arXiv subjects

H. Blas

Publications and source records attributed to H. Blas.

At least 19 recordsLinked to original sources

arXiv2026

Variable-mass sine-Gordon with point defects: integrability, soliton transmission, and quasi-conservation

We study integrable variable-mass sine-Gordon model (vmSG) with point defects. Using the Lax and B\"acklund-gauge formulations, we construct type-I and type-II defect matrices, derive the corresponding sewing conditions, and generate the bulk and defect contributions to an infinite hierarchy of conserved charges. The lowest members reduce to the standard sine-Gordon energy and momentum in the homogeneous limit, while for inhomogeneous backgrounds they define integrability-generated energy- and momentum-type quantities. Defect compatibility imposes matching conditions on the variable-mass functions across the defect. An analytical transmission factor $z$ characterizes soliton transmission, topological conversion, and absorption/emission processes. We then deform the type-I sewing conditions by parameters $\alpha$ and $\beta$ and derive the associated defect anomalies. Consistent one-soliton transmission is recovered for $\alpha \beta =1$, whereas for $\alpha \beta \neq 1$ parity-centered kink-kink and kink-antikink transmissions display vanishing lowest order integrated anomalies but no generic vanishing at higher order. Numerical simulations reproduce the integrable transmission/conversion regimes and show that non-integrable defects generate weak radiative tails and lowest order quasi-conserved charges, while quasi-conservation of the higher order charges remains unestablished. These results elucidate how exact defect integrability deforms into charge-dependent quasi-conservation and establish a framework for defect-controlled soliton transport in inhomogeneous media, with potential applications to nonuniform Josephson junctions, magnetic and nonlinear-optical systems, and effective molecular and DNA models.

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arXiv2026

Heun-function analysis of the Dirac spinor spectrum in a sine-Gordon soliton background

We study the Dirac spectrum in a sine-Gordon soliton background, where the induced position-dependent mass reduces the spectral problem to a Heun-type differential equation. Bound and scattering sectors are treated within a unified framework, with spectral data encoded in Wronskians matching local Heun solutions and exhibiting explicit dependence on the soliton parameters and the bare fermion mass. This formulation enables a systematic analysis of spinor bound and scattering states, supported by analytic and numerical verification of wave function matching across the soliton domain. The present work is related to arXiv:2512.07658 and emphasizes a pedagogical treatment of scattering states within the Heun-equation formalism.

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arXiv2025

Modified open Toda chain and quasi-integrability

We present a study of a quasi-integrable deformation of the three-particle open Toda chain, constructed by introducing a translation-invariant three-body interaction terms. Although this modification explicitly breaks the exact integrability of the standard Toda model, it retains fundamental structural properties, including energy and momentum conservation. Furthermore, we show that under a specific time-reflection and discrete symmetry among the chain coordinates, the system admits a quasi-conserved higher-order integral. Through analytic and numerical analysis of the deformed dynamics, we demonstrate the emergence and long-time persistence of quasi-conserved quantities, thereby establishing a controlled realization of quasi-integrability in a minimal nonlinear chain. Given the central role of integrable systems in elucidating the dynamics of classical and quantum models, this framework provides a concrete setting to investigate the mechanisms underlying the gradual breakdown of integrability and the onset of quasi-integrability in few-body systems.

nlin.SI↗

arXiv2024

Fermionic back-reaction on kink and topological charge pumping in the $sl(2)$ affine Toda coupled to matter

We explore the Faddeev-Jackiw (F-J) symplectic Hamiltonian reduction of the $sl(2)$ affine Toda model coupled to matter (ATM), which includes new parametrizations for a scalar field and a Grassmannian fermionic field. The structure of constraints and symplectic potentials primarily dictates the strong-weak dual coupling sectors of the theory, ensuring the equivalence between the Noether and topological currents. The analytical calculations encompass the fermion-kink classical solution, the excited fermion bound states localized on the kink, and the scattering states, all of which account for the fermion back-reaction on the soliton. The total energy, which includes the classical fermion-soliton interaction energy, the bound-state fermion energy, and the fermion vacuum polarization energy (VPE), is determined by the topological charge of the kink. This system satisfies first-order differential equations and a chiral current conservation equation. Our results demonstrate that the excited fermion bound states and scattering states significantly alter the properties of the kink. Notably, they give rise to a pumping mechanism for the topological charge of the in-gap kink due to fermionic back-reaction, as well as the appearance of kink states in the continuum (KIC).

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arXiv2023

Zero mode-soliton duality and pKdV kinks in Boussinesq system for non-linear shallow water waves

A Boussinesq system for a non-linear shallow water is considered. The nonlinear and topological effects are examined through an associated matrix spectral problem. It is shown an equivalence relationship between the bound states and topological soliton charge densities which resembles a formula of the Atiyah-Patodi-Singer-type index theorem. The zero mode components describe a topologically protected Kelvin wave of KdV-type and a novel Boussinesq-type field. We show that either the $1+1$ dimensional pKdV kink or the Kelvin mode can be mapped to the bulk velocity potential in $2+1$ dimensions.

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arXiv2022

Majorana zero mode-soliton duality and in-gap and BIC bound states in modified Toda model coupled to fermion

A two-dimensional field theory of a fermion chirally coupled to Toda field plus a scalar self-coupling potential is considered. Using techniques of integrable systems we obtain analytical zero modes, in-gap states and bound states in the continuum (BIC) for topological configurations of the scalar field. Fermion-soliton duality mappings are uncovered for the bound state spectrum, which interpolates the weak and strong coupling sectors of the model and give rise to novel Thirring-like and multi-frequency sine-Gordon models, respectively. The non-perturbative effects of the back-reaction of the fermion bound states on the kink are studied and it is shown that the zero mode would catalyze the emergence of a new kink with lower topological charge and greater slope at the center, in the strong coupling limit of the model. For special topological charges and certain relative phases of the fermion components the kinks can host Majorana zero modes. The Noether, topological and a novel nonlocal charge densities satisfy a formula of the Atiyah-Patodi-Singer-type. Our results may find applications in several branches of non-linear physics, such as confinement in QCD$_2$, braneworld models, high $T_c$ superconductivity and topological quantum computation. We back up our results with numerical simulations for continuous families of topological sectors.

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arXiv2022

Modified AKNS model, Riccati-type pseudo-potential approach and infinite towers of quasi-conservation laws

A dual Riccati-type pseudo-potential formulation is introduced for a modified AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws are uncovered. In addition, infinite towers of exact non-local conservation laws are uncovered in a linear formulation of the system. It is shown that certain modifications of the non-linear Schrödinger model (MNLS) can be obtained through a reduction process starting from the MAKNS model. So, the novel infinite sets of quasi-conservation laws and related anomalous charges are constructed by an unified and rigorous approach based on the Riccati-type pseudo-potential method, for the standard NLS and modified MNLS cases, respectively. The non-local properties, the complete list of towers of infinite number of anomalous charges and the (non-local) exact conservation laws of the quasi-integrable systems, such as the deformed Bullough-Dodd, Toda, KdV and SUSY sine-Gordon systems can be studied in the framework presented in this paper. Our results may find many applications since the AKNS-type system arises in several branches of non-linear physics, such as Bose-Einstein condensation, superconductivity and soliton turbulence.

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arXiv2021

Modified non-linear Schrödinger models, ${\cal C}{\cal P}_s{\cal T}_d$ invariant $N-$bright solitons and infinite towers of anomalous charges

Modifications of the non-linear Schrödinger model (MNLS) $ i \partial_{t} ψ(x,t) + \partial^2_{x} ψ(x,t) - [\frac{δV}{δ|ψ|^2} ] ψ(x,t) = 0,$ where $ψ\in C$ and $V: R_{+} \rightarrow R$, are considered. We show that the MNLS models possess infinite towers of quasi-conservation laws for soliton-type configurations with a special complex conjugation, shifted parity and delayed time reversion (${\cal C}{\cal P}_s{\cal T}_d$) symmetry. Infinite towers of anomalous charges appear even in the standard NLS model for ${\cal C}{\cal P}_s{\cal T}_d$ invariant $N-$bright solitons. The true conserved charges emerge through some kind of anomaly cancellation mechanism. A dual Riccati-type pseudo-potential formulation is introduced for a modified AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws are uncovered. In addition, infinite towers of exact non-local conservation laws are uncovered in a linear system formulation. Our analytical results are supported by numerical simulations of $2-$bright-soliton scatterings with potential $ V = - \frac{ 2η}{2+ ε} ( |ψ|^2 )^{2 + ε}, ε\in R, η>0$. Our numerical simulations show the elastic scattering of bright solitons for a wide range of values of the set $\{η, ε\}$ and a variety of amplitudes and relative velocities. The AKNS-type system is quite ubiquitous, and so, our results may find potential applications in several areas of non-linear physics, such as Bose-Einstein condensation, superconductivity, soliton turbulence and the triality among gauge theories, integrable models and gravity theories.

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arXiv2021

Modified non-linear Schrödinger models, ${\cal C}{\cal P}_s{\cal T}_d$ symmetry, dark solitons and infinite towers of anomalous charges

Some modified (defocusing) non-linear Schrödinger models (MNLS) possess infinite towers of anomalous conservation laws with asymptotically conserved charges. The so-called anomalies of the quasi-conservation laws vanish upon space-time integration for a special ${\cal C}{\cal P}_s{\cal T}_d$ symmetric field configurations. We verify numerically the degree of modifications of the charges around the dark-soliton interaction regions by computing numerically some representative anomalies related to lowest order quasi-conservation laws of the non-integrable cubic-quintic NLS model as a modified (defocusing) NLS model. This modification depends on the parameter $ε$, such that the standard NLS is recovered for $ε=0$. Here we present the numerical simulations for small values of $|ε|$, and show that the collision of two dark solitons are elastic. The NLS-type equations are quite ubiquitous in several areas of non-linear science.

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arXiv2020

Quasi-integrable KdV models, towers of infinite number of anomalous charges and soliton collisions

We found, through analytical and numerical methods, new towers of infinite number of asymptotically conserved charges for deformations of the Korteweg-de Vries equation (KdV). It is shown analytically that the standard KdV also exhibits some towers of infinite number of anomalous charges, and that their relevant anomalies vanish for $N-$soliton solution. Some deformations of the KdV model are performed through the Riccati-type pseudo-potential approach, and infinite number of exact non-local conservation laws is provided using a linear formulation of the deformed model. In order to check the degrees of modifications of the charges around the soliton interaction regions, we compute numerically some representative anomalies, associated to the lowest order quasi-conservation laws, depending on the deformation parameters $\{ε_1, ε_2\}$, which include the standard KdV ($ε_1=ε_2=0$), the regularized long-wave (RLW) ($ε_1=1,ε_2=0$), the modified regularized long-wave (mRLW) ($ε_1=ε_2=1$) and the KdV-RLW (KdV-BBM) type ($ε_2=0,\,ε\neq \{0,1\}$) equations, respectively. Our numerical simulations show the elastic scattering of two and three solitons for a wide range of values of the set $\{ε_1, ε_2\}$, for a variety of amplitudes and relative velocities. The KdV-type equations are quite ubiquitous in several areas of non-linear science, and they find relevant applications in the study of General Relativity on $AdS_{3}$, Bose-Einstein condensates, superconductivity and soliton gas and turbulence in fluid dynamics.

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arXiv2019

Riccati-type pseudopotentials, conservation laws and solitons of deformed sine-Gordon models

Deformed sine-Gordon (DSG) models $\partial_ξ\partial_η\, w + \frac{d}{dw}V(w) = 0$, with $V(w)$ being the deformed potential, are considered in the context of the Riccati-type pseudopotential approach. A compatibility condition of the deformed system of Riccati-type equations reproduces the equation of motion of the DSG models. Then, we provide a pair of linear systems of equations for DSG model, and provide an infinite tower of non-local conservation laws. Through a direct construction and supported by numerical simulations of soliton scatterings, we show that the DSG models, which have recently been defined as quasi-integrable in the anomalous zero-curvature approach [Ferreira-Zakrzewski, JHEP05(2011)130], possess new towers of infinite number of quasi-conservation laws. We compute numerically the first sets of non-trivial and independent charges (beyond energy and momentum) of the DSG model: the two third order conserved charges and the two fifth order asymptotically conserved charges in the pseudopotential approach, and the first four anomalies of the new towers of charges, respectively. We consider kink-kink, kink-antikink and breather configurations for the Bazeia {\sl et al.} potential $V_{q}(w) = \frac{64}{q^2} \tan^2{\frac{w}{2}} (1-|\sin{\frac{w}{2}}|^q)^2 \, (q \in R)$, which contains the usual SG potential $V_2(w) = 2[1- \cos{(2 w)}]$. The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.

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arXiv2018

Some results on natural numbers represented by quadratic polynomials in two variables

We consider a set of equations of the form $p_j (x,y) = (10 x+m_j)(10 y + n_j),\,\,x\geq 0, y\geq0$, $j=1,2,3$, such that $\{m_1=7, n_1=3\}$, $\{m_2=n_2=9\}$ and $\{m_3=n_3=1\}$, respectively. It is shown that if $(a(p_j),b(p_j)) \in N \times N$ is a solution of the $j'$th equation one has the inequality $\frac{p_j}{100}\leq A(p_j) B(p_j) \leq \frac{121}{10^4} p_j$, where $A(p_j)\equiv a(p_j)+1, B(p_j)\equiv b(p_j)+1\,$ and $p_{j}$ is a natural number ending in 1, such that $\{A(p_1)\geq 4, B(p_1)\geq 8\}$, $\{A(p_2) \geq 2, B(p_2)\geq 2\}$, and $\{A(p_3) \geq 10, B(p_3)\geq 10\}$ hold, respectively. Moreover, assuming the previous result we show that $1\leq ( \frac{A(p_j+10) B(p_j+10)}{A(p_j) B(p_j)})^{1/100} \leq e^{0,000201} x (1+ \frac{10}{p_j})^{(0,101)^2}$, with $\{A(p_1)\geq 31, B(p_1)\geq 71\}$, $\{A(p_2) \geq 11, B(p_2)\geq 11\}$, and $\{A(p_3) \geq 91, B(p_3)\geq 91\}$, respectively. Finally, we present upper and lower bounds for the relevant positive integer solution of the equation defined by $p_j = (10 A+m_j)(10 B + n_j)$, for each case $j=1,2,3$, respectively.

math.NT↗

arXiv2017

Quasi-integrable non-linear Schrödinger models, infinite towers of exactly conserved charges and bright solitons

Deformations of the focusing non-linear Schrödinger model (NLS) are considered in the context of the quasi-integrability concept. We strengthen the results of JHEP09(2012)103 for bright soliton collisions. We addressed the focusing NLS as a complement to the one in JHEP03(2016)005, in which the modified defocusing NLS models with dark solitons were shown to exhibit an infinite tower of exactly conserved charges. We show, by means of analytical and numerical methods, that for certain two-bright-soliton solutions, in which the modulus and phase of the complex modified NLS field exhibit even parities under a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved during the scattering process of the solitons. We perform extensive numerical simulations and consider the bright solitons with deformed potential $ V = \frac{ 2η}{2+ ε} $ |ψ|^2$^{2 + ε}, ε\in \IR, η<0$. However, for two-soliton field components without definite parity we also show numerically the vanishing of the first non-trivial anomaly and the exact conservation of the relevant charge. So, the parity symmetry seems to be a sufficient but not a necessary condition for the existence of the infinite tower of conserved charges. The model supports elastic scattering of solitons for a wide range of values of the amplitudes and velocities and the set $\{η, ε\}$. Since the NLS equation is ubiquitous, our results may find potential applications in several areas of non-linear science.

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arXiv2016

Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons

The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schrödinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sine-Gordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential $V =ηI^2 - \fracε{6} I^3 $ and the saturable type potential satisfying $V'[I] =2 ηI - \frac{εI^q}{1+ I^q},\,q \in \IZ_{+}$, with a deformation parameter $ε\in \IR$ and $I=|ψ|^2$. The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of $\{η, ε, n\}$. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.

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arXiv2012

New derivation of soliton solutions to the AKNS$_2$ system via dressing transformation methods

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schrödinger equation (AKNS$_r$)\,($r=1,2$). Using the dressing transformation (DT) method and the related tau functions we study the AKNS$_{r}$ system for the vanishing, (constant) non-vanishing and the mixed boundary conditions, and their associated bright, dark, and bright-dark N-soliton solutions, respectively. Moreover, we introduce a modified DT related to the dressing group in order to consider the free field boundary condition and derive generalized N-dark-dark solitons. As a reduced submodel of the AKNS$_r$ system we study the properties of the focusing, defocusing and mixed focusing-defocusing versions of the so-called coupled non-linear Schrödinger equation ($r-$CNLS), which has recently been considered in many physical applications. We have shown that two$-$dark$-$dark$-$soliton bound states exist in the AKNS$_2$ system, and three$-$ and higher$-$dark$-$dark$-$soliton bound states can not exist. The AKNS$_r$\,($r\geq 3$) extension is briefly discussed in this approach. The properties and calculations of some matrix elements using level one vertex operators are outlined.

nlin.SI↗

arXiv2011

Generalized AKNS System, Non-vanishing Boundary Conditions and N-Dark-Dark Solitons

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schrödinger equation (AKNS). Using the dressing transformation (DT) method and the related tau functions we study the AKNS$_{r}$ system for the vanishing, (constant) non-vanishing and the mixed boundary conditions, and their associated bright, dark and bright-dark N-soliton solutions, respectively. Moreover, we introduce a modified DT related to the dressing group in order to consider the free field boundary condition and derive generalized N-dark-dark solitons. We have shown that two$-$dark$-$dark$-$soliton bound states exist in the AKNS$_2$ system, and three$-$ and higher$-$dark$-$dark$-$soliton bound states can not exist. As a reduced submodel of the AKNS$_r$ system we study the properties of the focusing, defocusing and mixed focusing-defocusing versions of the so-called coupled non-linear Schrödinger equation ($r-$CNLS), which has recently been considered in many physical applications. The properties and calculations of some matrix elements using level one vertex operators are outlined.

nlin.SI↗

arXiv2010

Noncommutative (generalized) sine-Gordon/massive Thirring correspondence, integrability and solitons

Some properties of the correspondence between the non-commutative versions of the (generalized) sine-Gordon (NCGSG$_{1,2}$) and the massive Thirring (NCGMT$_{1,2}$) models are studied. Our method relies on the master Lagrangian approach to deal with dual theories. The master Lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM$_{1,2}$), in which the Toda field $g$ belongs to certain subgroups of $ GL(3)$, and the matter fields lie in the higher grading directions of an affine Lie algebra. Depending on the form of $g$ one arrives at two different NC versions of the NCGSG$_{1,2}$/NCGMT$_{1,2}$ correspondence. In the NCGSG$_{1,2}$ sectors, through consistent reduction procedures, we find NC versions of some well-known models, such as the NC sine-Gordon (NCSG$_{1,2}$) (Lechtenfeld et al. and Grisaru-Penati proposals, respectively), NC (bosonized) Bukhvostov-Lipatov (NCbBL$_{1,2}$) and NC double sine-Gordon (NCDSG$_{1,2}$) models. The NCGMT$_{1,2}$ models correspond to Moyal product extension of the generalized massive Thirring model. The NCGMT$_{1,2}$ models posses constrained versions with relevant Lax pair formulations, and other sub-models such as the NC massive Thirring (NCMT$_{1,2}$), the NC Bukhvostov-Lipatov (NCBL$_{1,2}$) and constrained versions of the last models with Lax pair formulations. We have established that, except for the well known NCMT$_{1,2}$ zero-curvature formulations, generalizations ($n_{F} \ge 2$, $n_F=$number of flavors) of the massive Thirring model allow zero-curvature formulations only for constrained versions of the models and for each one of the various constrained sub-models defined for less than $n_F$ flavors, in the both NCGMT$_{1,2}$ and ordinary space-time descriptions (GMT), respectively. The non-commutative solitons and kinks of the $ GL(3)$ NCGSG$_{1,2}$ models are investigated.

hep-th↗

arXiv2009

Some comments on the integrability of the noncommutative generalized massive Thirring model

Some properties of a non-commutative version of the generalized massive Thirring theory (NCGMT) are studied. We develop explicit calculations for the affine Lie algebra $gl(3)$ case. The NCGMT model is written in terms of Dirac type fields corresponding to the Moyal product extension of the ordinary multi-field massive Thirring model. We discuss the Lagrangian formulation, its zero-curvature representation and integrability property of certain submodels.

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