Conservation Laws and Non-Lie Symmetries for Linear PDEs

Abstract

We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.     

Birkhoff Normal Form for Some Nonlinear PDEs

 We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation

On the Inverse Scattering Method for Integrable PDEs on a Star Graph

We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schrödinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is introduced. For such conditions, the method is shown to be as efficient as the ISM on the full-line.     

Nonlinear Transformations and Integrable Evolution Equations

The transformation properties of the partial differential equations integrable by the inverse scattering transform method are considered. It is shown that the nonlinear transformations characteristic to the integrable equations (symmetry groups, Backlund-transformations) and integrable equations themselves are contained in certain universal nonlinear transformations group which is defined by linear spectral problem.     

Galilean-Invariant Nonlinear PDEs and their Exact Solutions

Abstract

All systems of (n+1)-dimensional quasilinear evolutional second-order equations invariant under the chain of algebras AG(1.n) ? AG ₁(1.n) ? AG ₂(1.n) are described. The obtained results are illustrated by examples of nonlinear Schrödinger equations.     

Van Kampen and Case Formalism Applied to Linear and Weakly Nonlinear Initial Value Problems in Unmagnetized Plasma

A unified account is given of the normal mode theory of homogeneous unmagnetized plasma. Electron (Langmuir), ion sound and electromagnetic waves are treated in stable and (Penrose) unstable plasma with the Van Kampen-Case formalism, using generalized functions. Simplified formulae are derived for second-order Langmuir waves, taking full account of free-streaming effects.     

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.     

Heisenberg-Langevin Formalism for Squeezing Dynamics of Linear Hybrid Optomechanical System

International Journal of Theoretical Physics - A hybrid quantum optomechanical system is an interface made from coupling between photons, excitons and mechanical oscillations. We use the quantum...     

The Quantum Formalism and the GRW Formalism

The Ghirardi–Rimini–Weber (GRW) theory of spontaneous wave function collapse is known to provide a quantum theory without observers, in fact two different ones by using either the matter density ontology (GRWm) or the flash ontology (GRWf). Both theories are known to make predictions different from those of quantum mechanics, but the difference is so small that no decisive experiment can as yet be performed. While some testable deviations from quantum mechanics have long been known, we provide here something that has until now been missing: a formalism that succinctly summarizes the empirical predictions of GRWm and GRWf. We call it the GRW formalism. Its structure is similar to that of the quantum formalism but involves different operators. In other words, we establish the validity of a general algorithm for directly computing the testable predictions of GRWm and GRWf. We further show that some well-defined quantities cannot be measured in a GRWm or GRWf world.     

Comment on “A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map“

Comment on a recent paper on Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 523-528.     

物理实验数据的线性与非线性拟合

Origin是一款制图和数据处理功能强大的应用软件。本文结合大学物理实验,以实例详细介绍如何利用Origin进行实验数据的拟合分析。包括线性回归拟合、多项式拟合和非线性拟合中的一阶指数衰减拟合。实践证明,利用Origin软件对实验数据进行曲线拟合和分析能够显著提高数据处理效率。     

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with N species, M reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs’ macroscopic chemical thermodynamics under G. N. Lewis’ kinetic law of entire equilibrium (detailed balance in nonlinear chemical kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy function and its associated balance equation with nonnegative source and sink are the major discoveries. The proof is based on the large deviation principle of this type of Markov processes. A general fluctuation dissipation theorem for stochastic reaction kinetics is also proved. The mathematical theory illustrates how a novel macroscopic dynamic law can emerges from the mesoscopic kinetics in a multi-scale system.     

Riemann-Hilbert method and N-soliton for two-component Gerdjikov-Ivanov equation

We consider the Riemann–Hilbert method for initial problem of the vector Gerdjikov–Ivanov equation, and obtain the formula for its N-soliton solution, which is expressed as a ratio of (N + 1) × (N + 1) determinant and N × N determinant. Furthermore, by applying asymptotic analysis, the simple elastic interactions of N-soliton are confirmed, and the shifts of phase and position are also explicitly displayed.     

Completely Integrable Equation for the Quantum Correlation Function of Nonlinear Schrödinger Equation

Correlation functions of exactly solvable models can be described by differential equations [1]. In this paper we show that for the non-free fermionic case, differential equations should be replaced by integro-differential equations. We derive an integro-differential equation, which describes a time and temperature dependent correlation function of the penetrable Bose gas. The integro-differential equation turns out to be the continuum generalization of the classical nonlinear Schr?dinger equation. Received: 15 January 1997 / Accepted: 21 March 1997     

线性化求解非线性记录光栅耦合波方程

采用折射率调制特性为上凸型和下凹型的调制曲线,提出了线性化求解非线性记录全息图耦合波方程的方法。以反射相位型全息光栅为例,用耦合波理论分析计算了记录介质折射率调制为上凸型和下凹型调制曲线的体全息图的衍射光谱特性。计算结果表明,折射率调制类型对光栅衍射光谱的带宽有较大影响,对次峰峰值也有影响,而对主峰峰值和峰值位置影响不大。这些结果可以解释实验测得的全息光栅衍射光谱带宽较宽和光栅衍射光谱的多样性。     

Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

Dynamical system theory is applied to the integrable nonlinear wave equation $u_t±(u^3−u^2)x+(u^3)xxx=0$. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation corresponds to the case of wave speed $c$=0. In the case of $c^6$≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.     

Extended Z-Invariance for Integrable Vector and Face Models and Multi-component Integrable Quad Equations

Journal of Statistical Physics - In a previous paper (Kels in J Phys A 50(49):495202, 2017), the author has established an extension of the Z-invariance property for integrable edge-interaction...