arXiv subjects

Vladimir S. Gerdjikov

Publications and source records attributed to Vladimir S. Gerdjikov.

At least 19 recordsLinked to original sources

arXiv2024

On the $N$-waves hierarchy with constant boundary conditions. Spectral properties

The paper is devoted to $N$-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators $L$, whose potentials $Q(x,t)$ tend to constants $Q_\pm$ for $x\to \pm \infty$. For special choices of $Q_\pm$ we outline the spectral properties of $L$, the direct scattering transform and construct its fundamental analytic solutions. We generalise Wronskian relations for the case of CBC -- this allows us to analyse the mapping between the scattering data and the $x$-derivative of the potential $Q_x$. Next, using the Wronskian relations we derive the dispersion laws for the $N$-wave hierarchy and describe the NLEE related to the given Lax operator.

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arXiv2024

On affine Toda field theories related to ${\bf D}_r$ algebras and their real Hamiltonian forms

The paper deals with affine 2-dimensional Toda field theories related to simple Lie algebras of the classical series ${\bf D}_r$. We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of $\mathfrak{g}$. This is illustrated on the examples ${\bf D}_{r+1}^{(1)} \to {\bf B}_r^{(1)}$ and ${\bf D}_4^{(1)} \to {\bf G}_2^{(1)}$ using external automorphisms of the corresponding extended Dynkin diagrams.

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arXiv2022

Real Hamiltonian forms of affine Toda field theories: spectral aspects

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.

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arXiv2022

On soliton solutions and soliton interactions of Kulish-Sklyanin and Hirota-Ohta systems

In this paper we consider a simplest two-dimensional reduction of the remarkable three-dimensional Hirota-Ohta system. The Lax pair of the Hirota-Ohta system was extended to a Lax triad by adding extra third linear equation, whose compatibility conditions with the Lax pair of the Hirota-Ohta imply another remarkable systems: the Kulish-Sklyanin system (KSS) together with its first higher commuting flow, which we can call as vector complex MKdV. This means that any common particular solution of these both two-dimensional integrable systems yields a corresponding particular solution of the three-dimensional Hirota-Ohta system. Using the dressing Zakharov-Shabat method we derive the $N$-soliton solutions of these systems and analyze their interactions, i.e. derive explicitly the shifts of the relative center-of-mass coordinates and the phases as functions of the discrete eigenvalues of the Lax operator. Next we relate to these NLEE system of Hirota--Ohta type and obtain its $N$-soliton solutions

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arXiv2021

Fundamental Analytic Solutions for the Kulish-Sklyanin Model with Constant Boundary Conditions

In the present paper we analyze the construction of fundamental analytic solutions (FAS) for the generalized Kulish-Sklyanin models (KSM) for vanishing (VBC) and constant boundary conditions (CBC). Using FAS one can reduce the direct and inverse scattering problems for the Lax operator to a Riemann-Hilbert problem (RHP). For VBC we find two FAS $χ^+(x,t,λ) $ and $χ^-(x,t,λ) $ analytic in the upper/lower $\mathbb{C}_\pm$ complex $λ$-plane. The RHP consists in: given the sewing function $G(x,t,λ)$ to constructing both $χ^\pm(x,t,λ) $ in their regions of analyticity. For CBC the problem becomes more complicated, because now the RHP must be formulated on a Riemannian surface of genus 1.

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arXiv2021

On the mKdV equations related to the Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$

We outline the derivation of the mKdV equations related to the Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$. First we formulate their Lax representations and provide details how they can be obtained from generic Lax operators related to the algebra $sl(6)$ by applying proper Mikhailov type reduction groups $\mathbb{Z}_h$. Here $h$ is the Coxeter number of the relevant Kac-Moody algebra. Next we adapt Shabat's method for constructing the fundamental analytic solutions of the Lax operators $L$. Thus we are able to reduce the direct and inverse spectral problems for $L$ to Riemann-Hilbert problems (RHP) on the union of $2h$ rays $l_ν$. They start from the origin of the complex $λ$-plane and close equal angles $π/h$. To each $l_ν$ we associate a subalgebra $\mathfrak{g}_ν$ which is a direct sum of $sl(2)$-subalgebras. Thus to each regular solution of the RHP we can associate scattering data of $L$ consisting of scattering matrices $T_ν\in \mathcal{G}_ν$ and their Gauss decompositions. The main result of the paper is to extract from $T_0$ and $T_1$ related to the rays $l_0$ and $l_1$ the minimal sets of scattering data $\mathcal{T}_k$, $k=1, 2$. We prove that each of the minimal sets $\mathcal{T}_1$ and $\mathcal{T}_2$ allows one to reconstruct both the scattering matrices $T_ν$, $ν=0, 1, \dots 2h$ and the corresponding potentials of the Lax operators $L$.

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arXiv2021

Multicomponent Fokas-Lenells equations on Hermitian symmetric spaces

Multi-component integrable generalizations of the Fokas-Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.

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arXiv2020

On Asymptotic Dynamical Regimes of Manakov $N$-soliton Trains in Adiabatic Approximation

We analyze the dynamical behavior of the $N$-soliton train in the adiabatic approximation of the Manakov model. %perturbed by gain/loss effects and also by several types of external potentials. The evolution of Manakov $N$-soliton trains is described by the complex Toda chain (CTC) which is a completely integrable dynamical model. Calculating the eigenvalues of its Lax matrix allows us to determine the asymptotic velocity of each soliton. So we describe sets of soliton parameters that ensure one of the two main types of asymptotic regimes: the bound state regime (BSR) and the free asymptotic regime (FAR). In particular we find explicit description of special symmetric configurations of $N$ solitons that ensure BSR and FAR. We find excellent matches between the trajectories of the solitons predicted by CTC with the ones calculated numerically from the Manakov system for wide classes of soliton parameters. This confirms the validity of our model.

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arXiv2019

Riemann-Hilbert problem, integrability and reductions

The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R \simeq \mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $\mathbb{D}_h$ symmetries are presented.

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arXiv2017

Kulish-Sklyanin type models: integrability and reductions

We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider two Riemann-Hilbert problems: the first formulated on the real axis $\mathbb{R}$ in the complex $λ$-plane; the second one is formulated on $\mathbb{R} \oplus i\mathbb{R}$. The first RHP for $s=1$ allows one to solve the Kulish-Sklyanin (KS) model; the second RHP is relevant for a new type of KS model. An important example for nontrivial deep reductions of KS model is given. Its effect on the scattering matrix is formulated. In particular we obtain new 2-component NLS equations. Finally, using the Wronskian relations we demonstrate that the inverse scattering method for KS models may be understood as a generalized Fourier transforms. Thus we have a tool to derive all their fundamental properties, including the hierarchy of equations and the hierarchy of their Hamiltonian structures.

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arXiv2017

On the soliton solutions of a family of Tzitzeica equations

We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of soliton solutions. The first type corresponds to a set of 6 symmetrically situated discrete eigenvalues of the Lax operator $L$; to each soliton of the second type one relates a set of 12 discrete eigenvalues of $L$. We also outline how one can construct general $N$ soliton solution containing $N_1$ solitons of first type and $N_2$ solitons of second type, $N=N_1+N_2$. The possible singularities of the solitons and the effects of change of variables that relate the different members of Tzitzeica family equations are briefly discussed. All equations allow quasi-regular as well as singular soliton solutions.

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arXiv2017

On Kaup-Kupershchmidt type equations and their soliton solutions

We start with the Lax representation for the Kaup-Kupersschmidt equation (KKE). We We outline the deep relation between the scalar Lax operator and the matrix Lax operators related to Kac-Moody algebras. Then we derive the MKdV equations gauge equivalent to the KKE. Next we outline the symmetry and the spectral properties of the relevant Lax operator. Using the dressing Zakharov-Shabat method we demonstrate that the MKdV and KKE have two types of one-soliton solutions and briefly comment on their properties.

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arXiv2017

On nonlocal models of Kulish-Sklyanin type and generalized Fourier transforms

A special class of multicomponent NLS equations, generalizing the vector NLS and related to the {\bf BD.I}-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic solutions are constructing thus reducing the inverse scattering problem to a Riemann-Hilbert problem. We introduce the minimal sets of scattering data $\mathfrak{T}$ which determines uniquely the scattering matrix and the potential $Q$ of the Lax operator. The elements of $\mathfrak{T}$ can be viewed as the expansion coefficients of $Q$ over the `squared solutions' that are natural generalizations of the standard exponentials. Thus we demonstrate that the mapping $\mathfrak{T} \to Q$ is a generalized Fourier transform. Special attention is paid to two special representatives of this MNLS with three-component and five components which describe spinor ($F=1$ and $F=2$, respectively) Bose-Einstein condensates.

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arXiv2017

New types of two component NLS-type equations

We study MNLS related to the D.III-type symmetric spaces. Applying to them Mikhailov reduction groups of the type $\mathbb{Z}_r\times \mathbb{Z}_2$ we derive new types of 2-component NLS equations. These are {\bf not} counterexamples to the Zakharov-Schulman theorem because the corresponding interaction Hamiltonians depend not only on $|q_k|^2$, but also on $q_1q_2^* +q_1^* q_2$.

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arXiv2016

On integrable wave interactions and Lax pairs on symmetric spaces

Multi-component generalizations of derivative nonlinear Schrodinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z_2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann-Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup-Newell (KN) and Gerdjikov-Ivanov (GI) types is derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundamental properties of the multi-component GI equations are briefly discussed at the end.

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arXiv2016

On the N-wave Equations with PT-symmetry

We study extensions of N-wave systems with PT-symmetry. The types of (nonlocal) reductions leading to integrable equations invariant with respect to P- (spatial reflection) and T- (time reversal) symmetries is described. The corresponding constraints on the fundamental analytic solutions and the scattering data are derived. Based on examples of 3-wave (related to the algebra sl(3,C)) and 4-wave (related to the algebra so(5,C)) systems, the properties of different types of 1- and 2-soliton solutions are discussed. It is shown that the PT symmetric 3-wave equations may have regular multi-soliton solutions for some specific choices of their parameters.

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arXiv2015

On MKdV Equations Related to the Affine Kac-Moody Algebra $A_{5}^{(2)}$

We have derived a new system of mKdV-type equations which can be related to the affine Lie algebra $A_{5}^{(2)}$. This system of partial differential equations is integrable via the inverse scattering method. It admits a Hamiltonian formulation and the corresponding Hamiltonian is also given. The Riemann-Hilbert problem for the Lax operator is formulated and its spectral properties are discussed.

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arXiv2015

Soliton equations related to the affine Kac-Moody algebra D^(1)_4

We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D^(1)_4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.

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