Nonlinear interaction of head-on solitary waves in integrable and nonintegrable systems

This study numerically investigates the nonlinear interaction of head-on solitary waves in a granular chain(a nonintegrable system)and compares the simulation results with the theoretical results in fluid(an integrable system).Three stages(the pre-in-phase traveling stage,the central-collision stage,and the post-in-phase traveling stage)are identified to describe the nonlinear interaction processes in the granular chain.The nonlinear scattering effect occurs in the central-collision stage,which decreases the amplitude of the incident solitary waves.Compared with the leading-time phase in the incident and separation collision processes,the lagging-time phase in the separation collision process is smaller.This asymmetrical nonlinear collision results in an occurrence of leading phase shifts of time and space in the post-in-phase traveling stage.We next find that the solitary wave amplitude does not influence the immediate space-phase shift in the granular chain.The space-phase shift of the post-in-phase traveling stage is only determined by the measurement position rather than the wave amplitude.The results are reversed in the fluid.An increase in solitary wave amplitude leads to decreased attachment,detachment,and residence times for granular chains and fluid.For the immediate time-phase shift,leading and lagging phenomena appear in the granular chain and the fluid,respectively.These results offer new knowledge for designing mechanical metamaterials and energy-mitigating systems.

An Eight Component Integrable Hamiltonian Hierarchy from a Reduced Seventh-Order Matrix Spectral Problem

We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.

Matrix integrable fifth-order mKdV equations and their soliton solutions

We consider matrix integrable fifth-order mKdV equations via a kind of group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems. Based on properties of eigenvalue and adjoint eigenvalue problems, we solve the corresponding Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues, and construct their soliton solutions, when there are zero reflection coefficients. Illustrative examples of scalar and two-component integrable fifthorder mKdV equations are given.

New Painlevé Integrable(3+1)-Dimensional Combined pKP–BKP Equation:Lump and Multiple Soliton Solutions

We introduce a new form of the Painlevé integrable(3+1)-dimensional combined potential Kadomtsev–Petviashvili equation incorporating the B-type Kadomtsev–Petviashvili equation(pKP–BKP equation). We perform the Painlevé analysis to emphasize the complete integrability of this new(3+1)-dimensional combined integrable equation. We formally derive multiple soliton solutions via employing the simplified Hirota bilinear method. Moreover, a variety of lump solutions are determined. We also develop two new(3+1)-dimensional pKP–BKP equations via deleting some terms from the original form of the combined p KP–BKP equation. We emphasize the Painlevé integrability of the newly developed equations, where multiple soliton solutions and lump solutions are derived as well. The derived solutions for all examined models are all depicted through Maple software.

Lie symmetry analysis and invariant solutions for the(3+1)-dimensional Virasoro integrable model

Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions for the(3+1)-dimensional Virasoro integrable model,including the interaction solution between a kink and a soliton,the lump-type solution and periodic solutions,have been studied analytically and graphically.

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2+ 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.

A New Integrable Lattice Hierarchy and Its Two Discrete Integrable Couplings

A new and efficient way is presented for discrete integrable couplings with the help of two semi-direct sum Lie algebras. As its applications, two discrete integrable couplings associated with the lattice equation are worked out. The approach can be used to study other discrete integrable couplings of the discrete hierarchies of solition equations.

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.

High-Order Binary Symmetry Constraints of a Liouville Integrable Hierarchy and Its Integrable Couplings

A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra A^-2, its integrable couplings are established.

Nonlinear Super Integrable Couplings of A Super Integrable Hierarchy and Its Super Hamiltonian Structures

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.

Nonlinear Super Integrable Couplings of a Super Integrable Hierarchy

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. And its super Hamiltonian structures were established by using super trace identity. As its reduction, special cases of this nonlinear super integrable coupling were obtained.

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti- Johnson （G J） hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their ttamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M- dimensional loop algebra ~X is produced. By taking advantage of ~X a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra ~FM of the loop algebra ~X is presented. Based on the ~FM, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.

Matrix Lie Algebras and Integrable Couplings

Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.

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 s/（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.

The Exact Solutions of Some (2 + 1)-Dimensional Integrable Systems

The equivalence of three (2 + 1)-dimensional soliton equations is proved, and the quite generalsolutionswitha some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, many specialtypes of the localized excitations like the solitoff solitons, multi-dromion solutions, lump, and multi-ring soliton solutionsare obtained.

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure

By using a Lie algebra, an integrable couplings of the classicai-Boussinesq hierarchy is obtained. Then, the Hamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity.

Modified Heisenberg Ferromagnet Model and Integrable Equation

We investigate some integrable modified Heisenberg ferromagnet models by using the prolongation structure theory. Through associating them with the motion of curve in Minkowski space, the corresponding coupled integrable equations are presented.

A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources

A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with sl（4） is presented by Yu. Based on this method, we construct a new integrable couplings of the classical-Boussinesq hierarchy with self-consistent sources by using of loop algebra sl（4）. In this paper, we also point out that there exist some errors in Yu＇s paper and have corrected these errors and set up new formula. The method can be generalized other soliton hierarchy with self-consistent sources.

GENERALIZED FRACTIONAL TRACE VARIATIONAL IDENTITY AND A NEW FRACTIONAL INTEGRABLE COUPLINGS OF SOLITON HIERARCHY

Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.