Darboux Matrices Research Articles

Exact solutions for two dimensional [formula omitted] Toda equation

For the two dimensional Toda equation corresponding to the Kac–Moody algebra Dn(1), the Darboux transformation is constructed. The coefficient matrices of the Lax pair of this equation are of even order. Comparing with the scalars in the Darboux matrices for the two dimensional A2n(2) , Cn(1) and Dn+1(2) Toda equations, the structure of the 2 × 2 blocks in the Darboux matrix for this equation is much more complicated. In the construction of Darboux matrices, it is demanded that the solutions of the Lax pair are in Lag(ℂ2n). With the help of a dense subset of Lag(ℂ2n), the nontrivial blocks of the Darboux matrices are represented by elements of O(n,ℂ). Quite a few algebraic techniques are used to simplify the Darboux matrices and to show that the form of the Darboux matrices only depends on the parity of n.

Rational solutions of Painlevé-II equation as Gram determinant

Under the Flaschka–Newell Lax pair, the Darboux transformation for the Painlevé-II equation is constructed by the limiting technique. With the aid of the Darboux transformation, the rational solutions are represented by the Gram determinant, and then we give the large y asymptotics of the determinant and the rational solutions. Finally, the solution of the corresponding Riemann–Hilbert problem is obtained from the Darboux matrices.

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

AbstractThe complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well‐known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.

The fully supersymmetric AKNS problem: Darboux transformations and discrete systems

The fully supersymmetric AKNS spectral problem with four superfields proposed by Morosi and Pizzocchero is considered. This is a hybrid problem of the classical AKNS and supersymmetric AKNS spectral problems. Under the assumption that Darboux matrices are linear with respect to the spectral parameter, three Darboux transformations and the corresponding Bäcklund transformations are constructed. Through proper reductions, we obtain Darboux transformations for the coupled supersymmetric nonlinear Schrödinger and coupled supersymmetric modified Korteweg–de Vries equations, and recover the Darboux transformations for Manin-Radul supersymmetric Korteweg–de Vries equation and supersymmetric AKNS hierarchy, respectively. The first Darboux-Bäcklund transformation is further applied to construct integrable discrete super systems, and both semi-discrete and fully discrete systems are obtained. The associated continuum limits are studied as well.

Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications

We construct so-called Darboux matrices and fundamental solutions in the important case of generalised Hamiltonian (or canonical) systems depending rationally on the spectral parameter. A wide class of explicit solutions is obtained in this way. Interesting results for dynamical systems depending on several variables and their explicit solutions follow. For these purposes we use our version of Bäcklund-Darboux transformation and square roots of the corresponding generalised matrix eigenvalues. Some new auxiliary results on the roots of matrices are included as well. An appendix is added to make the paper self-contained.

Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions

We propose three spectral problems for NLS-mKdV equation by combining three integrable coupling ways. Then we obtain three nonlinear perturbation terms to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation. We proved the Lax integrability of the integrable nonlinear perturbed hierarchies. On the basis of a special orthogonal group, we prove the Liouville integrability of a third-order integrable nonlinear perturbed hierarchy of NLS-mKdV equation by deriving its bi-Hamiltonian structures. We build three Darboux matrices for constructing the Darboux transformations of the first two equations. As applications of the Darboux transformation, we present explicit solutions of these equations, three-dimensional plots, and density profiles the evolution of solitary waves.

Interactions of localized wave and dynamics analysis in generalized derivative nonlinear Schrödinger equation

The generalized derivative nonlinear Schrödinger equation (g-DNLSE) is a very important integrable system in many fields of physics and applied mathematics. In this paper, we deduce a g-DNLSE which can be reduced to several famous equations. Firstly, for the g-DNLSE, we study the modulational instability which may be the reason for the formation of solitons and rogue waves. Secondly, we select the trivial seed solutions to obtain soliton solutions of the g-DNLSE through two Darboux matrices. Thirdly, using the generalized perturbation (1,N−1)-fold Darboux transformation, we derive smooth-positon solutions and rogue wave solutions of the g-DNLSE. In addition, we obtain the breather solutions and interaction solutions by the generalized perturbation (n,N−n)-fold Darboux transformation. It is worth noting that all of these solutions can change from a strong interaction to a weak interaction by choosing the parameters. This may also be one of the reasons why relevant wave structures presenting diversity. The dynamic behavior of rational soliton solution and multi-rogue wave solution is discussed through numerical simulation, and it may provide a theoretical basis for studying the physical mechanism of multi-rogue wave in optics.

Explicit solutions of rational integrable differential-difference equations

The rational integrable differential-difference equations are proposed from a different discrete spectral problem. We proved the Liouville integrability of rational integrable differential-difference equations by deriving its bi-Hamiltonian structures. Furthermore, infinitely many conservation laws of the hierarchy are listed based on the Lax pairs. We found Darboux matrices to construct their Darboux transformations (DT) of the first two nontrivial equations respectively. Explicit solutions of the first two nontrivial equations are presented and analyzed.

Asymptotic analysis of high-order solitons for the Hirota equation

In this paper, we mainly analyze the long-time asymptotics of high-order soliton for the Hirota equation. Two different Riemann–Hilbert representations of Darboux matrices with high-order soliton are given to establish the relationships between inverse scattering method and Darboux transformation. The asymptotic analysis with single spectral parameter is derived through the formulas of determinant directly. Furthermore, the long-time asymptotics with k spectral parameters is given by combining the iterated Darboux matrices and the result of high-order soliton with single spectral parameter, which discloses the structure of high-order soliton clearly and is possible to be utilized in the optic experiments.

Binary Darboux transformation for general matrix mKdV equations and reduced counterparts

Based on Lax pairs and adjoint Lax pairs, a Darboux transformation is constructed for a class of general matrix mKdV equations and a kind of symmetric integrable reductions of the general case is analyzed. A fundamental step is to formulate a new type of Darboux matrices, in which eigenvalues could be equal to adjoint eigenvalues. From the zero seed solution, the resulting binary Darboux transformation is used to generate soliton solutions for the general matrix mKdV equations and the corresponding reduced counterparts.

Yang–Baxter maps, Darboux transformations, and linear approximations of refactorisation problems

Yang–Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang–Baxter equation. For a set X = Ω × V, where V is a vector space and Ω is regarded as a space of parameters, a linear parametric YB map is a YB map Y: X × X → X × X such that Y is linear with respect to V and one has πY = π for the projection π: X × X → Ω × Ω. These conditions are equivalent to certain parametric nonlinear algebraic relations for the components of Y. Such a map Y may be nonlinear with respect to parameters from Ω. We present general results on such maps, including the clarification of the structure of the algebraic relations that define them and several transformations which allow one to obtain new such maps from known ones. Also, methods for constructing such maps are described. In particular, developing an idea from (Konstantinou-Rizos and Mikhailov 2013 J. Phys. A: Math. Theor. 46 425201), we demonstrate how to obtain linear parametric YB maps from nonlinear Darboux transformations of some Lax operators using linear approximations of matrix refactorisation problems corresponding to Darboux matrices. New linear parametric YB maps with nonlinear dependence on parameters are presented.

Entwining Yang–Baxter maps related to NLS type equations

We construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense.

Nonlinearizations of Darboux transformations of Kaup-Newell equation based on therestricted Kaup-Newell flow

Nonlinearizations of Darboux transformations of Kaup-Newell equation are investigated.Based on the assumption that the maps obtained by nonlinearizing of Darboux transformations of Kaup-Newell equationare just the Backlund transformations of the restricted Kaup-Newell flows, constraints betweenthe potentials in the Darboux matrices and the eigenfunctions can be simply established.Nonlinearizations of four fundamental Darboux transformations are carried out andfour integrable symplectic maps which share the same set of invariants are obtained.

Darboux polynomial matrices: the classical massive Thirring model as a study case

One way of constructing explicit expressions of solutions of integrable systems of partial differential equations goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to obtain such matrices in polynomial form. Our method is illustrated by applying it to the classical massive Thirring model, and by combining it with the Dihedral group of symmetries possessed by this model.

Darboux transformations, finite reduction groups and related Yang–Baxter maps

In this paper, we construct Yang–Baxter (YB) maps using Darboux matrices which are invariant under the action of finite reduction groups. We present six-dimensional YB maps corresponding to Darboux transformations for the nonlinear Schrödinger (NLS) equation and the derivative nonlinear Schrödinger (DNLS) equation. These YB maps can be restricted to four-dimensional YB maps on invariant leaves. The former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries (Fordy A and Kassotakis A 2013 J. Phys. A: Math. Theor. 46 205201). We give a six-dimensional YB-map corresponding to the Darboux transformation for a deformation of the DNLS equation. We also consider vector generalizations of the YB maps corresponding to the NLS and DNLS equations.

Algebraic construction of the Darboux matrix revisited

We present algebraic construction of Darboux matrices for 1+1-dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux–Bäcklund transformation, based on different λ-dependences of the Darboux matrix: polynomial, sum of partial fractions or the transfer matrix form. We derive symmetric N-soliton formulae in the general case. The matrix spectral parameter and dressing actions in loop groups are also discussed. We describe reductions to twisted loop groups, unitary reductions, the matrix Lax pair for the KdV equation and reductions of chiral models (harmonic maps) to SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent Darboux matrix generates the binary Darboux transformation. The paper is intended as a review of known results (usually presented in a novel context) but some new results are included as well, e.g., general compact formulae for N-soliton surfaces and linear and bilinear constraints on the nonisospectral Lax pair matrices which are preserved by Darboux transformations.

Elementary Darboux Transformations and Factorization

A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.

Darboux transformations for the twisted so(p, q) system and local isometric immersion of space forms

For the n-dimensional integrable system with a twisted reduction, Darboux transformations given by Darboux matrices of degree two are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly-independent curvature normals to give the explicit expression of the position vector. Some examples are given from the trivial solutions and standard imbedding .

Nonlinear dynamics of a continuum Heisenberg spin chain with an anisotropy and an external magnetic field

By stereographically projecting the spin vector onto a complex plane in the equations of motion for a continuum Heisenberg spin chain with an anisotropy (an easy plane and an easy axis) and an external magnetic field, the effect of the magnetic field for integrability of the system is discussed. Then, introducing an auxiliary parameter, the Lax equations for Darboux matrices are generated recursively. By choosing the constants, the Jost solutions are satisfied the corresponding Lax equations. The exact soliton solutions are investigated, then the total magnetic momentum and its z-component are obtained. These results show that the solitary waves depend essentially on two velocities which describe a spin configuration deviating from a homogeneous magnetization. The depths and widths of solitary waves vary periodically with time. The center of an inhomogeneity moves with a constant velocity, while the shape of soliton also changes with another velocity and this shape is not symmetrical with respect to the center. The total magnetic momentum and its z-component vary with time.

Solitons in a uniaxial anisotropic Heisenberg spin chain with Gilbert damping in an external magnetic field

Using the stereographic projection of the unit sphere of the spin field onto a complex plane for the equations of motion in a uniaxial anisotropic Heisenberg spin chain with Gilbert damping in an external magnetic field, the effect of the magnetic field for the integrability of the system is discussed. The effect of the Gilbert damping is also analyzed. Then, introducing an auxiliary parameter, the Lax equations for the Darboux matrices are generated recursively. The Jost solutions satisfy the corresponding Lax equations if constants are suitably chosen. The exact soliton solutions are then investigated. These results show that the solitary waves depend essentially on two velocities that describe a spin configuration deviated from a homogeneous magnetization, while the depths and widths of surface of solitary waves vary periodically with time. The center of an inhomogeneity moves with a constant velocity, while the shape of a soliton also changes with another constant velocity and it is not symmetrical with respect to the center. The z component of the total magnetic momentum and the total magnetic momentum vary with time. The asymptotic behavior of multisoliton solutions is also given.