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Envelope solitons in the weakly nonlinear Klein–Gordon equation in 1+1 dimensions are investigated by the asymptotic perturbation (AP) method. Two different types of solitons are possible according to the properties of the dispersion relation. In the first case, solitons propagate with the group velocity (less than the light speed) of the carrier wave, on the contrary in the second case solitons always move with the group velocity of the carrier wave, but now this velocity is greater than the light speed. Superluminal signals are then possible in classical relativistic nonlinear field equations.

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We investigate the interaction among small amplitude water waves, when the fluid motion is in a basin of arbitrary, uniform depth. Waves are supposed to be non-resonant, i.e., with different group velocities that are not close to each other. Starting from the isotropic pseudo-differential Milewski–Keller equation and using an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling, we show that the amplitude slow modulation of Fourier modes can be described by a model system of non-linear evolution equations. We demonstrate that the system is C-integrable, i.e., can be linearized through an appropriate transformation of the dependent and independent variables. A subclass of solutions gives rise to non-localized line-solitons and localized solitons (dromions). Each soliton propagates with the group velocity and during a collision maintains its shape, the only change being a phase shift.

The paper is a rather informal introduction to the concepts and results of the E-infinity Cantorian theory of quantum physics. The fundamental tools of complexity theory and non-linear dynamics (Hausdorff dimensions, fat fractals, etc.) are used to give what we think to be a new interpretation of high energy physics and to determine the corresponding mass-spectrum. Particular attention is paid to the role played by the VAK, KAM theorem, Arnold diffusion, Newhaus sinks and knot theory in determining the stability of an elementary “particle-wave” which emerges in self-organizatory manner out of sizzling vacuum fluctuation.

The book presents an introduction to the theory of solitons, with
emphasis on the background material and introductory concepts of current
research trends. Connections between a nonlinear partial differential
equation that exhibits soliton behavior (the Korteweg-de Vries equation)
and a linear eigenvalue problem are indicated, and one-dimensional
scattering theory and inverse scattering methods are discussed. The
Korteweg-de Vries equation is then treated by inverse scattering
techniques, and the other common soliton equations are introduced,
including the modified Korteweg-de Vries equation, the sine-Gordon
equation and the cubic Schroedinger equation. Applications of soliton
equations in physical situations are considered, with attention given to
shallow water waves, ion plasma waves, one-dimensional dislocation
theory vortex filaments, and coherent optical pulse propagation.
Backlund transformations for the Korteweg-de Vries and other evolution
equations are presented, and soliton perturbation theory is examined in
relation to the Korteweg-de Vries equation and the cubic Schroedinger
equation.

Analytical and numerical investigation of electron acoustic waves shows the existence of interacting dromions solutions. Using the asymptotic perturbation (AP) method, based on Fourier expansion and spatio-temporal rescaling, it is found that the amplitude slow modulation of Fourier modes is described by a system of nonlinear evolution equations. This system is C-integrable, i.e. can be linearized through an appropriate transformation of the dependent and independent variables. We demonstrate the existence of dromion solutions, which propagate with their own group velocity and during a collision maintain their shape, the only change being a phase shift. Numerical results are used to check the validity of the AP method.

Interaction among nonresonant waves of the nonlinear Klein–Gordon equation in ordinary (three-dimensional) space is investigated, by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. We show that the slow amplitude modulation of Fourier modes can be described by a system of nonlinear evolution equations. The system is C-integrable, i.e. can be linearized through an appropriate transformation of the dependent variables. N-period quasiperiodic solutions with a nonlinear dispersion relation are observed. Moreover, envelope solitons with fixed but arbitrary shapes and velocities connected to the group velocities of the carrier waves are possible. During a collision, solitons maintain their shape, but are subjected to a phase shift. The technique proposed in this paper can be applied to the description of soliton interactions in nonlinear dispersive media without using the complexity of the inverse scattering method.

How does a “standing” light wave that is perpendicular to the direction of motion of a receding observer, look to the observer? Both the relativistic Doppler effect and the relativistic conservation of lateral distances, implicit in the Lorentz transformation, are valid. Nevertheless, the size of all objects in the receding frame can be shown to be changed. To keep the observed light consistent with the observed nodal separation, a scale transformation is required. The factor is the same as governs the frequency change. The proposed result is consistent with a recent size-change result obtained in a gravitational setting.

The idea of complex time, as first proposed by El Naschie in 1995, not only provided a very important mathematical utility in clarifying the nature of nowness, but also opened a definite possibility for the instantaneous transmission of information through the theoretical prediction of massless particles travelling at velocities larger than the speed of light. Based on a very simple thought experiment, here we show that the complex nature of time arises when two independent inertial observers, in relative uniform motion, communicate via a light signal in order to compare their own proper time measurements for the same event. The observation that the time employed by the signal to go from one observer to the other is calculable, but not measurable, permits to build up a general expression for the complex time, which not only complies with the possibility of time decomposition into two dimensions, but also conciliates with the idea of a complex space. In particular, we find that El Naschie’s complex time can be interpreted as an asymptotic limit when the velocity of the moving observer equals that of light. Within this new formulation, the inverse Lorentz transformations of special relativity follow as a direct consequence of the complex time.

A weakly nonlinear Lorentz invariant complex field model in 3+1 dimensions is studied by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. It is shown that a nonlinear system of partial differential equations describes oscillation amplitudes of Fourier modes. This system is C-integrable, i.e., can be linearized through a suitable transformation of the dependent and independent variables. We resolve the Cauchy problem and demonstrate that localized nondispersive waves (envelope solitons) with finite energy exist under appropriate initial conditions. These particle-like solutions propagate with the group velocity of their carrier wave. During a collision solitons maintain their shape, because the only change is a phase shift. Energy E and momentum p of solitons are identical to those of a relativistic particle. If the Planck constant is connected to the spatial dimension of the envelope soliton, then we obtain at the lowest order of approximation the quantum relations E=ℏω, λ=h/p, where λ and ω are wavelength and frequency of the carrier wave. This work represents a possible way to achieve the Einstein–de Broglie soliton–particle concept.

The paper is a rather informal introduction to the concepts and results of the E-infinity Cantorian theory of quantum physics. The fundamental tools of complexity theory and non-linear dynamics (Hausdorff dimensions, fat fractals, etc.) are used to give what we think to be a new interpretation of high energy physics and to determine the corresponding mass-spectrum. Particular attention is paid to the role played by the VAK, KAM theorem, Arnold diffusion, Newhaus sinks and knot theory in determining the stability of an elementary “particle-wave” which emerges in self-organizatory manner out of sizzling vacuum fluctuation.

Faster than the speed of light. London: Heinemann

- J Magueijo

Magueijo J. Faster than the speed of light. London: Heinemann; 2003.

- A C Newell

A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, (1985).

- G L Lamb

G. L. Lamb Jr, Elements of Soliton Theory, John Wiley, New York, (1980).