Integrable vector isotropic equations admitting differential substitutions of first order

A symmetry classification of integrable vector evolution equations of third order admitting Miura-type transformations is presented. We obtain the Bäcklund autotransformation for the new equation as well as differential substitutions relating the solutions of some integrable isotropic equations.     

多分量AKNS方程的新可积分解

从一个任意阶矩阵谱问题出发,多分量AKNS方程的新可积分解被导出.通过利用迹恒等式建立了其双哈密顿结构.同时,证明了空间与时间的约束流在刘维尔意义下是两个完全可积的哈密顿系统.     

Decomposing a new nonlinear differential-difference system under a Bargmann implicit symmetry constraint

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.     

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.     

On the criteria for integrability of the Liénard equation

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Liénard equation using an approach based on nonlocal transformations. We also obtain some of the previously known criteria for integrability of the Liénard equation as a straightforward consequence of our approach’s application. We illustrate our results by several new examples of integrable Liénard equations.     

The Reduced Ostrovsky Equation: Integrability and Breaking

The reduced Ostrovsky equation is a modification of the Korteweg‐de Vries equation, in which the usual linear dispersive term with a third‐order derivative is replaced by a linear nonlocal integral term, which represents the effect of background rotation. This equation is integrable provided a certain curvature constraint is satisfied. We demonstrate, through theoretical analysis and numerical simulations, that when this curvature constraint is not satisfied at the initial time, then wave breaking inevitably occurs.     

Multiple-front solutions for the Burgers–Kadomtsev–Petviashvili equation

In this work, we study a completely integrable dissipative equation. The Burgers equation is extended by using the sense of the Kadomtsev–Petviashvili (KP) equation. The new established Burgers–KP equation is studied by using the tanh–coth method to obtain kink solutions and periodic solutions. We also apply the powerful Hirota’s bilinear method to establish exact N-soliton solutions for the derived integrable equation.     

Generalized Kadomtsev-Petviashvili equation with an infinite-dimensional symmetry algebra

A generalized Kadomtsev-Petviashvili equation, describing water waves in oceans of varying depth, density and vorticity is discussed. A priori, it involves 9 arbitrary functions of one, or two variables. The conditions are determined under which the equation allows an infinite-dimensional symmetry algebra. This algebra can involve up to three arbitrary functions of time. It depends on precisely three such functions if and only if it is completely integrable.     

Finite-Dimensional Integrable Hamiltonian Systems Related to the Nonlinear Schrödinger Equation

A complete treatment of the binary nonlinearizations of spectral problems of the nonlinear Schrödinger (NLS) equation with the choice of distinct eigenvalue parameters is presented. Two kinds of constraints between the potentials and the eigenfunctions of the NLS equation are considered. From the first constraint, a pair of new finite-dimensional completely integrable Hamiltonian systems which constitute an integrable decomposition of the NLS equation are obtained. From the second constraint, a novel finite-dimensional integrable Hamiltonian system, which includes the system of multiple three-wave interaction as a special case, is obtained. It is found that the eigenvalue parameters real or not can lead to completely different symplectic structures of the restricted NLS flows. In addition, a relationship between the binary restricted Ablowitz–Kaup–Newell–Segur flows and the restricted NLS flows is revealed.     

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint,we obtain a finite dimensional system from the Lax pair of the nonlinear Schr¨odinger equation.We show that this system is a completely integrable Hamiltonian system.     

Dark solitons in N-coupled higher order nonlinear Schrödinger equations

By considering the third order dispersion, self-steepening and stimulated Raman scattering effects, we analyse the dark soliton propagation in N-coupled higher order nonlinear Schrödinger equations. Using Painlevé analysis, we prove that this system is completely integrable. The result is confirmed further by the presentation of Lax pair. Using the Hirota method, the construction of soliton solution is discussed.     

First-order differential substitutions for equations integrable on \mathbb{S}^n

We determine necessary conditions under which integrable vector evolution equations of third order admit Miura-type transformations. For equations integrable on the n-dimensional sphere, we obtain first-order differential substitutions.     

N-dimensional characteristic problem for the Mangeron equation of non-integer order

In the paper we consider a n-dimensional characteristic problem for a certain partial differential equation of non-integer order.We prove the existence and uniqueness of a solution of the problem in the spaces of integrable and continious functions, respectively. Morever, we give sufficient conditions under which the set of solutions is not empty and relatively compact in the space of integrable functions     

Stability of the Camassa-Holm solitons

Summary. We consider the stability problem of the solitary wave solutions of a completely integrable equation that arises as a model for the unidirectional propagation of shallow water waves. We prove that the solitary waves possess the spectral properties of solitons and that their shapes are stable under small disturbances.     

C. Neumann System Associated with the Levi Hierarchy

51.Intr0ductionUsuallyasolitonequationcanbeexpressedasthecompatibleconditionofanover-determinedsystem,theso-calledLaxpair-Wehavea1readyknownthatthetwol1nearequationsintheLaxpaircanben0nlinearizedtobetw0c0mpatible,comp1etelyintegrableHamilt0niansystemsintheLioll"i1lesenseundersomecertainc0nstraintconditionsbetweentheeigenfunctionsandthepotentials(seeL1J).Am0ngtheseconstraintconditionsaretheretheBargmannconstraint{see[1,2,4j),theC.Neumannconstraint(see[l,4J),andthesymmetryconstra1nt(see[5,6j…     

Application of He's variational iteration method and Adomian's decomposition method to Sawada–Kotera–Ito seventh‐order equation

In this article, the Sawada–Kotera–Ito seventh‐order equation is studied. He's variational iteration method and Adomian's decomposition method (ADM) are applied to obtain solution of this equation. We compare these methods together. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 887–897, 2011     

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.     

Asymptotic behavior of solutions of defocusing integrable discrete nonlinear Schrödinger equation

We report our recent result about the long-time asymptotics for the defocusing integrable discrete nonlinear Schr?dinger equation of Ablowitz- Ladik. The leading term is a sum of two terms that oscillate with decay of order t^-1/2     

Stability Conditions for Solutions of Multidimensional Completely Integrable Differential Equations

New sufficient tests are given for the stability and asymptotic stability of the zero solution of a nonautonomous completely integrable equation on an arbitrary salient convex closed cone and on a finitely generated cone. The class of Lyapunov functions suitable for studying the asymptotic behavior of solutions of nonautonomous completely integrable equations is significantly extended by substantially weakening the sign negativeness condition, traditional in the Lyapunov second method, for the derivative of the Lyapunov function at the interior points of the cone.     

The Integrability of Dispersive Hunter-Saxton Equation

In this paper, we prove that the dispersive form of Hunter-Saxton equation is a completely integrable and bi-Hamiltonian system.