A Hierarchy of New Nonlinear Evolution Equations and Their Bi-Hamiltonian Structures

A hierarchy of new nonlinear evolution equations associated with a 3 × 3 matrix spectral problem with four potentials is proposed, in which two typical members are a new coupled Burgers equation and a new coupled KdV equation. The bi-Hamiltonian structures for the hierarchy of nonlinear evolution equations are established by using the trace identity.     

Two integrable generalizations of WKI and FL equations:Positive and negative flows,and conservation laws

With the aid of Lenard recursion equations, an integrable hierarchy of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed, in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati–Konno–Ichikawa(WKI) equation. Further, a new generalization of the Fokas–Lenells(FL) equation is derived from the negative flows. Resorting to these two Lax pairs and Riccati-type equations, the infinite conservation laws of these two corresponding equations are obtained.     

Algebro-Geometric Solutions with Characteristics of a Nonlinear Partial Differential Equation with Three-Potential Functions

With the help of a simple Lie algebra, an isospectral Lax pair, whose feature presents decomposition of element(1, 2) into a linear combination in the temporal Lax matrix, is introduced for which a new integrable hierarchy of evolution equations is obtained, whose Hamiltonian structure is also derived from the trace identity in which contains a constant γ to be determined. In the paper, we obtain a general formula for computing the constant γ. The reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions,the m Kd V equation and a generalized linear Kd V equation. The algebro-geometric solutions(also called finite band solutions) of the generalized nonlinear heat equation are obtained by the use of theory on algebraic curves. Finally, two kinds of gauge transformations of the spatial isospectral problem are produced.     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems （FDHS） in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

A novel hierarchy of differential integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 x 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.     

Conservation Laws and Darboux Transformation for Sharma-Tasso-Olver Equation

A hierarchy of new nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is derived.One of the nontrivial equations in this hierarchy is the famous Sharma-Tasso-Olver equation.Then infinitely many conservation laws of this equation are deduced.Darboux transformation for the Sharma-Tasso-Olver equation is constructed with the aid of a gauge transformation.     

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.     

An extension of integrable equations related to AKNS and WKI spectral problems and their reductions

A novel hierarchy of integrable nonlinear evolution equations related to the combined Ablowitz–Kaup–Newell–Segur(AKNS) and Wadati–Konno–Ichikawa(WKI) spectral problems is proposed,from which the Lax pair for a corresponding negative flow and its infinite many conservation laws are obtained.Furthermore,a reduction of this hierarchy is discussed,by which a generalized sinh-Gordon equation is derived on the basis of its negative flow.     

Two types of loop algebras and their expanding Lax integrable models

Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation U_t-V_x+[U,V]=0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator \widetilde{J} is presented by constructing a subalgebra \widetilde{G } of the loop algebra \widetilde A₂. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their (2+1)-dimensional forms are derived from a (2+1)-dimensional zero-curvature equation.     

On quasi-periodic solutions of the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation

The 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation is decomposed into systems of integrable ordinary differential equations resorting to the nonlinearization of Lax pairs. The Abel–Jacobi coordinates are introduced to straighten the flows, from which quasi-periodic solutions of the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation are obtained in terms of Riemann theta functions.     

The integrability of an extended fifth-order KdV equation with Riccati-type pseudopotential

The extended fifth-order KdV equation in fluids is investigated in this paper. Based on the concept of pseudopotential, a direct and unifying Riccati-type pseudopotential approach is employed to achieve Lax pair and singularity manifold equation of this equation. Moreover, this equation is classified into three categories: extended Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, extended Lax equation and extended Kaup–Kuperschmidt (KK) equation. The corresponding singularity manifold equations and auto-Bäcklund transformations of these three equations are also obtained. Furthermore, the infinitely many conservation laws of the extended Lax equation are found using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas.     

Lie point symmetries,conservation laws,and analytical solutions of a generalized time-fractional Sawada–Kotera equation

Under investigation in this work is the invariance properties of the time-fractional generalized Sawada–Kotera equation, which can describe motion of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice. With the help of the Lie symmetry analysis method of fractional differential equations, we strictly derive the vector fields and symmetry reductions of the equation. Furthermore, based on the power series theory, an effective method is presented to succinctly construct its analytical solutions. Finally, using the new conservation theorem, the conservation laws of the equation are well constructed with a detailed derivation.     

Algebro-Geometric Solutions with Characteristics of a Nonlinear Partial Differential Equation with Three-Potential Functions

With the help of a simple Lie algebra, an isospectral Lax pair, whose feature presents decomposition of element (1, 2) into a linear combination in the temporal Lax matrix, is introduced for which a new integrable hierarchy of evolution equations is obtained, whose Hamiltonian structure is also derived from the trace identity in which contains a constant γ to be determined. In the paper, we obtain a general formula for computing the constant γ. The reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions, the mKdV equation and a generalized linear KdV equation. The algebro-geometric solutions (also called finite band solutions) of the generalized nonlinear heat equation are obtained by the use of theory on algebraic curves. Finally, two kinds of gauge transformations of the spatial isospectral problem are produced.     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

On generalized Lax equation of the Lax triple of KP Hierarchy

In terms of the operator Nambu 3-bracket and the Lax pair (L, B_n) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B_n, B_m). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B_n, B_m). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.     

Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether's theorem

The hierarchy of integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of linearization of these equations and their conservation law in the terms of solutions of corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of linearized equation due to the Noether's theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and conservation laws explicitly expressed through the variables of the nonlinear equations are derived.     

Algebraic-Geometric Solution to （2＋1）-Dimensional Sawada-Kotera Equation

Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.     

Complexiton solutions for two nonlinear partial differential equations via modification of simplified Hirota method

In this study, we obtain complexiton solutions of Sawada–Kotera equation and ninth-order KdV equation. For this cause, we employ Wazwaz and Zhaqilao’s method which can be regarded as generalization of simplified Hirota method through extension real parameters into complex parameters. Special conditions to distinguish complexiton, soliton, and soliton–complexiton interaction solutions from each other are given.     

Prolongation structure of a new integrable system

We have obtained the inverse scattering equations associated with a new pair of coupled nonlinear evolution equations in two dimensions. The spectral parameter is introduced by invoking the invariance of the equation set, and imposing those on the Lax pair.