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Lecture Notes for the matter "Mathematical Methods II" of the second course of the Bachelor in Physics in Universidad Complutense. It treats linear partial differential equations, separation of variables, symmetric operators in Hilbert space, Fourier analysis, eigenfunction expansion method, special functions, and the Helmholtz equation in curvilinear coordinates

©IOP Publishing Ltd. The authors wish to thank the Spanish Ministerio de Educación y Ciencia (research project FIS2005-00319) and the European Science Foundation (MISGAM programme) for their support., Critical points of semiclassical expansions of solutions to the dispersionful Toda hierarchy are considered and a double scaling limit method of regularization is formulated. The analogues of the critical points characterized by the strong conditions in the Hermitian matrix model are analysed and the property of doubling of equations is proved. A wide family of sets of critical points is introduced and the corresponding double scaling limit expansions are discussed., Spanish Ministerio de Educación y Ciencia, European Science Foundation (MISGAM programme), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© Elsevier Science Bv. We thank Prof. A. Martínez Finkelshtein for useful conversations and for calling our attention to the work [20]. The financial support of the Ministerio de Ciencia e Innovación under project FIS2011-22566 is gratefully acknowledged., We apply the method we have described in a previous paper (2013) to determine the phase structure of asymptotic zero densities of the standard cubic model of non-Hermitian orthogonal polynomials. We provide a complete description of the two phases: the one cut phase and the two cut phase, and analyze the phase transition processes of the types: splitting of a cut, birth and death of a cut. (C) 2014 Elsevier B.V. All rights reserved., Ministerio de Ciencia e Innovacion, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing Ltd. The financial support of the Ministerio de Ciencia e Innovación under project IS2011-22566 is gratefully acknowledged, We present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of Penner random matrix models, to obtain their explicit forms in the exactly solvable cases, and to determine the coefficients of their perturbative expansions in the continuum limit. The method relies on identities satisfied by the resolvent of the Jacobi matrix in the three-term recursion relation of the associated families of orthogonal polynomials. These identities lead to a convenient formulation of the string equations. As an application, we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double Penner models can be written as a sum of Gaussian contributions plus linear terms. To illustrate the one-cut case we discuss the linear, double and cubic Penner models, and for the two- cut case we discuss theoretically and numerically the existence of a double-branch structure of the free energy for the Gaussian Penner model., Ministerio de Ciencia e Innovación (MICINN), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2005 Elsevier B.V. All rights reserved., A class of exact solutions of the dispersionless Toda hierarchy constrained by a string equation is obtained. These solutions represent deformations of analytic curves with a finite number of nonzero harmonic moments. The corresponding tau-functions are determined and the emergence of cusps is studied., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing . The authors wish to thank Prof. Y. Kodama for his interest and help during the elaboration of this work., A previously introduced scheme for describing integrable deformations of algebraic curves is completed. Lenard relations are used to characterize and classify these deformations in terms of hydrodynamic-type systems. A general solution of the compatibility conditions for consistent deformations is given and expressions for the solutions of the corresponding Lenard relations are provided., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©1985 The American Physical Society. The author wishes to thank G. Garcia Alcaine for helpful conversations. Partial financial support by Comision Asesora de Investigacion Cientifica y Tecnica is gratefully acknowledged., A method is given for finding the shifts in position of the solitons for the case of nonzero reflection coefficient. Expressions for boost generators in terms of scattering data play a prominent role in the analysis. Phase-shift formulas which show the effect of the radiation component on the soliton motion are deduced for the nonlinear Schrodinger equation, the Korteweg —de Vries equation, and the sine-Gordon equation., Comision Asesora de Investigacion Cientifica y Tecnica, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing Ltd. LMA and EM are grateful to G Alvarez for many useful conversations on the subject of spectral curves in gauge/string dualities. The financial support of the Universidad Complutense under project GR58/08-910556, the Comision Interministerial de Ciencia y Tecnología under project FIS2011-22566 and PRIN 2008 grant no. 28002K9KXZ are gratefully acknowledged., We study the moduli space of the spectral curves y ^2 = W ‘ (z) ^2 + f(z) which characterize the vacua of N = 1 U(n) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential W(z). The integrable structure of the Whitham equations is used to determine the spectral curves from their moduli. An alternative characterization of the spectral curves in terms of critical points of a family of polynomial solutions W to Euler-Poisson-Darboux equations is provided. The equations for these critical points are a generalization of the planar limit equations for one-cut random matrix models. Moreover, singular spectral curves with higher order branch points turn out to be described by degenerate critical points of W. As a consequence we propose a multiple scaling limit method of regularization and show that, in the simplest cases, it leads to the Painlevè-I equation and its multi-component generalizations., Universidad Complutense de Madrid, Comisión Interministerial de Ciencia y Tecnología, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2008 Elsevier Ltd. All rights reserved. The authors wish to thank the Spanish Ministerio de Educación y Ciencia (research project FIS2005-00319) and the European Science Foundation (MISGAM programme) for their support., Critical processes of ideal integrable models of Hele-Shaw flows are considered. A regularization method based on multiscaling expansions of solutions of the KdV and Toda hierarchies characterized by string equations is proposed. Examples are exhibited in which the tritronquee solution of the Painleve-I equation turns out to provide the leading term of the regularization., Spanish Ministerio de Educación y Ciencia, European Science Foundation (MISGAM programme), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©Springer. The financial support of the Universidad Complutense under project GR58/08-910556 and the Ministerio de Ciencia e Innovación under projects FIS2008-00200 and FIS2011-22566 are gratefully acknowledged., We study the superpotentials, quantum parameter space and phase transitions that arise in the study of large N dualities between N = 1 SUSY U(N) gauge theories and string models on local Calabi-Yau manifolds. The main tool of our analysis is a notion of spectral curve characterized by a set of complex partial ’t Hooft parameters and cuts given by projections on the spectral curve of minimal supersymmetric cycles of the underlying Calabi-Yau manifold. We introduce a prepotential functional via a variational problem which determines the complex density as an extremal constrained by the period conditions. This prepotential is shown to satisfy the special geometry relations of the spectral curve. We give a system of equations for the branch points of the spectral curves and determine the appropriate branch cuts as Stokes lines of a suitable set of polynomials. As an application, we use a combination of analytical and numerical methods to study the cubic model, determine the analytic condition satisfied by critical one-cut spectral curves, and characterize the transition curves between the one-cut and two-cut phases both in the space of spectral curves and in the quantum parameter space., Universidad Complutense de Madrid, Ministerio de Ciencia e Innovación (MICINN), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2011 IOP Publishing Ltd. The financial support of the Universidad Complutense under project GR35/10-A910556, the Comision Interministerial de Ciencia y Tecnología under projects FIS2008-00200 and FIS2008-00209 are gratefully acknowledged., We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlevé I and Painlevé II hierarchies associated with one-cut and two-cut critical models, respectively., Universidad Complutense, Comisión Interministerial de Ciencia y Tecnología, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©1989 American Institute of Physics. M. M. and L. M. A. wish to thank Professor P. C. Sabatier and the Laboratoire de Physique Mathematique de Montpellier for their warm hospitality., A multiseries integrable model (MSIM) is defined as a family of compatible flows on an infinite-dimensional Lie group of N-tuples of formal series around N given poles on the Riemann sphere. Broad classes of solutions to a MSIM are characterized through modules over rings of rational functions, called asymptotic modules. Possible ways for constructing asymptotic modules are Riemann-Hilbert and (j problems. When MSIM's are written in terms of the "group coordinates," some of them can be "contracted" into standard integrable models involving a small number of scalar functions only. Simple contractible MSIM's corresponding to one pole, yield the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. Two-pole contractible MSIM's are exhibited, which lead to a hierarchy of solvable systems of nonlinear differential equations consisting of (2 + 1 )-dimensional evolution equations and of quite strong differential constraints., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2004 Plenum Publishing Corporation. The authors thank Professor E. V. Ferapontov for the useful discussions. This work was supported by the Russian Foundation for Basic Research (Grant No. 04-01-00403, A. B. S.), the Presidential Program for Supporting Leading Scientific Schools (Grant No. NSh 1716.2003.1, A. B. S.), the Ministerio de Cultura y Deporte of Spain (grant to A. B. S.) and the DGCYT (Project No. BFM2002-01607, L. M. A.)., We investigate the diagonal hydrodynamic reductions of a hierarchy of integrable hydrodynamic chains and analyze their compatibility with previously introduced reductions of differential type., Russian Foundation for Basic Research, Presidential Program for Supporting Leading Scientific Schools, Ministerio de Cultura y Deporte of Spain, DGCYT, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©1995 American Institute of Physics. The authors would like to thank Professor A. Ibort for many helpful conversations. The financial support of the CICYT is also acknowledged., We consider the Manin-Radul and Jacobian supersymmetric KP hierarchies from the point of view of the tau-function formalism. Solutions of their associated systems of Sato equations are characterized in terms of correlation functions of supersymmetric vertex operators of superghost type. The expression of the wave functions of these hierarchies in terms of tau-functions is obtained and the corresponding bilinear identities are established. Explicit methods for generating soliton and rational solutions are given., CICYT, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2003 American Institute of Physics. The authors are grateful to Professor F. Guil for showing them the relevance of enerating functions for canonical transformations of twistor data to get solutions of the dKP hierarchy. This work was partially supported by CICYT Project No. BFM2002-01607., 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., CICYT, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2006 American Institute of Physics. Partial economical supports from the Spanish DGCYT Program No. FIS2005-00319 and from the European Science Foundation Program No. MISGAM are acknowledged, A scheme for solving Whitham hierarchies satisfying a special class of string equations is presented. The tau-function of the corresponding solutions is obtained and the differential expressions of the underlying Virasoro constraints are characterized. Illustrative examples of exact solutions of Whitham hierarchies are derived and applications to conformal maps dynamics are indicated., DGCYT, Spain, European Science Foundation (MISGAM programme), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing. The authors are grateful to Prof. Manuel Mañas for many useful discussions., Solutions of the Riemann-Hilbert problem implementing the twistorial structure of the dispersionless Toda (dToda) hierarchy are obtained. Two types of string equations are considered which characterize solutions arising in hodograph sectors and integrable structures of two-dimensional quantum gravity and Laplacian growth problems., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©Atlantis Press. The authors wish to thank the Spanish Ministerio de Educacion y Ciencia (research project FIS2005-00319) and the European Science Foundation (MISGAM programme) for their financial support., It is proved that the system of string equations of the dispersionless 2-Toda hierarchy which arises in the planar limit of the hermitian matrix model also underlies certain processes in HeleShaw flows., Spanish Ministerio de Educacion y Ciencia, European Science Foundation (MISGAM programme), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing. The authors wish to thank the Spanish Ministerio de Ciencia e Innovación, research projects FIS2005-00319 and FIS2008-00200, and acknowledge the support received from the European Science Foundation (ESF) and the activity entitled Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM). This paper was finished during the research visits of one of the authors (MM) to the Université Catholique de Louvain and to the Scuola Internazionale Superiore di Studi Avanzati/International School for Advanced Studies (SISSA) in Trieste, MM wish to thanks Prof. van Moerbeke and Prof. Dubrovin for their warm hospitality, acknowledge economical support from MISGAM and SISSA and reckons different conversations with P. van Moerbeke, T. Grava, G. Carlet and M. Caffasso., The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated with an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov-Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov-Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site-independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations., Ministerio de Ciencia e Innovación (MICINN), European Science Foundation (ESF), MISGAM, SISSA, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2003 IOP Publishing Ltd. This work originated during the stay of the authors at the Isaac Newton Institute for the Mathematical Sciences of Cambridge University as participants of the programme ‘Integrable Systems’. The authors are grateful to the organizers for the support provided. They also acknowledge S P Tsarev and A Mikhailov for useful comments and conversations. The work is partially supported by CICYT proyecto PB98–0821., A general scheme for analysing reductions of Whitham hierarchies is presented. It is based on a method for determining the S-function by means of a system of first-order partial differential equations. Compatibility systems of differential equations characterizing both reductions and hodograph solutions of Whitham hierarchies are obtained. The method is illustrated by exhibiting solutions of integrable models such as the dispersionless Toda equation (heavenly equation) and the generalized Benney system., CICYT, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©1985 The American Physical Society. This work was financially supported by the Comision Asesora de Investigacion Cientifica y Tecnica., The problem of the soliton motion in the case of a nonzero reflection coefficient is solved exactly within the framework of the inverse scattering method. A general method is given for obtaining the asymptotic expressions of the soliton angle variables. As illustrative examples of our procedure, we consider the Korteweg-de Vries equation and its higher analogs, the nonlinear Schrodinger equation, and the sine-Gordon equation., Comision Asesora de Investigacion Cientifica y Tecnica, Spain, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2010 Elsevier B.V. All rights reserved. The authors wish to thank the Spanish Ministerio de Educación y Ciencia (research project FIS2008-00200/FIS) for its finantial support. B. K. is thankful to the Departamento de Física Teórica II for the kind hospitality, The one-cut case of the Hermitian random matrix model in the large N limit is considered. Its singular sector in the space of coupling constants is analyzed from the point of view of the hodograph equations of the underlying dispersionless Toda hierarchy. A deep connection with the singular sector of the hodograph equations of the 1-layer Benney (classical long wave equation) hierarchy is stablished. This property is a consequence of the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler- Poisson-Darboux equations., Spanish Ministerio de Educación y Ciencia, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 1984 The American Physical Society. I am grateful to A. Fernandez Ranada for many helpful discussion. s and advice. This work was supported in part by the Cornision Asesora de Investigacion Cientifica y Tecnica., The relationship between the soliton dynamics provided by the classical sine-Gordon and massive Thirring models is exhibited. Solitons are characterized as classical relativistic particles through the consideration of their associated canonical realizations of the Poincare group. It is shown that the soliton in the massive Thirring model determines two different kinds of relativistic particles from which sine-Gordon kinks and breathers may be reproduced. In particular, sine-Gordon breathers are characterized as composite systems of two solitons of the massive Thirring model. Soliton scattering in the sine-Gordon equation is described in terms of soliton scattering in the massive Thirring model., Cornision Asesora de Investigacion Cientifica y Tecnica., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©Amer Physical Soc. It is a pleasure to thank Professor P. C. Sabatier and the group of the Laboratoire de Physique Mathematique of Montpellier University for warm hospitality while this work was in progress. Partial financial support from the Comision Asesora de Investigacion Cientifica y Tecnica, Spain is also acknowledged., The effect of the radiation modes on soliton motion in nonlinear lattices is investigated. A method based on the inverse scattering transform is developed, which enables us to characterize the position shifts of solitons due to their interaction with the radiation component. Applications to the Toda and, Comision Asesora de Investigacion Cientifica y Tecnica, Spain, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2002 Elsevier Science B.V. All rights reserved. A.B. Shabat was supported by the Russian Foundation for Basic Research (Grant Nos. 96-15-96093 and 98-01- 01161), INTAS (Grant No. 99-1782) and a Rotschild professorship. L. Martı́nez Alonso was supported by the Fundación Banco Bilbao Vizcaya Argentaria. Both authors wish to thank the organizers of the program Integrable Systems at the Newton Institute of Cambridge University for their warm hospitality., A hierarchy of infinite-dimensional systems of hydrodynamic type is considered and a general scheme for classifying its reductions is provided. Wide families of integrable systems including, in particular, those associated with energy-dependent spectral problems of Schrodinger type, are characterized as reductions of this hierarchy. N-phase type reductions and their corresponding Dubrovin equations are analyzed. A symmetry transformation connecting different classes of reductions is formulated., Russian Foundation for Basic Research, INTAS, Fundación Banco Bilbao Vizcaya Argentaria, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2005 Springer Science+Business Media, Inc. One of the authors (L. M. A.) thanks the members of the Physics Department of Lecce University for their warm hospitality. This work was supported in part by the DGCYT (Project No. BFM2002-01607), COFIN (Grant 2002 “Sintesi”), and the National Science Foundation (Grant No. DMS-0404931)., We present a general scheme for determining and studying integrable deformations of algebraic curves, based on the use of Lenard relations. We emphasize the use of several types of dynamical variables: branches, power sums, and potentials., DGCYT, COFIN, National Science Foundation, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2003 IOP Publishing Ltd. This work is partially supported by the DGESIC Proyecto PB98-0821., 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., DGESIC, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2005 American Institute of Physics. One of the authors (L.M.A.) wishes to thank the members of the Physics Department of Lecce University for their warm hospitality. This work is partially supported by DGCYT Project BFM 2002-01607 and by the grant COFIN 2004 “Sintesi” One of the authors (Y.K.) is partially supported by NSF Grant No. DMS 0404931, A general scheme for determining and studying hydrodynamic type systems describing integrable deformations of algebraic curves is applied to cubic curves. Lagrange resolvents of the theory of cubic equations are used to derive and characterize these deformations., DGCYT, COFIN, NSF, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing Ltd. The authors wish to thank the Spanish Ministerio de Educación y Ciencia (research project FIS2008-00200/FIS) for its finantial support. This work is also part of the MISGAM programme of the European Science Foundation., The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated with pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the phase space of the Whitham hierarchy of dispersionless integrable systems is provided. Applications to the analysis of the large-n limit of multiple orthogonal polynomials and their associated random matrix ensembles and models of non-intersecting Brownian motions are given., Spanish Ministerio de Educación y Ciencia, European Science Foundation (MISGAM programme), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing Ltd. The authors wish to thank the Spanish Ministerio de Educación y Ciencia (research project FIS2005-00319) for its finantial support. This work is also part of the MISGAM programme of the European Science Foundation., An iterative algorithm for determining a class of solutions of the dispersionful 2-Toda hierarchy characterized by string equations is developed. This class includes the solution which underlies the large N-limit of the Hermitian matrix model in the one-cut case. It is also shown how the double scaling limit can be naturally formulated in this scheme, Spanish Ministerio de Educación y Ciencia, European Science Foundation (MISGAM programme), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2011 Elsevier B.V. The financial support of the Universidad Complutense under project GR58/08-910556 and the Comisión Interministerial de Ciencia y Tecnología under projects FIS2008-00200 and FIS2008-00209 are gratefully acknowledged., We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher-Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach., Universidad Complutense, Comision Interministerial de Ciencia y Tecnologia, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©Springer. This work was supported by the Spanish Ministerio de Educación y Ciencia (Research Project No. FIS2008-00200/FIS)., We completely describe the singular sectors of the one-layer Benney system (classical long-wave equation) and dispersionless Toda system. The associated Euler-Poisson-Darboux equations E(1/2, 1/2) and E(-1/2,-1/2) are the main tool in the analysis. We give a complete list of solutions of the one-layer Benney system depending on two parameters and belonging to the singular sector. We discuss the relation between Euler-Poisson-Darboux equations E(ɛ, ɛ) with the opposite sign of ɛ., Ministerio de Educación y Ciencia, Spain, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing. We thank Prof. A. Martínez Finkelshtein for useful conversations and for calling our attention to the work [6]. The financial support of the Ministerio de Ciencia e Innovación under projects FIS2008-00200 and FIS2011-22566 is gratefully acknowledged., This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model., Ministerio de Ciencia e Innovación, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©1992 American institute of Physics. The authors would like to thank Professor M. Jaulent for many helpful conversations and the Laboratoire de Physique-Mathematique of Montpellier for kind hopitality while part of this work was done. Financial support from the Direcci6n General de Investigacibn Cientifica y Ticnica DGICYT is also aknowledged., The DaveyStewartson equation is considered from the point of view of the bilinear formalism of the Kyoto school. Multidromion solutions are constructed in terms of free fermions and their asymptotic properties are characterized. The dynamical properties of dromions are discussed., DGICYT, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©1998 Elsevier Science B.V., We show how Ramond free neutral Fermi fields lead to a Ƭ-function theory of BKP type which describes iso-orthogonal deformations of systems of orthogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©Atlantis Press. A B Shabat wish to thank A Ibort for his support during his stay as a visiting professor of the Carlos III University of Madrid. He also acknowledges the group of the Department of Theoretical Physics II of the Complutense University for their warm hospitality. L Martínez Alonso was partially supported by the DGCYT project BFM2002- 01607. A Shabat was partially supported by the RFFI project 01-01-00874 and Sc. Schools RF grant 06- 15-96093. Both authors are grateful to E Ferapontov for several useful and stimulating discussions., We present a theory of compatible differential constraints of a hydrodynamic hierarchy of infinite-dimensional systems. It provides a convenient point of view for studying and formulating integrability properties and it reveals some hidden structures of the theory of integrable systems. Illustrative examples and new integrable models are exhibited., DGCYT, RFFI, Sc. Schools RF, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing. The authors wish to thank the Spanish Ministerio de Ciencia e Innovaciòn, research project FIS2008- 00200, and acknowledge the support received from the European Science Foundation (ESF) and the activity entitled Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM). MM wish to thank Prof. van Moerbeke and Prof. Dubrovin for their warm hospitality, acknowledge economical support from MISGAM and SISSA and reckons different conversations with P. van Moerbeke, T. Grava, G. Carlet and M. Caffasso. MM also acknowledges to Prof. Liu for his invitation to visit the China Mining and Technology University at Beijing., The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov-Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and Toda pictures of the dispersionless Whitham hierarchy emerge in the dispersionless limit. Moreover, the additional symmetries and string equations for the dispersive Whitham hierarchy are studied in this limit., Ministerio de Ciencia e Innovacion, Spain, European Science Foundation (ESF), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2006 Elsevier B.V. Partially supported by MEC project FIS2005-00319, A scheme for solving quasiclassical-string equations is developed to prove that genus-zero hitham hierarchies describe the deformations of planar domains determined by rational conformal-maps. This property is applied in normal matrix models to show that deformations of simply-connected supports of eigenvalues under changes of coupling constants are governed by genus-zero Whitham hierarchies., Ministerio de Educación y Ciencia, Spain, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©2002 Plenum Publishing Corporation. International Conference on Nonlinear Evolution Equations and Dynamical Systems. (15. 2001. Cambridge, Inglaterrad). One of the authors (B. K.) is grateful to the organizers of the program “Integrable Systems” for the support and was also supported in part by the COFIN 2000 “Sintesi.” The stay of L. M. A. at Cambridge University as a BBV visiting professor was supported by the Fundaci´on Banco Bilbao Vizcaya Argentaria. E. M. was supported in part by the CICYT (Project No. PB98-0821)., A ∂¯formalism for studying dispersionless integrable hierarchies is applied to the dispersionless KP (dKP) hierarchy. We relate this formalism to the theory of quasiconformal mappings on the plane and present some classes of explicit solutions of the dKP hierarch., COFIN, Fundación Banco Bilbao Vizcaya Argentaria, CICYT, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2022 American Physical Society. The financial support of the Spanish Ministerio de Economia y Competitividad under Project No. PGC2018094898-B-I00 is gratefully acknowledged., We analyze the emergence of classical inflationary universes in a kinetic-dominated stage using a suitable class of solutions of the Wheeler-DeWitt equation with a constant potential. These solutions are eigenfunctions of the inflaton momentum operator that are strongly peaked on classical solutions exhibiting either or both a kinetic-dominated period and an inflation period. Our analysis is based on semiclassical WKB solutions of the Wheeler-DeWitt equation interpreted in the sense of Borel (to perform a correct connection between classically allowed regions) and on the relationship of these solutions to the solutions of the classical model. For large values of the scale factor the WKB Vilenkin tunneling wave function and the Hartle-Hawking no-boundary wave functions are recovered as particular instances of our class of wave functions., Ministerio de Economia y Competitividad (MINECO), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

We determine generalised asymptotic solutions for the inflaton field, the Hubble parameter, and the equation-of-state parameter valid during the oscillatory phase of reheating for potentials that close to their global minima behave as even monomial potentials. For the quadratic potential, we derive a generalised asymptotic expansion for the inflaton with respect to the scale set by inverse powers of the cosmic time. For the quartic potential, we derive an explicit, two-term generalised asymptotic solution in terms of Jacobi elliptic functions, with a scale set by inverse powers of the square root of the cosmic time. In the general case, we find similar two-term solutions where the leading order term is defined implicitly in terms of the Gauss hypergeometric function. The relation between the leading terms of the instantaneous equation-of-state parameter and different averaged values is discussed in the general case. Finally, we discuss the physical significance of the generalised asymptotic solutions in the oscillatory regime and their matching to the appropriate solutions in the thermalization regime., Ministerio de Economía y Competitividad (MINECO), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2020 American Institute of Physics. The financial support of the Spanish Ministerio de Economia y Competitividad under Project Nos. FIS2015-63966-P, PGC2018-094898B-I00, and PGC2018-098440-B-I00 is gratefully acknowledged. J.L.V. would like to thank the Departamento de Analisis Matematico y Matematica Aplicada of the Universidad Complutense de Madrid for his appointment as Honorary Professor., We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann-Lemaitre-Robertson-Walker universe. The existence and properties of separatrices are investigated in the framework of the Hamilton-Jacobi formalism, where the main quantity is the Hubble parameter considered as a function of the inflaton field. A wide class of inflaton models that have separatrix solutions (and include many of the most physically relevant potentials) is introduced, and the properties of the corresponding separatrices are investigated, in particular, asymptotic inflationary stages, leading approximations to the separatrices, and full asymptotic expansions thereof. We also prove an optimal growth criterion for potentials that do not have separatrices., Ministerio de Economía y Competitividad (MINECO), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2020 American Physical Society. The financial support of the Spanish Ministerio de Ciencia, Innovacion y Universidades under Projects No. FIS2015-63966-P and No. PGC2018-094898-B-I00 is gratefully acknowledged. We thank Professor Gabriel Alvarez Galindo for fruitful discussions., Single-field inflaton models in the kinetic dominance period admit formal solutions given by generalized asymptotic expansions called psi series. We present a method for computing psi series for the Hubble parameter as a function of the inflaton field in the Hamilton-Jacobi formulation of inflaton models. Similar psi series for the scale factor, the conformal time, and the Hubble radius are also derived. They are applied to determine the value of the inflaton field when the inflation period starts and to estimate the contribution of the kinetic dominance period to calculate the duration of inflation. These psi series are also used to obtain explicit two-term truncated psi series near the singularity for the potentials of the Mukhanov-Sasaki equation for curvature and tensor perturbations. The method is illustrated with wide families of inflaton models determined by potential functions combining polynomial and exponential functions, as well as with generalized Starobinsky models., Ministerio de Ciencia e Innovación (MICINN), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© IOP Publishing Ltd. We wish to thank G. Alvarez for his help and many useful conversations. The financial support of the Spanish Ministerio de Economía y Competitividad under Project No. FIS2015-63966-P is also gratefully acknowledged., We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles., Ministerio de Economía y Competitividad (MINECO), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

©IOP Publishing Ltd. We thank Prof. A. Martínez Finkelshtein for calling our attention to many nice results on zero asymptotics of Laguerre and Jacobi polynomials. The financial support of the Spanish Ministerio de Economía y Competitividad under Project No. FIS2015-63966-P is gratefully acknowledged., The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes G function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of n. We show that the planar limit of the free energy with 't Hooft sequences does not exist. Therefore we use a certain modification that uses Kuijlaars-McLaughlin sequences instead of 't Hooft sequences and leads to a well-defined planar free energy and to an associated two-dimensional phase space. We describe the different configurations of complex saddle points of the holomorphic matrix integral both to the left and to the right of the critical point, and interpret the phase transitions in terms of processes of gap closing, eigenvalue tunneling, and Bose condensation., Ministerio de Economía y Competitividad (MINECO), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub