Obliquely propagating shock-like structures and nonlinear solitary holes in weakly relativistic magneto-plasma

A theoretical investigation is carried out for understanding the properties of obliquely propagating shock-like structures in weakly relativistic magneto-plasma. By using the Sagdeev's pseudo-potential method, we have found the obliquely propagating shock-like solution and the relation between the amplitude, the inverse scale length, the relativistic effects and the effects of obliqueness. It is shown that the shock-like structure is nonlinear extension of the solitary hole having negative trapping parameter in weakly relativistic magneto-plasma.     

The formally variable separation approach for the dust-acoustic solitary waves with dust charge variation

The formally variable separation approach is used for handling the dust-acoustic solitary waves in a dusty plasma, including consideration of dust charge variation. New analytical solutions of nonlinear waves are formally derived for the governing equation of the system. We have triumphantly derived the exact analytical expressions and some approximate expressions of the nonlinear dust-acoustic waves in a dusty plasma under some special cases. The work introduces entirely new solutions and emphasizes the power of the newly developed method that can be used in problems with identical nonlinearities.     

Multidimensional dust acoustic solitary waves in inhomogeneous magnetized dusty plasmas with nonthermal ions

The linear dispersion relation and a modified variable coefficients Korteweg–de Vries (MKdV) equation governing the three-dimensional dust acoustic solitary waves are obtained in inhomogeneous dusty plasmas comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and nonthermally distributed ions. The numerical results show that the inhomogeneity, the nonthermal ions, the external magnetic field and the collision have strong influence on the frequency and the nonlinear properties of dust acoustic solitary waves and both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump) are excited.     

Nuclear solitary waves

This paper gives an overview of our recent work on the fission solitary wave reactor concept. In order to gain an insight into this problem a simple but appropriate model, namely a one–group diffusion equation coupled with simplified burn–up equations of e.g. the common U238–Pu239 conversion chain is studied. It is shown that this coupled system is analytically solvable in its one–dimensional case and a permanent plane solitary wave exists in a fertile medium, e.g. natural uranium. This wave mechanism could realize a nuclear reactor, which possess a constant reactivity and, more importantly, improves significantly the utilization of nuclear fuel. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attenuation of waves propagating in fluid mixtures

Wave propagation in fluid mixtures is investigated on the basis of effective models of block and layered media. These models are anisotropic fluids described by wave equations. In the equations, additional terms describing wave attenuation are introduced. The attenuation is related to a friction force proporitional to the difference of tangent displacements on the boundaries. Owing to attenuation, the total energy of the wave field decreases steadily and the amplitudes of waves are diminished expotentially with time, which is determined by attenuation coefficients. The attenuation coe.cients are found in the cases where two fluids are mixed completely and where the particles of one fluid are inclusions into the other. The approach suggested enables one to consider more complicated fluid mixtures as well. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 148–179.     

Analytical study of the nonlinear dust-acoustic waves in an unmagnetized dusty plasma

The nonlinear dust-acoustic waves in an unmagnetized dusty plasma, including consideration of the dust charge variation, is analytically investigated by using the formally variable separation approach. The exact analytical solutions in the general case are also obtained.     

Symmetry of solitary waves

It is shown that all supercritical solitary wave solutions to the equations for water waves are symmetric, and monotone on either side of the crest. The proof is based on the Alexandrov method of moving planes. Further a priori estimates, and asymptotic decay properties of solutions are derived     

Diffraction of solitary waves

Whitham's extension of geometrical optics to nonlinear diffraction is applied to solitary waves of reference amplitudea ₀ in water of uniform depthd ₀ on the hypotheses thata ₀d ₀ and that the angle through which a diffracted wave is turned is of the order of (a ₀/d ₀)^1/2. The equations governing the amplitude and direction of the waves are reduced to a quasi-linear, hyperbolic systemof two first-order partial differential equations. Explicit results are obtained for diffraction by a convex bend and by a concave corner, and it is found that a solitary wave of initial amplitudea ₀ cannot be turned through a convex angle greater than (3a ₀/d ₀)^1/2 without separating or otherwise losing its identity. An empirical generalization for larger amplitudes and turning angles is proposed. General solutions are obtained (in an appendix) through a hodograph transformation.

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Zusammenfassung Whitham's Erweiterung der geometrischen Optik auf nichtlineare Diffraktion wird auf Einzelwellen (solitary waves) angewendet. Dabei wird angenommen, dass die Referenzamplitudea ₀ viel kleiner als die konstante, ungestörte Wassertiefed ₀ sei und dass der Ablenkungswinkel die Grössenordnung (a ₀/d ₀)^1/2 habe. Die Gleichungen für Amplitude und Richtung der Wellen werden auf ein quasi-lineares, hyperbolisches System von zwei partiellen Differentialgleichungen erster Ordnung reduziert. Explizite Resultate für die Diffraktion an einer konvex gekrümmten Wand und an einer konkaven Ecke werden angegeben. Dabei wird gefunden, dass eine Welle der ursprünglichen Amplitudea ₀ durch eine konvexe Biegung nicht mehr als (3a ₀/d ₀)^1/2 abgelenkt werden kann, ohne dass sie ablöst oder ihre Identität verliert. Eine empirische Verallgemeinerung für grössere Amplituden und Ablenkwinkel wird vorgeschlagen. Im Anhang werden mit einer Hodographentransformation allgemeine Lösungen gegeben.

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Dedicated to my good friend Nikolaus Rott on the happy occasion of his sixtieth birthday.     

A study of Langmuir waves in plasmas

This paper talks about Langmuir waves in plasmas. The integration of the governing nonlinear Schrödinger’s equation with perturbation terms is carried out. Both Kerr law as well as the power law nonlinearity are considered.     

Quasi-stationary solitons for Langmuir waves in plasmas

The multiple-scale perturbation analysis is used to study the perturbed nonlinear Schrödinger’s equation, that describes the Langmuir waves in plasmas. The perturbation terms include the non-local term due to nonlinear Landau damping. The WKB type ansatz is used to define the phase of the soliton that captures the corrections to the pulse where the standard soliton perturbation theory fails.     

Irrotational solitary waves with surface-tension

Purely capillary solitary waves cannot be obtained under the systematic shallow water theory developed by Friedrichs. In fact, if we neglect the gravity and take into account the surface-tension only at the free surface and proceed on the lines of Keller, who obtained cnoidal and solitary waves using the Friedrichs’ shallow water systematic theory, we get nothing other than uniform flow. In this article, on the lines of Friedrichs and Hyers, we find the solitary wave motion when surface-tension is also taken into account along with the gravity andgh/U² < 1, whereg is the acceleration due to gravity,h and U are the depth and the horizontal velocity of the liquid at infinity. The solution is sought in the form of infinite series in ascending powers of a suitably defined parameter after giving a stretching in the horizontal direction.     

Two dimensional solitary waves in shear flows

In this paper we study existence and asymptotic behavior of solitary-wave solutions for the generalized Shrira equation, a two-dimensional model appearing in shear flows. The method used to show the existence of such special solutions is based on the mountain pass theorem. One of the main difficulties consists in showing the compact embedding of the energy space in the Lebesgue spaces; this is dealt with interpolation theory. Regularity and decay properties of the solitary waves are also established.     

On solitary waves in one-dimensional microstructured solids

In a microstructured solid with nonlinearities in macroscale and microscale, the propagation of 1D waves is governed by an extended KdV equation, which admits asymmetric solitary waves. An approximate solution is presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Instability of nonlinear dispersive solitary waves

We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula is used to find the instability criteria. These techniques have also been extended to study instability of periodic waves and of the full water wave problem.     

Complex solitary waves from one-soliton solution

Complex solitary wave solutions are obtained for higher-order nonlinear Schrödinger equation as a one-parameter, (C₁) family of solutions. These solutions are found to be stable in a certain range of the parameter. It is observed that for C₁<1, these stable waves propagate at faster bit rate than the solitons under the same input conditions. The complex solutions can also be obtained by the action of the nonlinear operator on the one-soliton solution.     

Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas

Plasmas are believed to be possibly the most abundant form of ordinary matter in the Universe, supporting a variety of the wave phenomena, while a dusty plasma is of interest as a non-Hamiltonian system of interacting particles. In this Letter, symbolic computation on an observationally/experimentally-supported (2+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation is done, for certain dust-acoustic, electron-acoustic, positron-acoustic, magneto-acoustic, dust-magneto-acoustic, ion-acoustic, dust-ion-acoustic and/or quantum-dust-ion-acoustic waves in one of the cosmic/laboratory dusty plasmas. Auto-Bäcklund transformation and families of the solitonic solutions are obtained, for the electrostatic wave potential, perturbation of the magnitude of the magnetic field, fluctuation of electron or ion density, or radial-direction component of the velocity of ions or dust particles, relying on such plasma coefficient functions as the nonlinearity, dispersion, dusty-fluid-viscosity/Burgers-dissipation, geometric-effect and diffraction/transverse-perturbation coefficients. Shock structures presented in this Letter are very close to the experimental results previously reported. Future plasma observations/experiments might verify some other effects offered by our analytic results with respect to those plasma coefficient functions.     

Stochastic perturbation of solitons for Alfven waves in plasmas

The stochastic perturbation of solitons due to Alfven waves in plasmas, is studied in this paper, in addition to the deterministic perturbation terms. The Langevin equations are derived and it is proved that the soliton travels through the plasma with a fixed mean velocity.     

Generalized solitary waves in the gravity-capillary Whitham equation

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave.