Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 35–48, July, 2009.     

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.     

Multiple complex soliton solutions for integrable negative-order KdV and integrable negative-order modified KdV equations

In this work we show that the integrable negative-order Korteweg–de Vries (nKdV) and the integrable negative-order modified Korteweg–de Vries (nMKdV) equation admit multiple complex soliton solutions. To achieve this goal, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the nKdV equation, and the other form is nicely applicable for the nMKdV equation. We believe that the newly proposed complex forms and the obtained findings will shed light on complex solitons of other integrable equations.     

The integrability and parametric representation of a hierarchy of soliton equations

In this paper after having obtained the Lax pair of a hierarchy of soliton equations,we discuss the parametric representation for finite-band solutions of the stationary solitonequation, and prove it can be represented as a Hamiltonian system which is integrable inLiouville sense. The nonconfocal involutive integral representations {Fm} are obtained also.In the condition of finite-band solutions of the soliton equation, the time and space can bedevided inio two Hamiltonian systems, so the fi…     

The method of generalized Cole-Hopf substitutions and new examples of linearizable nonlinear evolution equations

We propose a new approach for constructing nonlinear evolution equations in matrix form that are integrable via substitutions similar to the Cole-Hopf substitution linearizing the Burgers equation. We use this new approach to find new integrable nonlinear evolution equations and their hierarchies. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 58–71, January, 2009     

Commutator identities on associative algebras and the integrability of nonlinear evolution equations

We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations. In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear integrable equation itself and its Lax pair from such a commutator identity. The problem of constructing new integrable nonlinear evolution equations thus reduces to the problem of constructing commutator identities on associative algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 477–491, March, 2008.     

The Hirota’s bilinear method and the tanh–coth method for multiple-soliton solutions of the Sawada–Kotera–Kadomtsev–Petviashvili equation

In this work we derive a new completely integrable dispersive equation. The equation is obtained by combining the Sawada–Kotera (SK) equation with the sense of the Kadomtsev–Petviashvili (KP) equation. The newly derived Sawada–Kotera–Kadomtsev–Petviashvili (SK–KP) equation is studied by using the tanh–coth method, to obtain single-soliton solution, and by the Hirota bilinear method, to determine the N-soliton solutions. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations.     

A family of integrable differential-difference equations and its Bargmann symmetry constraint

A family of integrable differential-difference equations is constructed through discrete zero curvature equation. The Hamiltonian structures of the resulting differential-difference equations are established by the discrete trace identity. The Bargmann symmetry constraint of the resulting family is presented. Under this symmetry constraint, every differential-difference equation in the resulting family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.     

Integrability conditions for a certain class of nonlinear evolution equations and Kähler geometry

Multicomponent evolution equations associated with linear connections on complex manifolds are considered. It is proved that under some general assumptions an equation from this class is integrable by inverse scattering method if the corresponding linear connection is the Levi-Civita connection of an indefinite Kählerian metric of constant holomorphic sectional curvature. This result is based on a certain characterization of the above-mentioned Levi-Civita connections. It is shown that the obtained integrable equations are generalized ferromagnetics, and recurrent formulas for their local conservation laws are given.     

Nonlinear self-adjointness of the Krichever–Novikov equation

It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie–Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever–Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever–Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.     

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.     

Some Exact Solutions of Two Fifth Order KdV-type Nonlinear Partial Differential Equations

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries (fKdV) equation. The extended Tanh method has been used rigorously, by computational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.     

On classification of second-order differential equations with complex coefficients

This paper is concerned with the deficiency index problem of second-order differential equations with complex coefficients. It is known that this class of equations is classified into cases I, II, and III according to the number of linearly independent solutions in suitable weighted square integrable spaces. In this study, the original equation is reformulated into a new formally self-adjoint differential system by introducing a new spectral parameter and the relationship between the classifications of the equation and the system is obtained. Moreover, the exact dependence of cases II and III on the corresponding half planes is given and some criteria of the three cases are established.     

Integrable discretization and numerical simulations of the generalized coupled integrable dispersionless equations

In this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.     

Stretching and Shrinking of the Loop Soliton Interacting with a External Field

The integrable equation of motion of the loop soliton interacting with an external field is considered from the standpoint of stretching and/or shrinking of the loop. To study the role of the elastic force and the nonlinear forces, the basic equation is divided into three equations. We obtain stationary solutions for these equations and numerically solve their initial value problems to seek stability of the loop soliton.     

Periodic Liénard-type delay equations with state-dependent impulses

Conditions are obtained for Liénard-type equations with delay and state-dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation (and consequently with Lebesgue integrable second derivative). The results are applied to Josephson's equation and the nonconservative forced pendulum equation.     

Effect of inhomogeneity in energy transfer through alpha helical proteins with interspine coupling

The dynamics of homogeneous and inhomogeneous alpha helical proteins with interspine coupling is under investigation in this paper by proposing a suitable model Hamiltonian. For specific choice of parameters, the dynamics of homogeneous alpha helical proteins is found to be governed by a set of completely integrable three coupled derivative nonlinear Schrödinger (NLS) equations (Chen–Lee–Liu equations). The effect of inhomogeneity is understood by performing a perturbation analysis on the resulting perturbed three coupled NLS equation. An equivalent set of integrable discrete three coupled derivative NLS equations is derived through an appropriate generalization of the Lax pair of the original Ablowitz–Ladik lattice and the nature of the energy transfer along the lattice is studied.     

Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation

Two hierarchies of integrable positive and negative lattice equations in connection with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions respectively of Lax operators with respect to the spectral parameter, and each equation in the resulting hierarchies is Liouville integrable. Moreover, infinitely many conservation laws of corresponding positive lattice equations are obtained in a direct way. Finally, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations, by means of which the exact solutions are given.     

Construction of a hierarchy of negative‐order integrable Burgers equations of higher orders

In this work, we employ the recursion operator, the Burgers equation and its inverse operator, for constructing a hierarchy of negative‐order integrable Burgers equations of higher orders. The complete integrability of each established equation emerges by virtue of the correlation between integrability and recursion operators. We use the simplified Hirota's method to obtain multiple kink solutions for some of the derived equations, and in particular, for the generalized negative‐order Burgers equation.     

二阶多项式系统的可积条件

本文推广了 Liouville关于方程可积性的定义 ,定义二阶多项式系统 ( * )的可积性为首次积分可由P( x,y) ,Q( x,y)通过有限次代数运算 ,积分 ,微分 ,指数运算和解代数方程得到 .证明了与二阶多项式系统相对应的一阶算子具有由定理给出的某种“特征”是该系统可积的充分条件 .最后 ,利用此结果给出了Burgers-K-d V方程的行波解的首次积分 .