Nonlinear integrable couplings of a nonlinear Schrdinger-modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B 2,a new extended Lie algebra F is constructed.Based on the Lie algebras B 2 and F,the nonlinear Schro¨dinger-modified Korteweg de Vries(NLS-mKdV) hierarchy with self-consistent sources as well as its nonlinear integrable couplings are derived.With the help of the variational identity,their Hamiltonian structures are generated.     

Two new integrable couplings of the soliton hierarchies with self-consistent sources

A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with sl(4) has been presented (Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613). Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl(4). In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.     

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations,we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem.The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair.By Lax pair,an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found.The solutions with specific properties are distinct from those of the local Hirota equation.In order to further describe the properties and the dynamic features of the solutions explicitly,several kinds of graphs are depicted.     

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.     

Multi-Soliton Solutions for the Coupled Fokas–Lenells System via Riemann–Hilbert Approach

We aim to construct multi-soliton solutions for the coupled Fokas–Lenells system which arises as a model for describing the nonlinear pulse propagation in optical fibers. Starting from the spectral analysis of the Lax pair, a Riemann–Hilbert problem is presented. Then in the framework of the Riemann–Hilbert problem corresponding to the reflectionless case, N-soliton solutions to the coupled Fokas–Lenells system are derived explicitly.     

A Chain of Type Ⅱ and Its Exact Solutions

The exact solutions of a chain of type Ⅱ are investigated.The chain of type E is first transformed to an integrable differential-difference equation,which has the Kaup-Newell spectral problem as its continuous spatial spectral problem and a Darboux transformation of the Kaup-Newell equation as its discrete temporal spectral problem.Then,with these spectral problems,a Darboux transformation of the transformed equation is constructed.Finally,as an application of the Darboux transformation,an exact solution of the transformed equation and thus the chain of type Ⅱ are presented.     

A hierarchy of lattice soliton equations and its Darboux transformation

A new discrete spectral problem is introduced and the corresponding hierarchy of the lattice soliton equations are derived by means of the trace identity. We find a new Darboux transformation of the lattice soliton equation, through which the explicit solutions are shown.     

An extension of the modified Sawada—Kotera equation and conservation laws

Based on the modified Sawada-Kotera equation, we introduce a 3 × 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawada-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawada-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawada-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.     

Construction of Multi-soliton Solutions of the N-Coupled Hirota Equations in an Optical Fiber

This work aims to study the N-coupled Hirota equations in an optical fiber under the zero boundary condition at infinity. By analyzing the spectral problem, a matrix Riemann–Hilbert problem on the real axis is strictly established. Then, by solving the presented matrix Riemann–Hilbert problem under the constraint of no reflection,the bright multi-soliton solutions to the N-coupled Hirota equations are explicitly gained.     

Multisoliton Solutions for the Isospectral and Nonisospectral BKP Equation with Self-Consistent Sources

The isospectral and nonisospectral BKP equation with self-consistent sources is derived to study the linear problem of the BKP system. The bilinear form of the nonisospeetral BKP equation with self-consistent sources is given and the N-soliton solutions are obtained with the Hirota method and Pfaffian technique, respectively.     

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.     

Integrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem

Based on a new discrete three-by-three matrix spectral problem, a hierarchy of integrable lattice equations with three potentials is proposed through discrete zero-curvature representation, and the resulting integrable lattice equation reduces to the classical Toda lattice equation. It is shown that the hierarchy possesses a HamiItonian structure and a hereditary recursion operator. Finally, infinitely many conservation laws of corresponding lattice systems are obtained by a direct way.     

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.     

New Positive and Negative Hierarchies of Integrable Differential-Difference Equations and Conservation Laws

Two hierarchies of nonlinear integrable positive and negative lattice equations are derived from a discrete spectrak problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws about the positive hierarchy.     

Multiply Kink and Anti-Kink Solutions for a Coupled Camassa–Holm Type Equation

This article amis at revealing dynamical behavior of a coupled Camassa–Holm type equation, which was proposed by Geng and Wang based on a 4×4 matrix spectral problem with two potentials. Its kink and anti-kink solutions are presented explicitly. In particular, some exact multi-kink and anti-kink wave solutions are discussed and under some conditions, the kink and anti-kinks look like hat-shape solitons. The dynamic characters of the obtained solutions are investigated by figures. The method used in this paper can be widely applied to looking for the multi-kinks for Camassa–Holm type equations possessing cubic nonlinearity.     

Hierarchy property of traffic networks

The flourishing complex network theory has aroused increasing interest in studying the properties of real-world networks. Based on the traffic network of Chang--Zhu--Tan urban agglomeration in central China, some basic network topological characteristics were computed with data collected from local traffic maps, which showed that the traffic networks were small-world networks with strong resilience against failure; more importantly, the investigations of assortativity coefficient and average nearest-neighbour degree implied the disassortativity of the traffic networks. Since traffic network hierarchy as an important basic property has been neither studied intensively nor proved quantitatively, the authors are inspired to analyse traffic network hierarchy with disassortativity and to finely characterize hierarchy in the traffic networks by using the n-degree--n-clustering coefficient relationship. Through numerical results and analyses an exciting conclusion is drawn that the traffic networks exhibit a significant hierarchy, that is, the traffic networks are proved to be hierarchically organized. The result provides important information and theoretical groundwork for optimal transport planning.     

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.     

Inverse Scattering Transform of the Coupled Sasa–Satsuma Equation by Riemann–Hilbert Approach

The inverse scattering transform of a coupled Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Firstly, the spectral analysis is performed for the coupled Sasa–Satsuma equation, from which a Riemann–Hilbert problem is formulated. Then the Riemann–Hilbert problem corresponding to the reflection-less case is solved.As applications, multi-soliton solutions are obtained for the coupled Sasa–Satsuma equation. Moreover, some figures are given to describe the soliton behaviors, including breather types, single-hump solitons, double-hump solitons, and two-bell solitons.     

Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.     

Multi-soliton solutions of a two-component Camassa–Holm system:Darboux transformation approach

We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.