Gerdjikov-Ivanov Research Articles

Exact solutions of the Gerdjikov-Ivanov equation using Darboux transformations

We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transformation for it. The solution is given in terms of quasideterminants. Further, the parabolic, soliton and breather solutions of the GI equation are given as explicit examples.

Algebro-geometric solutions for the Gerdjikov-Ivanov hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the Gerdjikov-Ivanov (GI) hierarchy. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the Baker-Akhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire GI hierarchy.

The rogue wave and breather solution of the Gerdjikov-Ivanov equation

The Gerdjikov-Ivanov (GI) system of q and r is defined by a quadratic polynomial spectral problem with 2 × 2 matrix coefficients. Each element of the matrix of n-fold Darboux transformation (DT) for this system is expressed by a ratio of (n + 1) × (n + 1) determinant and n × n determinant of eigenfunctions, which implies the determinant representation of q[n] and r[n] generated from known solution q and r. By choosing some special eigenvalues and eigenfunctions according to the reduction conditions q[n] = −(r[n])*, the determinant representation of q[n] provides new solutions of the GI equation. As examples, the breather solutions and rogue wave of the GI are given explicitly by the two-fold DT from a periodic “seed” with a constant amplitude.

Two kinds of new integrable decompositions of the Gerdjikov-Ivanov equation

The Gerdjikov-Ivanov(GI) equation can be given from the first non-trivial flow of the generalized Kaup-Newell equations under a reduction condition. In this paper, we perform binary nonlinearization for the generalized Kaup-Newell equations under two symmetry constraints. By imposing the reduction condition on this nonlinearization, we obtain two kinds of new integrable decompositions of the GI equation.

Two expanding integrable systems of the GI soliton hierarchy and a generalized GI hierarchy with self-consistent sources as well as its extension form

By using a Lie algebra G and its loop algebra G∼, a soliton hierarchy of evolution equations is derived from which the well-known Gerdjikov–Ivanov (GI) hierarchy with two potential functions is obtained. With the help of different choices of the modified terms, two expanding integrable systems are produced, one of them has complex Hamiltonian structure. Again by making a linear transformation between G and itself, a complex Lie algebra H is obtained, the corresponding loop algebra H∼ is defined for which a generalized GI hierarchy with self-consistent sources and its expanding integrable model are worked out, respectively.

Uniformly constructing finite-band solutions for a family of derivative nonlinear Schrödinger equations

Abstract Based a spectral problem with an arbitrary parameter and Lenard operator pairs, we derive a generalized Kaup–Newell type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the Kaup–Newell (KN) equation, the Chen–Lee–Liu (CLL) equation, the Gerdjikov–Ivanov (GI) equation, the Burgers equation, the MKdV equation and the Sharma–Tasso–Olver equation. Furthermore, the separation of variables for x - and t m -constrained flows of the the generalized Kaup–Newell hierarchy is shown. Especially the Kundu, KN, CLL and GI equations are uniformly decomposed into systems of solvable ordinary differential equations. A hyperelliptic Riemann surface and Abel–Jacobi coordinates are introduced to straighten the associated flow, from which the algebro-geometric solutions of these equations are explicitly constructed in terms of the Riemann theta functions by standard Jacobi inversion technique.

A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations

By introducing a spectral problem with an arbitrary parameter, we derive a Kaup–Newell-type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the Kaup–Newell (KN) equation, the Chen–Lee–Liu (CLL) equation, the Gerdjikov–Ivanov (GI) equation, the Burgers equation, the modified Korteweg-deVries (MKdV) equation and the Sharma–Tasso–Olver equation. It is shown that the hierarchy is integrable in Liouville’s sense and possesses multi-Hamiltonian structure. Under the Bargann constraint between the potentials and the eigenfunctions, the spectral problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system. The involutive representation of the solutions for the Kaup–Newell-type hierarchy is also presented. In addition, an N-fold Darboux transformation of the Kundu equation is constructed with the help of its Lax pairs and a reduction technique. According to the Darboux transformation, the solutions of the Kundu equation is reduced to solving a linear algebraic system and two first-order ordinary differential equations. It is found that the KN, CLL, and GI equations can be described by a Kundu-type derivative nonlinear Schrödinger equation involving a parameter. And then, we can construct the Hamiltonian formulations, Lax pairs and N-fold Darboux transformations for the Kundu, KN, CLL, and GI equations in explicit and unified ways.

Integrable systems of derivative nonlinear Schrödinger type and their multi-Hamiltonian structure

A spectral problem and the associated hierarchy of Schrodinger type equations are proposed. It is shown that the hierarchy is integrable in Liouville's sense and possesses multi-Hamiltonian structure. It is found that several kinds of important equation such as the Kaup-Newell (KN) equation, the Chen-Lee-Liu (CLL) equation, the Gerdjikov-Ivanov (GI) equation, the modified Korteweg-de Vries equation and the Sharma-Tasso-Olever equation are members in the hierarchy as its special reductions. Moreover, KN, CLL and GI equations are described by using a unified generalized derivative Schrodinger equation involving a parameter, and their Hamiltonian structure and Lax pairs are also given by unified and explicit formulae.

Bi-Hamiltonian Structure and Liouville Integrabilityfor a Gerdjikov-Ivanov Equation Hierarchy

A new spectral problem is introduced and theassociated hierarchy of nonlinear evolution equations is derivedwith the help of the zero curvature equation as well as recursive operators.It is found that the Gerdjikov-Ivanov (GI) equation, which is oneof three important derivative nonlinear Schrödinger equations,exactly belongs to the hierarchy as a special reduction.A powerful tool of the trace identityis used to establish the bi-Hamiltonian structure for the whole GI hierarchy.Moreover, it is shown that GI hierarchy admits an infinite common setof conserved quantities which are in involution in pairs underPoisson's bracket. This indicates that the whole GI hierarchyis completely integrable in Liouville's sense. PACS: 03. 40. Kf, 02. 30. Jr

Integrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation

A spectral problem and the associated Gerdjikov–Ivanov (GI) hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known GI equation of derivative nonlinear Schrödinger equations is obtained. It is shown that the GI hierarchy is integrable in a Liouville sense and possesses bi-Hamiltonian structure. Moreover, the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenfunctions. In particular, an explicit N-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique. Some explicit solitonlike solutions of the GI equation are given by applying its Darboux transformation.

Explicit exact solutions for the Lienard equation and its applications

A kind of explicit exact solution for the Lienard equation is obtained. By means of this solution, we obtain a kind of explicit exact solitary wave solution to the Rangwala-Rao (RR) equation, the Ablowitz (A) equation, the Gerdjikov-Ivanov (GI) equation, etc.

Explicit exact solutions for the Lienard equation and its applications

A kind of explicit exact solution for the Lienard equation is obtained. By means of this solution, we obtain a kind of explicit exact solitary wave solution to the Rangwala-Rao (RR) equation, the Ablowitz (A) equation, the Gerdjikov-Ivanov (GI) equation, etc.