Narrow Results

In this paper, we define and explore the Darboux and binary Darboux transformations for a semi-discrete short pulse equation. By iterating the binary Darboux transformation, we obtain quasi-Grammian solutions. Furthermore, we derive explicit matrix solutions for the binary Darboux matrix and reduce them to the elementary Darboux matrix. Finally, we plot the dynamics of bright, dark, double breather, and peak soliton solutions.

The regularized long-wave ($ℝ𝕃𝕎$) equation is a fundamental model in shallow water wave theory, with extended relevance to other physical systems, such as plasma physics. The $ℝ𝕃𝕎$ equation governs key nonlinear wave phenomena, including the propagation and interaction of solitons, or localized solitary waves. In plasma physics, it describes ion-acoustic waves, which are low-frequency compressional waves arising from the interaction between ions and electrons. While the ion-acoustic wave exhibits Korteweg–de Vries soliton behavior in the cold plasma limit, the $ℝ𝕃𝕎$ equation offers a more accurate representation of ion-acoustic solitons and peaked structures in warm plasmas by accounting for thermal effects.

This research presents novel analytical approximations for solitary wave solutions within the $ℝ𝕃𝕎$ equation, which is critical for modeling nonlinear shallow water and plasma waves. The equation’s ability to describe peaked solitary waves, known as “peakons”, represents wave-breaking phenomena. By employing the improved Riccati expansion and modified Fan expansion techniques, this study derives periodic peakon solutions, offering new insights into the equation’s behavior.

A thorough analysis of the $ℝ𝕃𝕎$ equation is provided, emphasizing its importance for nonlinear wave modeling in both fluid and plasma systems. This paper also includes numerical validation using He’s variational iteration method, which enhances the transparency of the findings by addressing the underlying assumptions and limitations. The principal findings contribute to the broader understanding of nonlinear solitary wave theory, with practical implications for nonlinear wave dynamics across multiple physical domains. Suggested extensions are outlined for further investigation within this established theoretical framework. This study maintains consistent notation and terminology to facilitate clear communication of ideas in adherence to academic standards.

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

We provide a pedagogical introduction to some aspects of integrability, dualities and deformations of physical systems in $0+1$ and in $1+1$ dimensions. In particular, we concentrate on the T-duality of point particles and strings as well as on the Ruijsenaars duality of finite many-body integrable models, we review the concept of the integrability and, in particular, of the Lax integrability and we analyze the basic examples of the Yang–Baxter deformations of nonlinear ${\sigma }$-models. The central mathematical structure which we describe in detail is the ${ℰ}$-model which is the dynamical system exhibiting all those three phenomena simultaneously. The last part of the paper contains original results, in particular, a formulation of sufficient conditions for strong integrability of non-degenerate ${ℰ}$-models.

We study the bilinearization of $N=1$ Supersymmetric Coupled Dispersionless (SUSY-CD) integrable system by introducing transformations defined in terms of bosonic and fermionic superfields. From the bosonic and fermionic superfield bilinear equations, we obtain the superfield soliton solutions of SUSY-CD integrable system.

The Joyce integrable system and the corresponding Bridgeland–Toledano-Laredo connections are fundamental objects associated with suitable abelian categories or, more generally, with a class of continuous families of stability data. We offer an overview of some of our work, mostly joint with M. Garcia Fernandez, focusing on equations of TBA type as a useful tool in the analysis of these objects and their deformations, and as a means to establish a connection with tropical geometry.

A two-dimensional integrable system being a deformation of the rational Calogero–Moser system is constructed via the symplectic reduction, performed with respect to the Sklyanin algebra action. We explicitly resolve the respective classical equations of motion via the projection method and quantize the system.

We modify the algebraic structure of a Poisson bialgebra by considering the deformed coproduct and deformations of an algebra e(2). These modifications lead to the quantum groups and provide new classes of completely integrable Hamiltonian systems.

Owing to the analogy between the Connes–Kreimer theory of the renormalization and the integrable systems, we derive the differential equations of the unit mass for the renormalized character ϕ₊ and the counter term ϕ_-. We give another proof of the scattering type formula of ϕ_-. The differential equation of ϕ_- of the coordinate ε on ℙ¹ is also given. The hierarchy of the renormalization groups is defined as the integrable systems.

Marginal β deformations of super-Yang–Mills theory are known to correspond to a certain class of deformations of the S⁵ background subspace of type IIB string theory in AdS₅×S⁵. An analogous set of deformations of the AdS₅ subspace is reviewed here. String energy spectra computed in the near-pp-wave limit of these backgrounds match predictions encoded by discrete, asymptotic Bethe equations, suggesting that the twisted string theory is classically integrable in this regime. These Bethe equations can be derived algorithmically by relying on the existence of Lax representations, and on the Riemann–Hilbert interpretation of the thermodynamic Bethe ansatz. This letter is a review of a seminar given at the Institute for Advanced Study, based on research completed in collaboration with McLoughlin.

In this paper we present Darboux transformation for the principal chiral and WZW models in two dimensions and construct multi-soliton solutions in terms of quasideterminants. We also establish the Darboux transformation on the holomorphic conserved currents of the WZW model and expressed them in terms of the quasideterminant. We discuss the model based on the Lie group SU(n) and obtain explicit soliton solutions for the SU(2) model.

We study Darboux transformations for the two boson (TB) hierarchy both in the scalar as well as in the matrix descriptions of the linear equation. While Darboux transformations have been extensively studied for integrable models based on SL(2, R) within the AKNS framework, this model is based on SL(2, R)⊗U(1). The connection between the scalar and the matrix descriptions in this case implies that the generic Darboux matrix for the TB hierarchy has a different structure from that in the models based on SL(2, R) studied thus far. The conventional Darboux transformation is shown to be quite restricted in this model. We construct a modified Darboux transformation which has a much richer structure and which also allows for multi-soliton solutions to be written in terms of Wronskians. Using the modified Darboux transformations, we explicitly construct one-soliton/kink solutions for the model.

We study the conserved quantities of the generalized Heisenberg magnet (GHM) model. We derive the nonlocal conserved quantities of the model using the iterative procedure of Brezin et al. [Phys. Lett. B82, 442 (1979).] We show that the nonlocal conserved quantities Poisson commute with local conserved quantities of the model.

The dressing method of Zakharov and Shabat [Funct. Anal. Appl.8, 226 (1974) and ibid.13, 166 (1980)] has been employed to the generalized coupled dispersionless integrable system in two dimensions. The dressed solutions to the Lax pair and to the nonlinear matrix equation have been obtained in terms of Hermitian projectors. The dressing method has been related with the quasi-determinant solutions obtained by using the standard matrix Darboux transformation. The iteration of dressing procedure has been shown to give N-soliton solutions of the system. At the end, the explicit soliton solution has been obtained for the system based on Lie group SU(2).

We present a construction of integrable hierarchies without or with boundary, starting from a single R-matrix, or equivalently from a ZF algebra. We give explicit expressions for the Hamiltonians and the integrals of motion of the hierarchy in term of the ZF algebra. In the case without boundary, the integrals of motion form a quantum group, while in the case with boundary they form a Hopf coideal subalgebra of the quantum group.

We present an integral representation to the quantum Knizhnik–Zamolodchikov equation associated with twisted affine symmetry for massless regime |q| = 1. Upon specialization, it leads to a conjectural formula for the correlation function of the Izergin–Korepin model in massless regime |q| = 1. In a limiting case q → -1, our conjectural formula reproduce the correlation function for the Izergin–Korepin model^1,2 at critical point q = -1.

The behavior of solitons in integrable theories is strongly constrained by the integrability of the theory, that is by the existence of an infinite number of conserved quantities that these theories are known to possess. As a result, the soliton scattering of such theories is expected to be trivial (with no change of direction, velocity or shape). In this paper we present an extended review on soliton scattering of two spatial dimensional integrable systems which have been derived as dimensional reductions of the self-dual Yang–Mills equations and whose scattering properties are highly nontrivial.

We introduce a ℤ₂-graded version of the nonlinear Schrödinger equation that includes one fermion and one boson at the same time. This equation is shown to possess a supersymmetry which proves to be itself part of a super-Yangian symmetry based on gl(1|1). The solution exhibits a super version form of the classical Rosales solution. Then, we second quantize these results, and give a Lax pair formulation (based on gl(2|1)) for the model.

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and nonlocal families of R-matrix solutions to the modified Yang–Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.

We extend the exactly solvable Hamiltonian describing f quantum oscillators considered recently by J. Dorignac et al. We introduce a new interaction which we choose to be quasi-exactly solvable. The properties of the spectrum of this new Hamiltonian are studied as functions of the new coupling constant. We point out that both the original and the by us modified Hamiltonians are related to adequate Lie structures.