Conserved quantities of some Hamiltonian wave equations after full discretization

2006 ◽  
Vol 103 (2) ◽  
pp. 197-223 ◽  
Author(s):  
B. Cano
Keyword(s):  
Wave Equations ◽  
Conserved Quantities ◽  
Full Discretization

Soliton Solutions, Conservation Laws, and Reductions of Certain Classes of NonlinearWave Equations

2012 ◽  
Vol 67 (10-11) ◽  
pp. 613-620 ◽  
Author(s):  
Richard Morris ◽  
Abdul Hamid Kar ◽  
Abhinandan Chowdhury ◽  
Anjan Biswas
Keyword(s):  
Conservation Laws ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lie Symmetry ◽  
Nonlinear Wave Equations ◽  
Conserved Quantities ◽  
Symmetry Approach ◽  
Lie Symmetry Approach

In this paper, the soliton solutions and the corresponding conservation laws of a few nonlinear wave equations will be obtained. The Hunter-Saxton equation, the improved Korteweg-de Vries equation, and other such equations will be considered. The Lie symmetry approach will be utilized to extract the conserved densities of these equations. The soliton solutions will be used to obtain the conserved quantities of these equations.

scholarly journals Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

2016 ◽  
Vol 54 (2) ◽  
pp. 1093-1119 ◽  
Author(s):  
Rikard Anton ◽  
David Cohen ◽  
Stig Larsson ◽  
Xiaojie Wang
Keyword(s):  
Wave Equations ◽  
Multiplicative Noise ◽  
Full Discretization ◽  
Stochastic Wave Equations

scholarly journals A SOLUTION OF THE CONSERVATION LAW FORM OF THE SERRE EQUATIONS

2016 ◽  
Vol 57 (4) ◽  
pp. 385-394 ◽  
Author(s):  
C. ZOPPOU ◽  
S. G. ROBERTS ◽  
J. PITT
Keyword(s):  
Conservation Law ◽  
Wave Equations ◽  
Second Order ◽  
Finite Volume Scheme ◽  
Conserved Quantities ◽  
Shallow Water Wave Equations ◽  
Temporal Derivative ◽  
Second Order Elliptic Equation ◽  
Spatial Terms ◽  
Temporal And Spatial

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations.

scholarly journals Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 850
Author(s):  
Xiao-Qun Cao ◽  
Ya-Nan Guo ◽  
Shi-Cheng Hou ◽  
Cheng-Zhuo Zhang ◽  
Ke-Cheng Peng
Keyword(s):  
Shallow Water ◽  
Nonlinear Equations ◽  
Wave Equations ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Conserved Quantities ◽  
Long Wave ◽  
Complex Models ◽  
Lagrange Functional ◽  
Variational Formulations

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation.

scholarly journals SPATIAL HAMILTONIAN IDENTITIES FOR NONLOCALLY COUPLED SYSTEMS

2018 ◽  
Vol 6 ◽  
Author(s):  
BENTE BAKKER ◽  
ARND SCHEEL
Keyword(s):  
Differential Equations ◽  
Wave Equations ◽  
Broad Class ◽  
Hamiltonian Formalism ◽  
Coupled Systems ◽  
Conserved Quantities ◽  
Gradient Flows ◽  
Lagrange Equations ◽  
Finite Dimensional ◽  
Pattern Forming

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler–Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether’s theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler–Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.

scholarly journals Conservation laws of coupled semilinear wave equations

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640004 ◽  
Author(s):  
Stephen C. Anco ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Wave Equations ◽  
Gordon Equation ◽  
Complete Classification ◽  
Conserved Quantities ◽  
Semilinear Wave Equations ◽  
Two Component ◽  
Klein Gordon ◽  
Klein Gordon Equation

A complete classification of all low-order conservation laws is carried out for a system of coupled semilinear wave equations which is a natural two-component generalization of the nonlinear Klein–Gordon equation. The conserved quantities defined by these conservation laws are derived and their physical meaning is discussed.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Sign in / Sign up

Export Citation Format

Share Document