New exact solutions of two-dimensional integrable equations using the $\bar \partial $-dressing method

We review new classes of exact solutions with functional parameters with constant asymptotic values at infinity of the Nizhnik-Veselov-Novikov equation and new classes of exact solutions with functional parameters of two-dimensional generalizations of the Kaup-Kupershmidt and Sawada-Kotera equations, constructed using the Zakharov-Manakov \(\bar \partial \)-dressing method. We present subclasses of multisoliton and periodic solutions of these equations and give examples of linear superpositions of exact solutions of the Nizhnik-Veselov-Novikov equation.     

An extended simplest equation method and its application to several forms of the fifth-order KdV equation

In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada-Kotera, Sawada-Kotera-Parker-Dye, Caudrey-Dodd-Gibbon, Kaup-Kupershmidt, Kaup-Kupershmidt-Parker-Dye, and the Ito forms.     

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     

New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

N-soliton solutions for the integrable bidirectional sixth-order Sawada-Kotera equation

In this work, the integrable bidirectional sixth-order Sawada-Kotera equation is examined. The equation considered is a KdV6 equation that was derived from the fifth order Sawada-Kotera equation. Multiple soliton solutions and multiple singular soliton solutions are formally derived for this equation. The Cole-Hopf transformation method combined with the Hirota’s bilinear method are used to determine the two sets of solutions, where each set has a distinct structure.     

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.     

The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation

We obtain new exact solutions to generalized Sawada-Kotera equation. Using the variational iteration method combined with the improved generalized tanh-coth method, we construct new traveling wave solutions for the standard Sawada-Kotera equation and, by means of scaling, we obtain new solutions to general Sawada-Kotera equation. Periodic and soliton solutions are formally derived for both models.     

New Miura type transformations between integrable dispersive wave equations

In this paper, an algebraic method is devised to construct new Miura type transformations between integrable dispersive wave equations. The characteristic feature of our method lies in that the travelling wave solutions of an aimed equation can be determined by the solutions of a simpler equation directly. Our work is an attempt in searching for travelling solutions of complicate nonlinear equations.     

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.     

From the AKNS system to the matrix Schrödinger equation with vanishing potentials: Direct and inverse problems

We relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schrödinger equation. The corresponding Miura transformation, which allows this connection, converts the focusing matrix nonlinear Schrödinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.     

Integrable Boundary Value Problem for the Boussinesq Equation

We discuss a method for seeking integrable boundary conditions for nonlinear equations. For the Boussinesq equation, we find a new boundary condition that is compatible with the Lax pair and has an infinite set of higher symmetries and a Bäcklund transformation. We construct a class of explicit partial solutions of an equation satisfying this boundary condition.     

一类高阶非线性波方程的子方程与精确行波解

结合子方程和动力系统分析的方法研究了一类五阶非线性波方程的精确行波解.得到了这类方程所蕴含的子方程, 并利用子方程在不同参数条件下的精确解, 给出了研究这类高阶非线性波方程行波解的方法, 并以Sawada Kotera方程为例, 给出了该方程的两组精确谷状孤波解和两组光滑周期波解.该研究方法适用于形如对应行波系统可以约化为只含有偶数阶导数、一阶导数平方和未知函数的多项式形式的高阶非线性波方程行波解的研究.     

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.     

Nonlocal Symmetries of the Camassa-Holm Type Equations

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.     

Geometric finite difference schemes for the generalized hyperelastic-rod wave equation

Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa-Holm equation, the BBM equation and the hyperelastic-rod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity.     

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.     

Hirota-type Equations,Soliton Solutions,Backlund Transformations and Conservation Laws

In this paper, first a class of Hirota-type equations Σ^l_{k=1}H_k(D_x,D_t,D_y)[F_k(D_x,D_t,D_y)f • f] • [G_k(D_x,D_t,D_y)f • f] = 0 are considered. By imposing certain conditions on F_k,G_k and H_k, we show that the abovementioned equation possesses one-soliton solution. Secondly we present a new integrable equation which is an extention of Novikov-Veselov equation and Ito equation. We obtain a Băcklund transformation (BT) of this equation. Finally we consider a generalized equation of Ramani and Sawada-Kotera, and obtain its BT. Starting with the BT an infinite number of conservation laws are derived.