Evolution of solitary waves for a perturbed nonlinear Schrödinger equation

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.     

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)     

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.     

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     

Bifurcation and nonsmooth dynamics of solitary waves in the generalized long-short wave equations

Generalized wave equations, which model the resonant interaction between the long wave and the short wave, are considered. To understand the underlying complex dynamics, the bifurcations and nonsmooth behaviors of solitary waves for this system are investigated by qualitative techniques in dynamical systems. These complex behaviors may serve as mechanisms for fascinating physical phenomena such as solitons, chaos and turbulence.     

Long‐time dynamics of KdV solitary waves over a variable bottom

We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ?_tu = ??_x(?u + f(u) ? b(t,x)u), where f is a nonlinearity and b is a small, bounded, and slowly varying function related to the varying depth of a channel of water. Many variable‐coefficient KdV‐type equations, including the variable‐coefficient, variable‐bottom KdV equation, can be rescaled into the bKdV. We study the long‐time behavior of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b(t,x) plus an H¹(?)‐small fluctuation. © 2005 Wiley Periodicals, Inc.     

Elliptical flows perturbed by shear waves

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).     

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.     

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.     

Adiabatic invariants of nonlinear dynamic systems with small parameter

We develop a symplectic method of finding the adiabatic invariants of nonlinear dynamic systems with small parameter. We show that a necessary and sufficient condition for the existence of quasi-Hamiltonian adiabatic invariants of nonlinear dynamic systems with regular dependence on a small parameter is that the Cauchy problem be well-posed for an equation of Lax type in the class of nongradient local functionals on the cotangent manifold of the phase space. It is established that scalar nonlinear dynamic systems always have a priori complete evolution invariants, not only adiabatic invariants. We also consider typical applications in hydrodynamics and oscillatory systems of mathematical physics.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 179–185.     

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.     

A variational approach to surface solitary waves

Two-dimensional flow of an incompressible, inviscid fluid in a region with a horizontal bottom of infinite extent and a free upper surface is considered. The fluid is acted on by gravity and has a non-diffusive, heterogeneous density which may be discontinuous. It is shown that the governing equations allow both periodic and single-crested progressing waves of permanent form, the analogues, respectively, of the classical cnoidal and solitary waves. These waves are shown to be critical points of flow related functionals and are proved to exist by means of a variational principle.     

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)     

Orbital stability of solitary waves for Kundu equation

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c₃+s₂υ<0, while Guo and Wu (1995) only considered the case 2c₃+s₂υ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.     

Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

     

耦合Schrdinger和KdV方程组孤波解的存在性

主要研究了耦合的非线性Schrdinger和KdV方程孤波解的存在性.文章利用集中紧性原理找到预紧性的极小化序列,通过平移的方式来寻找方程组对应泛函在H~1(R)的极小值函数,从而得到原方程非平凡解的存在性.     

Instability of solitary waves in non-linear composite media

A problem on the dynamic instability of soliton solutions (solitary waves) of Hamilton's equations, describing plane waves in non-linear elastic composite media with or without anisotropy, is considered. In the anisotropic case, there are two two-parameter families of solitary waves: fast and slow and, when there is no anisotropy, there is one three-parameter family. A classification of the instability of solitary waves of the fast family in the anisotropic case and of representatives of families of solitary waves, the velocities of which lie outside of the range of stability when there is anisotropy and when there is no anisotropy, is given on the basis of a numerical solution of a Cauchy problem for the model equations. In this paper, the fundamental equations describing plane waves in non-linear, anisotropic, elastic composites are derived, the Hamilton form of the basic equations is presented, the symmetries in the anisotropic and isotropic cases are considered, the conserved quantities and the soliton solutions, that is, the solitary waves are presented, the nature of the instability of representatives of all three families is investigated, brief formulation of the results is presented and problems of the instability of the fast family in the anisotropic case and of representatives of the families, the velocities of which lie outside of the range of stability in the presence and absence of anisotropy (explosive instability), are discussed.