THE NONLINEARIZATION OF A (2+1)-DIMENSIONAL SOLITON EQUATION

2008 ◽  
Vol 22 (32) ◽  
pp. 3179-3194
Author(s):  
QIANG LIU ◽  
DIAN-LOU DU
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Hamiltonian System ◽  
Eigenvalue Problem ◽  
Hamiltonian Systems ◽  
Soliton Equation ◽  
Completely Integrable ◽  
Finite Dimensional ◽  
Dimensional Soliton

Based on a 2 × 2 eigenvalue problem, a new (2+1)-dimensional soliton equation is proposed. Moreover, we obtain a finite-dimensional Hamiltonian system. Then we verify it is completely integrable in the Liouville sense. In the end, we introduce a set of Hk polynomial integrable, by which we can separate the solition equation into three compantiable Hamiltonian systems of ordinary differential equation.

scholarly journals Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation

2003 ◽  
Vol 2003 (8) ◽  
pp. 409-427 ◽  
Author(s):  
Robert Willie
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Equation ◽  
Asymptotic Behaviour ◽  
Nonlinear Term ◽  
Damped Wave Equation ◽  
Order Ordinary Differential Equation ◽  
Finite Dimensional ◽  
Type Boundary ◽  
Large Diffusion

We study the effects of large diffusivity in all parts of the domain in a linearly damped wave equation subject to standard zero Robin-type boundary conditions. In the linear case, we show in a given sense that the asymptotic behaviour of solutions verifies a second-order ordinary differential equation. In the semilinear case, under suitable dissipative assumptions on the nonlinear term, we prove the existence of a global attractor for fixed diffusion and that the limiting attractor for large diffusion is finite dimensional.

scholarly journals Numerical simulation of finite dimensional multibody nonsmooth mechanical systems

2002 ◽  
Vol 55 (2) ◽  
pp. 107-150 ◽  
Author(s):  
B Brogliato ◽  
AA ten Dam ◽  
L Paoli ◽  
F Ge´not ◽  
M Abadie
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Ordinary Differential Equation ◽  
Mechanical Systems ◽  
Difficult Problem ◽  
Review Article ◽  
Differential Algebraic Equation ◽  
Complementarity Conditions ◽  
Finite Dimensional ◽  
Impact Phenomena

This review article focuses on the problems related to numerical simulation of finite dimensional nonsmooth multibody mechanical systems. The rigid body dynamical case is examined here. This class of systems involves complementarity conditions and impact phenomena, which make its study and numerical analysis a difficult problem that cannot be solved by relying on known Ordinary Differential Equation (ODE) or Differential Algebraic Equation (DAE) integrators only. The main techniques, mathematical tools, and existing algorithms are reviewed. The article utilizes 233 references.

scholarly journals On Nonlinear Profile Decompositions and Scattering for an NLS–ODE Model

2018 ◽  
Vol 2020 (18) ◽  
pp. 5679-5722
Author(s):  
Scipio Cuccagna ◽  
Masaya Maeda
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Hamiltonian System ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Golden Rule ◽  
Simplified Model ◽  
Radial Solutions ◽  
Fermi Golden Rule ◽  
Ode Model

Abstract In this paper, we consider a Hamiltonian system combining a nonlinear Schrödinger equation (NLS) and an ordinary differential equation. This system is a simplified model of the NLS around soliton solutions. Following Nakanishi [33], we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishi’s framework and Fermi Golden Rule estimates on $L^4$ in time norms.

Decomposing the Toda hierarchy by an implicit symmetry constraint

2019 ◽  
Vol 33 (03) ◽  
pp. 1950028
Author(s):  
Xi-Xiang Xu ◽  
Min Guo ◽  
Ning Zhang
Keyword(s):  
Hamiltonian System ◽  
Toda Lattice ◽  
Lattice Equation ◽  
Symmetry Constraint ◽  
Completely Integrable ◽  
Toda Hierarchy ◽  
Finite Dimensional ◽  
Symplectic Map ◽  
Toda Lattice Hierarchy

An implicit symmetry constraint of the famous Toda lattice hierarchy is presented. Using this symmetry constraint, every lattice equation in the Toda hierarchy is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.

COMPLETELY AND PARTIALLY INTEGRABLE HAMILTONIAN SYSTEMS IN THE NONCOMPACT CASE

2004 ◽  
Vol 01 (03) ◽  
pp. 167-183 ◽  
Author(s):  
EMANUELE FIORANI
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Integrable Hamiltonian System ◽  
Time Dependent ◽  
Integrable Hamiltonian Systems ◽  
Invariant Submanifold ◽  
Completely Integrable ◽  
Completely Integrable Hamiltonian System

Action-angle coordinates are shown to exist around an instantly compact invariant submanifold of a time-dependent completely integrable Hamiltonian system. Partially integrable Hamiltonian systems are also considered in the noncompact case; a comparison is made with other possible approaches. Results on symplectically complete foliations contained in the Appendix A can be used to give alternative proofs of some propositions.

scholarly journals A finite-dimensional integrable system associated with a polynomial eigenvalue problem

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
Taixi Xu ◽  
Weihua Mu ◽  
Zhijun Qiao
Keyword(s):  
Eigenvalue Problem ◽  
Integrable System ◽  
Hamiltonian Systems ◽  
Second Order ◽  
Integrable Hamiltonian Systems ◽  
Hamiltonian Structures ◽  
Polynomial Eigenvalue Problem ◽  
Constrained Hamiltonian Systems ◽  
Finite Dimensional ◽  
Order Polynomial

M. Antonowicz and A. P. Fordy (1988) introduced the second-order polynomial eigenvalue problemLφ=(∂2+∑i=1nviλi)φ=αφ(∂=∂/∂x,α=constant)and discussed its multi-Hamiltonian structures. Forn=1andn=2, the associated finite-dimensional integrable Hamiltonian systems (FDIHS) have been discussed by Xu and Mu (1990) using the nonlinearization method and Bargmann constraints. In this paper, we consider the general case, that is,nis arbitrary, provide the constrained Hamiltonian systems associated with the above-mentioned second-order polynomial ergenvalue problem, and prove them to be completely integrable.

scholarly journals Precise Asymptotics for Bifurcation Curve of Nonlinear Ordinary Differential Equation

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1272 ◽  
Author(s):  
Tetsutaro Shibata
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Nonlinear Ordinary Differential Equation ◽  
Precise Asymptotics ◽  
Bifurcation Curve ◽  
The Third ◽  
Precise Asymptotic

We study the following nonlinear eigenvalue problem −u″(t)=λf(u(t)),u(t)>0,t∈I:=(−1,1),u(±1)=0, where f(u)=log(1+u) and λ>0 is a parameter. Then λ is a continuous function of α>0, where α is the maximum norm α=∥uλ∥∞ of the solution uλ associated with λ. We establish the precise asymptotic formula for λ=λ(α) as α→∞ up to the third term of λ(α).

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