Mathematical aspects of classical nonlinear field equations

Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation.

By employing the terminology of functional differential calculus, Hamilton-Jacobi theory is extended to apply to classical field equations. It is shown that an asymptotic solution to the Hamilton-Jacobi functional differential equation provides an asymptotic general solution to the associated nonlinear classical field equations.

Relativistic solitons and superluminal signals

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

We discuss two aspects of the behavior of solitary waves in non-integrable classical field theories in one spatial dimension and contrast this behavior with that observed in closely related integrable theories. First, we examine the nature of kink/anti-kink interactions in the φ 4 theory, described by the classical field equation $$ {\phi _{tt}} - {\phi _{xx}} - \phi + {\phi ^3} = 0.$$

We study the solvability of one class of nonlinear equations with a small parameter in a Banach space. The main difficulty is that the principal linear part of this equation is non-invertible. To study the solvability of the considered class of equations we apply a new method combining the Pontryagin method from the theory of autonomous systems on the plane and the methods of calculating the rotations of vector fields. We also employ a scheme for matrix representations of split operators known in the bifurcation theory of solutions to nonlinear equations. In contrast to the Pontryagin method, we do not assume a differentiability for a nonlinear mapping and apply methods for calculating the rotations of vector fields. On the base of the proposed method we formulate and prove a theorem on solvability conditions for the considered class of nonlinear equations. As an application, we study two periodic problems for nonlinear differential equations with a small parameter, namely, a periodic problem for the system of ordinary differential equations in a resonance case and a periodic problem for a nonlinear elliptic equations with a non-invertible linear part.

Exact solutions for a generalized Higgs equation

Classical Yang-Mills field equations on Minkowski space with gaugegroup SU(2) are determined. Solutions invariant up to a gauge transformation under a four-dimensional subgroup of the conformal group are evaluated for the Yang-Mills field in the absence of matter. Some properties of the solutions of the equations such as their asymptotic behaviour are obtained.PACS No.: 03.50-z

A space-time functional formalism for classical field equations

The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations.

In this article we present the quaternionic formulation of classical Maxwell-like field equations using complex quaternions and show that in this formalism they get an elegant, economical and compact form. In the quaternionic description of classical fields two quantities are essential: the field quaternion embracing all field variables and the differential quaternionic operator embracing the time and space differential operators. Applying the quaternionic operator on the field quaternion one gets an equation which contains many individual equations. In this formalism the eight equations of the Maxwell equations can be written as one quaternionic. We describe different types of quaternionic fields and discuss their possible occurrence in nature.

In the general relativity theory, using Einstein’s gravity field equation, we discover the spherical solution of the classical quantum gravity. The careful point is that this theory is different from the other quantum theory. This theory is made by the Einstein’s classical field equation.

In the general relativity theory, using Einstein’s gravity field equation, we discover the spherical solution of the classical quantum gravity. The careful point is that this theory is different from the other quantum theory. This theory is made by the Einstein’s classical field equation.

This work presents a predictive and today's most complete model for electrostatic gating of nanofluidic channels. The model is based on the classical mean field equations and accounts for the surface reactivity. The model is successfully verified against previously published experimental data [1]. The fabrication concept of a novel device is introduced and its predicted performances are discussed with the help of the presented model. A ~40x performance improvement, in terms of tunability of the ionic conductivity in the nanochannel, in comparison to the result presented in [1], is expected.