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    Authors: Álvarez Nodarse, Renato; Arvesú Carballo, Jorge; Yáñez, Rafael José;

    27 pages, no figures.-- MSC2000 code: 33D45. MR#: MR1824666 (2002f:33025) Zbl#: Zbl 0983.33009 In the present paper, starting from the second-order difference hypergeometric equation on the non-uniform lattice x(s) satisfied by the set of discrete hypergeometric orthogonal q-polynomials {pn}, we find analytical expressions of the expansion coefficients of any q-polynomial rm(x(s)) on x(s) and of the product rm(x(s))qj(x(s)) in series of the set {pn}. These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric q-polynomials. This work has been partially supported by the European project INTAS-93-219-ext as well as by the Dirección General de Enseñanza Superior (DGES) of Spain under Grant BHA 2000-0206-C04-02 (R.A.N., J.A.) and PB 95-1205 (R.J.Y.) and by the Junta de Andalucía (R.J.Y.) under Grant FQM207. Publicado

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    In this work we construct the wobble exact solution of sine-Gordon equation by means of B acklund Transformations. We nd the parameters of the transformations corresponding to the Bianchi diagram for the wobble as a particular 3-soliton solutions. We show that this solution agrees with the wobbles obtained by K albermann and Segur by means of the Inverse Scattering Transform, and by Ferreira et al. using the Hirota method. The new formulation introduced allows to identify easily the parameters that de ne the building blocks of this solution - a kink and a breather, and can be used in further studies of this solution in the perturbed sine-Gordon equation. We acknowledge nancial support from MICINN through grants MOSAICO (SC, AS) and FIS2008-02380/FIS (NRQ), and grants FQM207 and P06-FQM-01735 (NRQ) (from Junta de Andalucía, Spain), and SIMUMAT-CM (AS) (from Comunidad de Madrid, Spain). Publicado

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    Discrete & Continuous Dynamical Systems - S
    Article . 2011 . Peer-reviewed
    License: CC BY
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      Discrete & Continuous Dynamical Systems - S
      Article . 2011 . Peer-reviewed
      License: CC BY
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    We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation with an external time-independent force of the form f(x)=rexp(−iKx). Here the solitons travel with an oscillating velocity and all other characteristics of the solitons (amplitude, width, momentum, and phase) also oscillate. This behavior was predicted by a collective variable theory and confirmed by simulations. However, the reason for these oscillations remains unclear. Moreover, the spectrum of the oscillations exhibits a second strong peak, in addition to the intrinsic soliton peak. We show that the additional frequency belongs to a certain extended linear mode (which we refer to as a phonon for short) close to the lower band edge of the phonon continuum. Initially the soliton is at rest. When it starts to move it is deformed, begins to oscillate, and excites the above phonon mode such that the total momentum in a certain moving frame is conserved. In this frame the phonon does not move. However, not only does the soliton move in the homogeneous, time-periodic field of the phonon, but it also oscillates. Junta de Andalucia IAC11-III-11965 MICINN FIS2011-24540, Junta de Andalucia Projects No. FQM207, No. P11-FQM7276, and No. P09-FQM-4643

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    Physical Review E
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    Physical Review E
    Article . 2013 . Peer-reviewed
    License: APS Licenses for Journal Article Re-use
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      Physical Review E
      Article . 2013 . Peer-reviewed
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    16 pages, no figures.-- MSC1991 codes: 33C45; 42C05. MR#: MR1625951 (99i:33012) Zbl#: Zbl 0944.33011 Let us consider an arbitrary hypergeometric polynomial $q_j(x)$ and a set of orthogonal hypergeometric polynomials $\{p_n(x)\}$ in the domain of orthogonality $\Gamma$. Here the expansion coefficients of $x^m$ and $x^mq_j(x),\ m\in{\bf N}_0$, in series of the set $\{p_n(x)\}$ are found in terms of the polynomials $\sigma(x)$ and $\tau(x)$ characterizing the second-order differential equations satisfied by the hypergeometric polynomials involved. The resulting general expressions, which are given in an explicit and compact form, are used to produce known (for checking) and unknown expansions for various concrete classical orthogonal polynomials. This work has been partially supported by the European project INTAS-93-219-ext. The first author also acknowledges the partial financial support of the Fundació Aula (Barcelona, Spain). The second author has been also partially supported by the Dirección General de Enseñanza Superior (DGES) of Spain under grant PB 95-1205 and by the Junta de Andalucía FQM207. Publicado

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    Journal of Computational and Applied Mathematics
    Article . 1998 . Peer-reviewed
    License: Elsevier Non-Commercial
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      Journal of Computational and Applied Mathematics
      Article . 1998 . Peer-reviewed
      License: Elsevier Non-Commercial
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    Authors: Quintero, Niurka R.; Cuesta, José A.; Álvarez Nodarse, Renato;

    Equations describing the evolution of particles, solitons, or localized structures, driven by a zero-average, periodic, external force, and invariant under time reversal and a half-period time shift, exhibit a ratchet current when the driving force breaks these symmetries. The biharmonic force f(t)=ϵ1 cos(qωt+ϕ1)+ϵ2 cos(pωt+ϕ2) does it for almost any choice of ϕ1 and ϕ2, provided p and q are two coprime integers such that p+q is odd. It has been widely observed, in experiments in semiconductors, in Josephson junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape v∝ϵp1ϵq2 cos(pϕ1−qϕ2+θ0) for small amplitudes, where θ0 depends on the damping (θ0=π/2 if there is no damping, and θ0=0 for overdamped systems). We rigorously prove that this precise shape can be obtained solely from the broken symmetries of the system and is independent of the details of the equation describing the system. Ministerio de Educación y Ciencia MTM2006-1300-C03-01 Ministerio de Educación y Ciencia MTM2009-12740-C03-02 (R.A.N.) Ministerio de Educación y Ciencia FIS2008-02380/FIS(N.R.Q.), and MOSAICO(J.A.C.)) Junta de Andalucía FQM262(R.A.N.) Junta de Andalucía FQM207(N.R.Q.) Junta de Andalucía P06-FQM-01735(N.R.Q. and R.A.N.) Cominidad de Madrid MODELICO-CM (J.A.C.)

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    Authors: Fred Cooper; Fred Cooper; Sihong Shao; Franz G. Mertens; +3 Authors

    We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar selfinteraction in the presence of external forces as well as damping of the form f (x, t) - ιμγͦψ, where both f and ψ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. This approach predicts intrinsic oscillations of the solitary waves, i.e. the amplitude, width and phase all oscillate with the same frequency. The translational motion is also a ected, because the soliton position oscillates around a mean trajectory. We then compare the results of the variational approximation with numerical simulations of the NLD equation, and nd a good agreement, if we take into account a certain linear excitation with speci c wavenumber that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved. We also solve explicitly the CC equations of the variational approximation in the non-relativistic regime for a homogeneous external force and obtain excellent agreement with the numerical solution of the CC equations. Alexander von Humboldt Foundation (Germany) through Research Fellowship for Experienced Researchers SPA 1146358 STP Junta de Andalucía (Spain) under Projects No. FQM207, No. P06-FQM-01735, and No. P09-FQM-4643 Theoretical Division and Center for Nonlinear Studies at Los Alamos National Laboratory National Natural Science Foundation of China (Nos. 11471025, 91330110, 11421101) Mathematical Institute of the University of Seville (IMUS) Department of Atomic Energy, Government of India Plan Propio of the University of Seville MICINN (Spain) through FIS2014-54497-P United States Department of Energy

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    Journal of Physics A Mathematical and Theoretical
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    Journal of Physics A Mathematical and Theoretical
    Article . 2016 . Peer-reviewed
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    Authors: Álvarez Nodarse, Renato; Arvesú Carballo, Jorge; Yáñez, Rafael José;

    27 pages, no figures.-- MSC2000 code: 33D45. MR#: MR1824666 (2002f:33025) Zbl#: Zbl 0983.33009 In the present paper, starting from the second-order difference hypergeometric equation on the non-uniform lattice x(s) satisfied by the set of discrete hypergeometric orthogonal q-polynomials {pn}, we find analytical expressions of the expansion coefficients of any q-polynomial rm(x(s)) on x(s) and of the product rm(x(s))qj(x(s)) in series of the set {pn}. These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric q-polynomials. This work has been partially supported by the European project INTAS-93-219-ext as well as by the Dirección General de Enseñanza Superior (DGES) of Spain under Grant BHA 2000-0206-C04-02 (R.A.N., J.A.) and PB 95-1205 (R.J.Y.) and by the Junta de Andalucía (R.J.Y.) under Grant FQM207. Publicado

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    In this work we construct the wobble exact solution of sine-Gordon equation by means of B acklund Transformations. We nd the parameters of the transformations corresponding to the Bianchi diagram for the wobble as a particular 3-soliton solutions. We show that this solution agrees with the wobbles obtained by K albermann and Segur by means of the Inverse Scattering Transform, and by Ferreira et al. using the Hirota method. The new formulation introduced allows to identify easily the parameters that de ne the building blocks of this solution - a kink and a breather, and can be used in further studies of this solution in the perturbed sine-Gordon equation. We acknowledge nancial support from MICINN through grants MOSAICO (SC, AS) and FIS2008-02380/FIS (NRQ), and grants FQM207 and P06-FQM-01735 (NRQ) (from Junta de Andalucía, Spain), and SIMUMAT-CM (AS) (from Comunidad de Madrid, Spain). Publicado

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    Discrete & Continuous Dynamical Systems - S
    Article . 2011 . Peer-reviewed
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      Discrete & Continuous Dynamical Systems - S
      Article . 2011 . Peer-reviewed
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    We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation with an external time-independent force of the form f(x)=rexp(−iKx). Here the solitons travel with an oscillating velocity and all other characteristics of the solitons (amplitude, width, momentum, and phase) also oscillate. This behavior was predicted by a collective variable theory and confirmed by simulations. However, the reason for these oscillations remains unclear. Moreover, the spectrum of the oscillations exhibits a second strong peak, in addition to the intrinsic soliton peak. We show that the additional frequency belongs to a certain extended linear mode (which we refer to as a phonon for short) close to the lower band edge of the phonon continuum. Initially the soliton is at rest. When it starts to move it is deformed, begins to oscillate, and excites the above phonon mode such that the total momentum in a certain moving frame is conserved. In this frame the phonon does not move. However, not only does the soliton move in the homogeneous, time-periodic field of the phonon, but it also oscillates. Junta de Andalucia IAC11-III-11965 MICINN FIS2011-24540, Junta de Andalucia Projects No. FQM207, No. P11-FQM7276, and No. P09-FQM-4643

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    Physical Review E
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    Physical Review E
    Article . 2013 . Peer-reviewed
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      Physical Review E
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    16 pages, no figures.-- MSC1991 codes: 33C45; 42C05. MR#: MR1625951 (99i:33012) Zbl#: Zbl 0944.33011 Let us consider an arbitrary hypergeometric polynomial $q_j(x)$ and a set of orthogonal hypergeometric polynomials $\{p_n(x)\}$ in the domain of orthogonality $\Gamma$. Here the expansion coefficients of $x^m$ and $x^mq_j(x),\ m\in{\bf N}_0$, in series of the set $\{p_n(x)\}$ are found in terms of the polynomials $\sigma(x)$ and $\tau(x)$ characterizing the second-order differential equations satisfied by the hypergeometric polynomials involved. The resulting general expressions, which are given in an explicit and compact form, are used to produce known (for checking) and unknown expansions for various concrete classical orthogonal polynomials. This work has been partially supported by the European project INTAS-93-219-ext. The first author also acknowledges the partial financial support of the Fundació Aula (Barcelona, Spain). The second author has been also partially supported by the Dirección General de Enseñanza Superior (DGES) of Spain under grant PB 95-1205 and by the Junta de Andalucía FQM207. Publicado

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    Journal of Computational and Applied Mathematics
    Article . 1998 . Peer-reviewed
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      Journal of Computational and Applied Mathematics
      Article . 1998 . Peer-reviewed
      License: Elsevier Non-Commercial
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    Authors: Quintero, Niurka R.; Cuesta, José A.; Álvarez Nodarse, Renato;

    Equations describing the evolution of particles, solitons, or localized structures, driven by a zero-average, periodic, external force, and invariant under time reversal and a half-period time shift, exhibit a ratchet current when the driving force breaks these symmetries. The biharmonic force f(t)=ϵ1 cos(qωt+ϕ1)+ϵ2 cos(pωt+ϕ2) does it for almost any choice of ϕ1 and ϕ2, provided p and q are two coprime integers such that p+q is odd. It has been widely observed, in experiments in semiconductors, in Josephson junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape v∝ϵp1ϵq2 cos(pϕ1−qϕ2+θ0) for small amplitudes, where θ0 depends on the damping (θ0=π/2 if there is no damping, and θ0=0 for overdamped systems). We rigorously prove that this precise shape can be obtained solely from the broken symmetries of the system and is independent of the details of the equation describing the system. Ministerio de Educación y Ciencia MTM2006-1300-C03-01 Ministerio de Educación y Ciencia MTM2009-12740-C03-02 (R.A.N.) Ministerio de Educación y Ciencia FIS2008-02380/FIS(N.R.Q.), and MOSAICO(J.A.C.)) Junta de Andalucía FQM262(R.A.N.) Junta de Andalucía FQM207(N.R.Q.) Junta de Andalucía P06-FQM-01735(N.R.Q. and R.A.N.) Cominidad de Madrid MODELICO-CM (J.A.C.)

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    Authors: Fred Cooper; Fred Cooper; Sihong Shao; Franz G. Mertens; +3 Authors

    We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar selfinteraction in the presence of external forces as well as damping of the form f (x, t) - ιμγͦψ, where both f and ψ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. This approach predicts intrinsic oscillations of the solitary waves, i.e. the amplitude, width and phase all oscillate with the same frequency. The translational motion is also a ected, because the soliton position oscillates around a mean trajectory. We then compare the results of the variational approximation with numerical simulations of the NLD equation, and nd a good agreement, if we take into account a certain linear excitation with speci c wavenumber that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved. We also solve explicitly the CC equations of the variational approximation in the non-relativistic regime for a homogeneous external force and obtain excellent agreement with the numerical solution of the CC equations. Alexander von Humboldt Foundation (Germany) through Research Fellowship for Experienced Researchers SPA 1146358 STP Junta de Andalucía (Spain) under Projects No. FQM207, No. P06-FQM-01735, and No. P09-FQM-4643 Theoretical Division and Center for Nonlinear Studies at Los Alamos National Laboratory National Natural Science Foundation of China (Nos. 11471025, 91330110, 11421101) Mathematical Institute of the University of Seville (IMUS) Department of Atomic Energy, Government of India Plan Propio of the University of Seville MICINN (Spain) through FIS2014-54497-P United States Department of Energy

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    Journal of Physics A Mathematical and Theoretical
    Article
    License: CC BY NC ND
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    Journal of Physics A Mathematical and Theoretical
    Article . 2016 . Peer-reviewed
    Data sources: Crossref
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