Papers by Vladimir Gerdjikov
arXiv (Cornell University), Dec 4, 2015
We have derived a new system of mKdV-type equations which can be related to the affine Lie algebr... more We have derived a new system of mKdV-type equations which can be related to the affine Lie algebra A (2) 5. 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
arXiv (Cornell University), Sep 16, 2000
Reductions of N-wave type equations related to simple Lie algebras and the hierarchy of their Ham... more Reductions of N-wave type equations related to simple Lie algebras and the hierarchy of their Hamiltonian structures are studied. The reduction group G R is realized as a subgroup of the Weyl group of the corresponding algebra. Some of the reduced equations are of physical interest.
Springer eBooks, 2010
The hierarchy of integrable equations are considered. The dynamical approach to theory of nonline... more The hierarchy of integrable equations are considered. The dynamical approach to theory of nonlinear waves is proposed. The special solutions (nonlinear waves) of considered equations are derived. We use powerful methods of computer algebra such as differential resultant and others.
arXiv (Cornell University), Mar 28, 2006
The reductions of the multi-component nonlinear Schrödinger (MNLS) type models related to C.I and... more The reductions of the multi-component nonlinear Schrödinger (MNLS) type models related to C.I and D.III type symmetric spaces are studied. We pay special attention to the MNLS related to the sp(4), so(10) and so(12) Lie algebras. The MNLS related to sp(4) is a three-component MNLS which finds applications to Bose-Einstein condensates. The MNLS related to so(12) and so(10) Lie algebras after convenient Z 6 or Z 4 reductions reduce to three and four-component MNLS showing new types of χ (3)-interactions that are integrable. We briefly explain how these new types of MNLS can be integrated by the inverse scattering method. The spectral properties of the Lax operators L and the corresponding recursion operator Λ are outlined. Applications to spinor model of Bose-Einstein condensates (BEC's) are discussed.
arXiv (Cornell University), Mar 24, 2008
The multi-component nonlinear Schrödinger equation related to C.I Sp(2p)/U (p) and D.III SO(2p)/U... more The multi-component nonlinear Schrödinger equation related to C.I Sp(2p)/U (p) and D.III SO(2p)/U (p)-type symmetric spaces with non-vanishing boundary conditions is solvable with the inverse scattering method (ISM). As Lax operator L we use the generalized Zakharov-Shabat operator. We show that the ISM for the Lax operator L(x, λ) is a nonlinear analog of the Fourier-transform method. As appropriate generalizations of the usual Fourierexponential functions we use the so-called "squared solutions", which are constructed in terms of the fundamental analytic solutions (FAS) χ ± (x, λ) of L(x, λ) and the Cartan-Weyl basis of the Lie algebra, relevant to the symmetric space. We derive the completeness relation for the "squared solutions" which turns out to provide spectral decomposition of the recursion (generating) operators Λ±, a natural generalizations of 1 i d dx in the case of nonlinear evolution equations (NLEE).
We first review some recent developments [2] of the N-wave equations and their gauge equivalent o... more We first review some recent developments [2] of the N-wave equations and their gauge equivalent ones related to the so(5) algebra. These include the form of the equations and the relevant recursion operators, the properties of the scattering data etc. Next we generalize these results to the multicomponent nonlinear Schrödinger equations and their gauge equivalent ones known as the Heisenberg ferromagnet type equations. The explicit form of the so(5) HF equations and the relevant recursion operators are derived.
Physical Review A, Nov 1, 1996
Zakharov-Shabat-Ablowitz-Kaup-Newel-Segur (ZS-AKNS) representation for Stokes-anti-Stokes stimula... more Zakharov-Shabat-Ablowitz-Kaup-Newel-Segur (ZS-AKNS) representation for Stokes-anti-Stokes stimulated Raman scattering (SRS) is proposed. Periodical waves, solitons and self-similarity solutions are derived. Transient and bright threshold solitons are discussed.
Теоретическая и математическая физика, Jul 30, 2022
arXiv (Cornell University), Mar 16, 2017
We start with the Lax representation for the Kaup-Kupersschmidt equation (KKE). We We outline the... more 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.
arXiv (Cornell University), Jun 11, 2021
We outline the derivation of the mKdV equations related to the Kac-Moody algebras A (1) 5 and A (... more We outline the derivation of the mKdV equations related to the Kac-Moody algebras A (1) 5 and A (2) 5. 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 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 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 ∈ 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 T k , k = 1, 2. We prove that each of the minimal sets T 1 and T 2 allows one to reconstruct both the scattering matrices T ν , ν = 0, 1,. .. 2h and the corresponding potentials of the Lax operators L.
arXiv (Cornell University), Sep 16, 2000
The adiabatic N-soliton train interactions for the scalar nonlinear Schrödinger (NLS) equation an... more The adiabatic N-soliton train interactions for the scalar nonlinear Schrödinger (NLS) equation and its perturbed versions are well studied. Here we briefly outline how they can be generalized for the higher NLS-type equations and for the multicomponent NLS equations. It is shown that in all these cases the complex Toda chain plays fundamental role.
arXiv (Cornell University), Oct 13, 2012
We propose new types of integrable spinor models, generalizing the well known ones of: i) Nambu-J... more We propose new types of integrable spinor models, generalizing the well known ones of: i) Nambu-Jona-Lasinio-Vaks-Larkin models, related to SU (N); ii) the Gross-Neveu models-SP (2N); and the iii) Zakharov-Mikhailov models-SO(N). We propose a method for constructing their Lax representation and outline the spectral properties of the Lax operators.
arXiv (Cornell University), Apr 2, 2006
Brief review of the methods for solving the multicomponent nonlinear Schrödinger (MNLS) equations... more Brief review of the methods for solving the multicomponent nonlinear Schrödinger (MNLS) equations and analysis of their Hamiltonian structures is given. Main attention is paid to the MNLS related to the C.IIand D.III-types symmetric spaces with nonvanishing (constant) boundary conditions. The spectral properties of their Lax operators are described. The derivation of the trace identities is outlined. The involutivity of their integrals of motion is proved using the method of the classical R-matrix.
arXiv (Cornell University), Apr 13, 2012
We start with a Riemann-Hilbert Problems (RHP) with canonical normalization whose sewing function... more We start with a Riemann-Hilbert Problems (RHP) with canonical normalization whose sewing functions depends on several additional variables. Using Zakharov-Shabat theorem we are able to construct a family of ordinary differential operators for which the solution of the RHP is a common fundamental analytic solution. This family of operators obviously commute. Thus we are able to construct new classes of integrable nonlinear evolution equations.
Theoretical and Mathematical Physics, Sep 1, 1992
The gauge equivalence between the inhomogeneous versions of the nonlinear Schrödinger and the Hei... more The gauge equivalence between the inhomogeneous versions of the nonlinear Schrödinger and the Heisenberg ferromagnet equations is studied. An unexplicit criterion for integrability is proposed. Examples of gauge equivalent inhomogeneous nonlinear evolution equations are presented. It is shown that in the nonintegrable cases the M-operators in their Lax representations possess unremovable pole singularities lying on the spectrum of the L-operators.
Physical review, Oct 25, 2001
Using the Karpman-Solov'ev method we derive the equations for the 2soliton adiabatic interaction ... more Using the Karpman-Solov'ev method we derive the equations for the 2soliton adiabatic interaction for solitons of the modified nonlinear Schrödinger equation (MNSE). Then we generalize these equations to the case of N interacting solitons with almost equal velocities and widths. On the basis of this result we prove that the N MNSE-soliton train interaction (N > 2) can be modeled by the completely integrable complex Toda chain (CTC). This is an argument in favor of universality of the complex Toda chain which was previously shown to model the soliton train interaction for nonlinear Schrödinger solitons. The integrability of the CTC is used to describe all possible dynamical regimes of the N-soliton trains which include asymptotically free propagation of all N solitons, N-soliton bound states, various mixed regimes, etc. It allows also to describe analytically the manifolds in the 4N-dimensional space of initial soliton parameters which are responsible for each of the regimes mentioned above. We compare the results of the CTC model with the numerical solutions of the MNSE for 2 and 3-soliton interactions and find a very good agreement.
Letters in Mathematical Physics, Sep 1, 1982
The main steps of constructing the spectral theory of the integro-differential operator A generat... more The main steps of constructing the spectral theory of the integro-differential operator A generating a class of exactly soluble nonlinear evolution equations (NLEE) related to the linear matrix first-order problem L, are outlined. Compact expressions for the diagonal of the resolvent of operator L through A are obtained.
arXiv (Cornell University), Dec 31, 2009
We analyze the properties of the soliton solutions of a class of models describing one-dimensiona... more We analyze the properties of the soliton solutions of a class of models describing one-dimensional BEC with spin F. We describe the minimal sets of scattering data which determine uniquely both the corresponding potential of the Lax operator and its scattering matrix. Next we give several reductions of these MNLS, derive their N-soliton solutions and analyze the soliton interactions. Finally we prove an important theorem proving that if the initial conditions satisfy the reduction then one gets a solution of the reduced MNLS.
arXiv (Cornell University), Apr 24, 2012
The inverse scattering transform for a special case of the 3-wave resonant interaction equations ... more The inverse scattering transform for a special case of the 3-wave resonant interaction equations with non-vanishing boundary conditions is studied. The Jost solutions and the fundamental analytic solutions (FAS) for the associated spectral problem are constructed. The inverse scattering problem for the Lax operator is formulated as a Riemann-Hilbert problem on a Riemannian surface. The spectral properties of the Lax operator are formulated. 1. the description of the whole class of NLEE related to a given spectral problem (Lax operator L(λ) in the form (2.1)) solvable by the ISM;
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Papers by Vladimir Gerdjikov