This review paper explores the Riccati-type pseudo-potential formulation applied to the quasi-integrable sine-Gordon, KdV, and NLS models. The proposed framework provides a unified methodology for analyzing quasi-integrability properties across various integrable systems, including deformations of the sine-Gordon, Bullough–Dodd, Toda, KdV, pKdV, NLS, and SUSY sine-Gordon models. Key findings include the emergence of infinite towers of anomalous conservation laws within the Riccati-type approach and the identification of exact non-local conservation laws in the linear formulations of deformed models. As modified integrable models play a crucial role in diverse fields of nonlinear physics—such as Bose–Einstein condensation, superconductivity, gravity models, optics, and soliton turbulence—these results may have far-reaching applications.

We explore the Faddeev-Jackiw symplectic Hamiltonian reduction of the sl(2) affine Toda model coupled to matter, 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 backreaction 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, 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 backreaction, as well as the appearance of kink states in the continuum.

We study a particular deformation of the potential KdV model (pKdV) and construct the quasi-conservation laws by a direct method. The charge densities, differing from their integrable counterpart with homogeneous degree terms, exhibit mixed scale dimension terms. The modifications of the charges around the soliton interaction regions are examined by numerically simulating some representative anomalies. We show numerically the elastic scattering of two kinks for a wide range of values of the deformation parameters. It discussed an anomaly cancellation mechanism to define an exact conservation law of the usual pKdV model, and a renormalization procedure is introduced for some divergent charges by subtracting the continuous linear background contribution. The KdV-type equations are quite ubiquitous in several areas of non-linear science, such as the study of General Relativity in Ads_3 , Bose-Einstein condensates, superconductivity, and fluid dynamics.

Abstract We study a non-Hermitian (NH) sl(2) affine Toda model coupled to fermions through soliton theory techniques and the realizations of the pseudo-chiral and pseudo- Hermitian symmetries. The interplay of non-Hermiticity, integrability, nonlinearity, and topology significantly influence the formation and behavior of a continuum of bound state modes (CBM) and extended waves in the localized continuum (ELC). The non-Hermitian soliton-fermion duality, the complex scalar field topological charges and winding numbers in the spectral topology are uncovered. The biorthogonal Majorana zero modes, dual to the NH Toda solitons with topological charges $$ \frac{2}{\pi}\arg \left(z=\pm i\right)=\pm 1 $$ 2 π arg z = ± i = ± 1 , appear at the complex-energy point gap and are pinned at zero energy. The zero eigenvalue λ(z = ± i) = 0, besides being a zero mode, plays the role of exceptional points (EPs), and each EP separates a real eigenvalue $$ \mathcal{A} $$ A -symmetric and $$ \mathcal{A} $$ A -symmetry broken regimes for an antilinear symmetry $$ \mathcal{A}\in \left\{\mathcal{PT},{\gamma}_5\mathcal{PT}\right\} $$ A ∈ PT γ 5 PT . Our findings improve the understanding of exotic quantum states, but also paves the way for future research in harnessing non-Hermitian phenomena for topological quantum computation, as well as the exploration of integrability and NH solitons in the theory of topological phases of matter.

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 QCD2, braneworld models, high Tc superconductivity and topological quantum computation. We back up our results with numerical simulations for continuous families of topological sectors.

In this paper, 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 nonlocal conservation laws are uncovered in a linear formulation of the system. It is shown that certain modifications of the nonlinear 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 nonlocal properties, the complete list of towers of infinite number of anomalous charges and the (nonlocal) 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 nonlinear physics such as Bose–Einstein condensation, superconductivity and soliton turbulence.

Modifications of the nonlinear Schrödinger (MNLS) model [Formula: see text] where [Formula: see text] and [Formula: see text], 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 ([Formula: see text]) symmetry. Infinite towers of anomalous charges appear even in the standard NLS model for [Formula: see text] invariant [Formula: see text]-bright solitons. The true conserved charges emerge through some kind of anomaly cancellation mechanism. Our analytical results are supported by numerical simulations of two-bright-soliton scatterings with potential [Formula: see text]. Our numerical simulations show the elastic scattering of bright solitons for a wide range of values of the set [Formula: see text] and a variety of amplitudes and relative velocities. The MNLS-type systems are quite ubiquitous, and so, our results may find potential applications in several areas of nonlinear physics, such as Bose–Einstein condensation, superconductivity, soliton turbulence and the triality among gauge theories, integrable models and gravity theories.

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 quasiconservation laws vanish upon space-time integration for a special CPsTd 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 ǫ, such that the standard NLS is recovered for ǫ = 0. Here we present the numerical simulations for small values of |ǫ|, 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.

We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather configurations of the Bazeia et al. potential Vqw=64q2tan2w21−sinw2q2,q∈R, which contains the usual SG potential V2w=21−cos2w.

We consider a set of equations of the form p_j (x,y) = (10 x+m_j)(10 y + n_j), x≥ 0, y≥0, 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)) ∈ N × N is a solution of the j'th equation one has the inequality p_j/100≤ A(p_j) B(p_j) ≤121/10^4 p_j, where A(p_j)≡ a(p_j)+1, B(p_j)≡ b(p_j)+1 and p_j is a natural number ending in 1, such that {A(p_1)≥ 4, B(p_1)≥ 8}, {A(p_2) ≥ 2, B(p_2)≥ 2}, and {A(p_3) ≥ 10, B(p_3)≥ 10} hold, respectively. Moreover, assuming the previous result we show that 1≤ ( A(p_j+10) B(p_j+10)/A(p_j) B(p_j))^1/100≤ e^0,000201 x (1+ 10/p_j)^(0,101)^2, with {A(p_1)≥ 31, B(p_1)≥ 71}, {A(p_2) ≥ 11, B(p_2)≥ 11}, and {A(p_3) ≥ 91, B(p_3)≥ 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.

Deformed sine-Gordon (DSG) models ∂ξ∂ηw+ddwV(w)=0, with V(w) being the deformed potential, are considered in the context of the Riccati-type pseudo-potential 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 the DSG model and an associated 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 and Zakrzewski, 2011 [1]), 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 pseudo-potential 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 et al. potential Vq(w)=64q2tan2⁡w2(1−|sin⁡w2|q)2(q∈R), which contains the usual SG potential V2(w)=2[1−cos⁡(2w)]. The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.

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 {ϵ1, ϵ2}, which include the standard KdV (ϵ1 = ϵ2 = 0), the regularized long-wave (RLW) (ϵ1 = 1, ϵ2 = 0), the modified regularized long-wave (mRLW) (ϵ1 = ϵ2 = 1) and the KdV-RLW (KdV-BBM) type (ϵ2 = 0, ≠ = {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 {ϵ1, ϵ2}, 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 AdS3, Bose-Einstein condensates, superconductivity and soliton gas and turbulence in fluid dynamics.

Some deformations of the integrable Korteweg-de Vries model (KdV) are associated to several towers of infinite number of asymptotically conserved charges. It has been shown that the standard KdV also exhibits infinite number of anomalous charges. In [9] there have been verified numerically the degrees of modifications of the charges around the soliton interaction regions, by computing numerically some representative anomalies, related to lowest order quasi-conservation laws, depending on the deformation parameters {∈1,∈2} , such that the standard KdV is recovered for (∈1=∈2=0) . Here we present the numerical simulations for some values of the pair {∈1,∈2} around ∈1≈0,∈2≈0 , and show that the collision of two and three solitons are elastic. The KdV-type equations are quite ubiquitous and find many applications in several areas of non-linear science.

In this article using the functions f1(k) = 10k + 1, k 6= ◦ 3+2; f2(k) = 10k + 3, k 6= ◦ 3−{0}; f3(k) = 10k + 7, k 6= ◦ 3+2, k 6= ◦ 7−{0}; and f4(k) = 10k + 9, k 6= ◦ 3, where k ∈ N0, we obtain two important results on prime numbers. The first result indicates that if p is a prime number that ends in 7, then p + 10l will be a prime number under certain conditions. The second result states that if k is a number ending in 7, then k+10 will be also a prime number under certain conditions. AMS Subject Classification: 11A41, 11A51, 11D72

We have studied the space-reflection symmetries of some soliton solutions of deformed sine-Gordon models in the context of the quasi-integrability concept. Considering a dual pair of anomalous Lax representations of the deformed model we compute analytically and numerically an infinite number of alternating conserved and asymptotically conserved charges through a modification of the usual techniques of integrable field theories. The charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings with equal and opposite velocities, split into two infinite towers of conserved and asymptotically conserved charges. For two-solitons without definite parity under space-reflection symmetry (kink-kink and kink-antikink scatterings with unequal and opposite velocities) our numerical results show the existence of the asymptotically conserved charges only. However, we show that in the center-of-mass reference frame of the two solitons the parity symmetries and their associated set of exactly conserved charges can be restored. Moreover, the positive parity breather-like (kink-antikink bound state) solution exhibits a tower of exactly conserved charges and a subset of charges which are periodic in time. We back up our results with extensive numerical simulations which also demonstrate the existence of long lived breather-like states in these models. The time evolution has been simulated by the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.

We show that the quasi-integrability concept 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 fi eld components are eigenstates of a space-reflction symmetry, the fi rst four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential V = eta I-2 - epsilon/6I(3) and the saturable type potential satisfying V'[I] = 2 eta I - epsilon I-q/1+Iq; q epsilon Z(+), with a deformation parameter epsilon epsilon IR and I = vertical bar psi vertical bar(2). The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of {eta, epsilon, q}. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.

Deformations of the focusing non-linear Schrödinger model (NLS) are considered in the context of the quasi-integrability concept. We strengthen the results of JHEP 09 (2012) 103 for bright soliton collisions. We addressed the focusing NLS as a complement to the one in JHEP 03 (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 \( V=\frac{2\eta }{2+\upepsilon}{\left({\left|\psi \right|}^2\right)}^{2+\upepsilon},\upepsilon \in \mathbb{R},\eta <0 \). 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.

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 \( V=\eta {I}^2-\frac{\in }{6}{I}^3 \) and the saturable type potential satisfying Open image in new window , with a deformation parameter ϵ ∈ Open image in new window 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, q}. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.