xm y )x is throughly analyzed. The Painlevé test yields a positive result. The Bäcklund transformations are found and the Darboux—Moutard—Matveev formalism arises in the context of this analysis. Some solutions and their interactions are also analyzed. The singular manifold equations are also used to determin...

A generalization of the direct method of Clarkson and Kruskal for fiuding similarity reluctions of a PDE is found and discussed. The generalization incorporates the singular manifold method largely based upon the Painlevé property. The symmetries found in this way are shown to be those corresponding to the so-called nonclassical symmetries by Blume...

A thorough analysis on the integrability of the anharmonic oscillator with variable damping coefficients is carried out. Using Painlevé analysis we find the most general form of the damping that allows for integrability of the oscillator. We present a novel method that yields exact and explicit solutions. These solutions are presented and classifie...

New and generalised solutions of the perturbed KdV equation u t +λ 1 uu x +λ 2 u xxx +λ 3 u xxxx +λ 4 (uu x ) x +λ 5 u xx =0, which describes the evolution of long shallow waves in a convecting fluid, are presented using a full Painlevé analysis.

I explicitly show that a statement made recently by Nucci and Clarkson in this journal on whether the non-classical method is more general than the direct method for the Fitzhugh-Nagumo equation can be completed using the Painlevé analysis. My conjecture is that the direct method combined with the singular manifold method can be equivalent to the n...

A generalization of the damped kink equation with a spacetime dependent damping factor is considered. The exact general solutions are obtained and some particular features concerning an infinite set of multikink solutions arising from the general solution are described. The physical significance of such a multikink solution is also discussed.

The most general classical solution with finite action of the N = 1 Gross-Neveu model is found. The use of conformal invariance and the toroidal (S1 ⊗ S1)-formalism based upon it are crucial for obtaining these general spinor field configurations.

Exact expressions for the Q-ball configurations in (1+1) dimensions are given using an infinite family of potentials that adiabatically approach a situation exhibiting spontaneous breakdown of the symmetry. The energy and Q-charge are also given analytically. Some interesting features appearing near the transition point to the kink phase are discus...

The general elliptic solution for the kink equation with damping is presented. Some properties of the damped kink are discussed with special emphasis on the relationship between friction and the kink velocity.

The general vacuum solutions of the Brans-Dicke theory in a cosmological Robertson-Walker-type metric are explicitly given. Several families of solutions have properties which essentially differ from the conventional Einstein theory. The geometry is not uniquely determined by the equations of motion, raising doubts about the Machian character of th...

We propose a fully conformal invariant describing gravity as a spontaneously broken theory. Newton's constant is automatically generated. We find through the study of classical solutions of the equations of motion that the breakdown of conformal symmetry can take place at the tree approximation without introducing arbitrary forms for the scalar pot...

The long-standing problem of cosmological constants arising from unified theories in particle physics, on the one side, and from purely gravitational phenomena, on the other side, is studied in the framework of the gauge-covariant theory of gravitation. The notion of system of units depending upon the corresponding interaction as well as the system...

In this paper the Singular Manifold Method has allowed us to obtain the Lax pair, Darboux transformations and τ functions for a non-linear Schrodiger equation in 2 + 1 dimensions. In this way we can iteratively build different kind of solutions with solitonic behavior.

We report here on how to improve the singular manifold method based on Painlevé analysis when the equation has more than one Painlevé branch. In particular, we apply it to an equation in (2+1)-dimensions. We obtain the Darboux transformations and an iterative procedure to obtain multisolitonic solutions, closely related to the Hirota method.