Chaos problems in observer's mathematics

This work considers the Hamilton equations of general relativity and quantum theory in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics, see [1], [2], [3]. The probability appears automatically, without apriori assumptions. Certain results and communications pertaining to solutions of these problems are also provided.

When we consider and analyze physical events with the purpose of creating corresponding models we often assume that the mathematical apparatus used in modeling is infallible. In particular, this relates to the use of infinity in various aspects and the use of Newton's definition of a limit in analysis. We believe that is where the main problem lies in contemporary study of nature. This work considers Physical aspects in a setting of arithmetic, algebra, geometry, analysis, topology provided by Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided. In particular, we prove the following Theorems, which give Observer's Mathematics point of view on Einstein photoelectric effect theory and Lamb-Scully and Hanbury-Brown-Twiss experiments: Theorem 1. There are some values of light intensity where anticorrelation parameter A ∈ [0,1). Theorem 2. There are some values of light intensity where anticorrelation parameter A = 1. Theorem 3. There are some values of light intensity where anticorrelation parameter A > 1.

In this short communication, we consider a discrete example of how to perform multiple scale expansions and by starting from the discrete nonlinear Schroodinger equation (DNLS) as well as the Ablowitz-Ladik nonlinear Schrodinger equation (AL-NLS), we obtain the corresponding discrete versions of a Korteweg-de Vries (KdV) equation. We analyze in particular the equation obtained from the AL-NLS and discuss its integrability, as well as its connections with previously studied discrete versions of the KdV equation.

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2 L^2 -based Sobolev spaces H s H^s where local well-posedness is presently known, apart from the H 1 4 ( R ) H^{\frac {1}{4}} (\mathbb {R} ) endpoint for mKdV and the H − 3 4 H^{-\frac {3}{4}} endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura’s transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

Water-wave experiments are presented showing the evolution of finite amplitude waves in relatively shallow water when no solitons are present. In each case, the initial wave is rectangular in shape and wholly below the still water level; the amplitude of the wave is varied. The asymptotic solution of the Korteweg-de Vries (KdV) equation in the absence of solitons (Ablowitz & Segur 1976) is compared with observed evolution. In addition, the asymptotic solution of the linearized KdV equation (a linear dispersive model) is compared with both the KdV solution and experiments. This comparison is a natural consequence of the fact that, in the absence of solitons, the asymptotic solutions of the KdV equation and its linearized version are qualitatively similar. Both the experiments and the model equations suggest that the asymptotic wave structure consists of a negative triangular wave, travelling with a speed (gh)½, followed by a train of modulated oscillatory waves which travel more slowly. Quantitative comparisons are made for the amplitude, shape and decay rate of the leading wave and the amplitude, dominant wavenumbers and velocities of the trailing wave groups. Over the parameter range of the experiments, asymptotic KdV theory predicts more closely the observed behaviour. The leading wave is observed to decay more rapidly than the trailing wave groups; hence the leading wave becomes less prominent with time. This is in agreement with the KdV solution, whereas just the opposite is predicted by linear theory. Linear predictions for the trailing wave groups are accurate only when they agree with the KdV predictions. Both models predict the evolution of short waves in the trailing wave region. When the short waves are unstable (k gt; 1·36), either group disintegration or focusing into envelope solitons is possible. Both of these phenomena are observed in the experiments; neither is predicted by long-wave models. The nonlinear Schrödinger equation is reviewed and tested as a model of these unstable wave groups. There is some evidence that the KdV equation and the nonlinear Schrödinger equation can be patched together to provide an asymptotic description of these unstable groups.

In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator—Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations; however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.

An application of the decomposition method for the generalized KdV and RLW equations

A Hamiltonian formalism has been introduced in (~,2) for periodic and solitonlike solutions of KdV (Korteweg-de Vries) equation. This allows one to use general theorems of classical dynamics (as done, for example, in (a) to obtain explicit solutions of the linearized evolution equations). Poisson's theorem (see e.g. ref, (4)) states that the Yoisson bracket of two constants of motion is also a constant of motion. This fascinating statement leads very often to trivial or dependent constants of motion. In the case of KdV, any two constants chosen among the well-known infinite sequence (5) of polynomials have zero Poisson brackets, as proved by G ~ D ~ , R (~) (see also LAX (e)). Fortunately, this does not hold for the only known (5) constant of motion depending explicitly on space and time, as we shall see here. KdV equation is given by

The Korteweg-de Vries (KdV) equation is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is a most remarkable nonlinear partial differential equation in 1 + 1 dimensions whose solutions can be exactly specified; it has a soliton like solution or solitary wave of sech2 form. Various physical systems of dispersive waves admit solutions in the form of generalized solitary waves. The study of this equation is the archetype of an integrable system and is one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. This article will cover in detail: the KdV equation and the inverse scattering method (based on Schrodinger and Gelfand-Levitan equations) used to solve it exactly.

The excitation of nonlinear electron-acoustic solitary waves (SWs) is studied theoretically in an unmagnetized, uniform, and collisionless plasma. The plasma is composed of two types of electrons referred to as inertial cold electrons and inertialess Kappa (κ) distributed superthermal hot electrons along with stationary ions. Two parameters κ and β are introduced in the governing equations, where the density related parameter is defined by β = n co /n ho (n co and n ho are the cold and hot electron number densities, respectively). Using a reductive perturbation technique, nonlinear Korteweg-de Vries (KdV), modified KdV (mKdV), and standard Gardner (sG) equations have been derived, which admit SW solutions. The KdV equation allows to analyze both compressive (negative) and rarefactive (positive) solitons. But a significant solitary structure vanishes around a critical region corresponding to critical κ c and β c . To look a solitary structure in the critical region, it is proceeded to the next approach of mKdV solution with more higher order calculation of perturbation. But only rarefactive soliton is found in this approach. Furthermore, having higher order perturbation, the next approach of sG equation allows to rescue both compressive and rarefactive solitons even in the critical region. It is observed that the excitation of soliton pulses with various shapes depends on both parameters κ and β.

This study investigates the impact of multiplicative white noise on soliton solutions of the Korteweg–de Vries (KdV) equation, a classical model for nonlinear wave propagation. While the deterministic KdV framework effectively captures soliton dynamics in idealized settings, real-world systems often experience stochastic fluctuations that can significantly affect wave behavior. By introducing space–time-dependent multiplicative noise, we formulate a stochastic KdV (SKdV) equation and derive analytical expressions for both single and multi-soliton solutions using a novel wave transformation in conjunction with the Hirota bilinear method. Numerical simulations across varying noise intensities reveal that stochastic perturbations lead to soliton deformation, amplitude modulation, phase shifts, and disrupted coherence. These effects become more pronounced with stronger noise, altering soliton interactions and long-term stability. The findings emphasize the importance of incorporating stochastic effects into nonlinear wave models, with implications for applications in fluid dynamics, plasma physics, and optical communications.

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation.

The time dependent Schrodinger equation is frequently ‘derived’ by postulating the energy E → iℏ(∂/∂t) and momentum p⃗ → (ℏ/i)∇⃗ operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely ‘a particle is what a particle detector detects.’ This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation.

In this article we propose a reduced model of the input-output behaviour of an arterial compartment, including the short systolic phase where wave phenomena are predominant. The objective is to provide basis for model-based signal processing methods for the estimation from non-invasive measurements and the interpretation of the characteristics of these waves. Standard space discretizations of distributed models of the flow lead to high order models for the pressure wave transfer function, and low order rational transfer functions approximations give poor results. The main idea developed here to circumvent these problems is to explicitly use a propagation delay in the reduced model. Due to phenomena such that peaking and steepening, the considered pressure pulse waves behave more like solitons generated by a Korteweg de Vries (KdV) equation than like linear waves. So we start with a quasi-1D Navier-Stokes equation that takes into account a radial acceleration of the wall, in order to be able to recover, during the reduction process, the dispersive term of KdV equation which, combined with the nonlinear transport term gives rise to solitons. The radial and axial acceleration terms being supposed small, a multiscale singular perturbation technique is used to separate the fast wave propagation phenomena taking place in a boundary layer in time and space described by a KdV equation from the slow phenomena represented by a parabolic equation leading to two-elements windkessel models.

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1088/1751-8113/49/41/415501">2016 <i>J. Phys. A: Math. Theor.</i> <b>49</b> 415501</ext-link>), and also gives the existence space of the local solutions.