Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

The approximate inertial manifolds (AIMs) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigenfunctions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin’s procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.     

Decomposition of a hierarchy of nonlinear evolution equations

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

A nonlinear finite element framework for flexible multibody dynamics

We present a uniform treatment of rigid body dynamics and nonlinear structural dynamics. The advocated approach is based on a rotationless formulation of rigid bodies, nonlinear beams and shells. In this connection, the specific kinematic assumptions are taken into account by the explicit incorporation of holonomic constraints. This approach facilitates the straightforward extension to flexible multibody dynamics by including additional constraints due to the interconnection of rigid and flexible bodies. We further address the design of energy-momentum schemes for the stable numerical integration of the underlying finite-dimensional Hamiltonian systems. To demonstrate the superior numerical performance of the proposed methodology, the numerical examples deals with a multibody system containing both rigid and flexible bodies undergoing large deformations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ermakov–Ray–Reid Systems in (2+1)‐Dimensional Rotating Shallow Water Theory

A (2+1)‐dimensional rotating shallow water system with an underlying circular paraboloidal bottom topography is shown to admit a multiparameter integrable nonlinear subsystem of Ermakov–Ray–Reid type. The latter system, which describes the time evolution of the semi‐axes of the elliptical moving shoreline on the paraboidal basin, is also Hamiltonian. The complete solution of the generic eight‐dimensional dynamical system governing the reduction is obtained in terms of an elliptic integral representation.     

A New Hierarchy Soliton Equations Associated with a Schrdinger Type Spectral Problem and the Corresponding Finite-dimensional Integrable System

By introducing a Schrodinger type spectral problem with four potentials, we derive a new hierarchy nonlinear evolution equations. Through the nonlinearization of eigenvalue problems, we get a new finite-dimensional Hamiltonian system, which is completely integrable in the Liouville sense.     

Dissipative and Hamiltonian Systems with Chaotic Behavior: An Analytic Approach

Some classes of dissipative and Hamiltonian distributed systems are described. The dynamics of these systems is effectively reduced to finite-dimensional dynamics which can be unboundedly complex in a sense. Yarying the parameters of these systems, we can obtain an arbitrary (to within the orbital topological equivalence) structurally stable attractor in the dissipative case and an arbitrary polynomial weakly integrable Hamiltonian in the conservative case. As examples, we consider Hopfield neural networks and some reaction–diffusion systems in the dissipative case and a nonlinear string in the Hamiltonian case.     

Nonlinear set-valued boundary value problems with continuous feedback

We obtain sufficient conditions for the existence of at least one absolutely continuous solution of a nonlinear functional-differential inclusion in a finite-dimensional space with nonlinear set-valued functional boundary conditions. The set-valuedness of the dynamics may be due to the presence of a control. In addition, we consider the case in which the set-valuedness of the system dynamics and the boundary conditions is specified by inequalities. We assume the presence of a continuous feedback and impose the requirements of solvability of the open-loop problem. Statements of problems of this type arise in connection with the analysis of conflict-control systems.     

Finite dimensionality and compactness of attractors for von Karman equations with nonlinear dissipation

Asymptotic behaviour of dynamics governed by PDE system describing nonlinear vibrations of a shell immersed in a supersonic gas is considered. The undelying dynamics is modeled by a nonlinear hyperbolil-like shell equation with a nonlinear dissipation. It is shown that all finite energy ("weak") solutions converge to a global, compact attractor which is also finite-dimensional.     

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.     

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I. Finite-dimensional discretization

We consider discretized Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite-dimensional Birkhoff normal form result, we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level of space approximation, provided the Sobolev norm of the initial data is small enough. This result holds under generic non-resonance conditions on the frequencies of the linear operator and on the step size. We apply these results to nonlinear Schrödinger equations as well as the nonlinear wave equation.     

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.     

Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy

The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map.     

DECOMPOSITION OF(1+1)-DIMENTIONAL,(2+1)-DIMENTIONAL SOLITON EQUATIONS AND THEIR QUASI-PERIODIC SOLUTIONS

A new spectral problem is proposed, and nonlinear differential equations of the corresponding hierarchy are obtained. With the help of the nonlinearization approach of eigenvalue problems, a new finite-dimensional Hamiltonian system on R2 nis obtained. A generating function approach is introduced to prove the involution of conserved integrals and its functional independence, and the Hamiltonian flows are straightened by introducing the Abel-Jacobi coordinates. At last, based on the principles of algebra curve, the quasi-periodic solutions for the corresponding equations are obtained by solving the ordinary differential equations and inversing the Abel-Jacobi coordinates.     

与一族(1+1)维孤子方程相联系的有限维可积系（英文）

In this paper, a new spectral problem is proposed and the corresponding soliton equations hierarchy are also obtained. Under a constraint between the potentials and the eigenfunctions, the eigenvalue problem is nonlinearized so as to be a new finite-dimensional Hamiltonian system. By resotring to the generating function approach, we obtain conserved integrals and the involutivity of the conserved integrals. The finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Finally, we show the decomposition of the soliton equations.     

Interacting Wave Fronts and Rarefaction Waves in a Second Order Model of Nonlinear Thermoviscous Fluids

A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation.