Logarithmic nonlinear Schro••dinger equation and irrotational, compressible flows: An exact solution

A class of irrotational, isentropic, and compressible flows is studied theoretically by formulating the density and the velocity potential in a Madelung transformation. The resulting nonlinear Schrödinger equation is solved in terms of similarity variables. One particular family of exact solutions, valid for any ratio of the specific heat capacities of the gas, permits explicit expressions of the fluid properties and velocities in terms of time and spatial coordinates. Analytically, the density is a Gaussian function of the similarity variable, while the temperature is a function of time only. This method is applicable in one (1D), two, and three dimensional geometries. As a simple example, a 1D gas column, with mass injection on one side and a steadily translating wall on the other, can be formulated exactly. The connection with the evolution of an unsteady velocity potential will also be examined. © 2011 American Physical Society.published_or_final_versio

Comment: Further sufficient conditions for an inverse relationship between productivity and employment

Extant empirical studies document that productivity gains due to technological progress often lead to reductions in employment. This paper rationalizes the stated empirical finding within the context of the theory of the competitive firm under price uncertainty. We show that technological progress affects employment adversely if the firm's coefficient of relative risk aversion is no less than unity and its production technology exhibits non-decreasing returns to scale. On the other hand, technological progress unambiguously increases output if the firm's preference has non-increasing absolute risk aversion. © 1999 Board of Trustees of the University of Illinois. All rights reserved.postprin

Periodic and solitary waves in systems of coherently coupled nonlinear envelope equations

Exact solutions for two classes of coherently coupled nonlinear envelope equations are derived in terms of products of Jacobi elliptic functions. Physical applications are illustrated in the context of nonlinear optics, namely, polarization of light beams and quadratic (or parametric) solitons. Stabilities of these double-humped solitary pulses are studied by direct numerical simulations. The use of computer is crucial, both in terms of symbolic manipulation in the derivation process and in the implementation of numerical schemes in stability consideration. © 2010 Taylor & Francis.postprin

Vortex arrays for sinh-Poisson equation of two-dimensional fluids: Equilibria and stability

The sinh-Poisson equation describes a stream function configuration of a stationary two-dimensional (2D) Euler flow. We study two classes of its exact solutions for doubly periodic domains (or doubly periodic vortex arrays in the plane). Both types contain vortex dipoles of different configurations, an elongated "cat-eye" pattern, and a "diagonal" (symmetric) configuration. We derive two new solutions, one for each class. The first one is a generalization of the Mallier-Maslowe vortices, while the second one consists of two corotating vortices in a square cell. Next, we examine the dynamic stability of such vortex dipoles to initial perturbations, by numerical simulations of the 2D Euler flows on periodic domains. One typical member from each class is chosen for analysis. The diagonally symmetric equilibrium maintains stability for all (even strong) perturbations, whereas the cat-eye pattern relaxes to a more stable dipole of the diagonal type. © 2004 American Institute of Physics.published_or_final_versio

Localized and periodic wave patterns for a nonic nonlinear Schrodinger equation

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Localized pulses for the quantic derivative nonlinear Schrödinger equation on a continuous-wave background

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Periodic solutions for systems of coupled nonlinear Schrödinger equations with three and four components

Periodic solutions of systems of coupled nonlinear Schrödinger equations (CNLS) was discussed. Hirota bilinear method and elliptic functions were used. It was found that each component of the CNLS may have multiple peaks within one period.published_or_final_versio

The superposition of algebraic solitons for the modified Korteweg-de Vries equation

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Localized pulses for quintic derivative nonlinear Schrodinger equation on a continuous wave background

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Periodic solutions for systems of coupled nonlinear Schrödinger equations with five and six components

Systems of coupled nonlinear Schrödinger (CNLS) equations arise in several branches of physics, e.g., optics and plasma physics. Systems with two or three components have been studied intensively. Recently periodic solutions for CNLS systems with four components are derived. The present work extends the search of periodic solutions for CNLS systems to those with five and six components. The Hirota bilinear method, theta and elliptic functions are employed in the process. The long wave limit is studied, and known results of solitary waves are recovered. The validity of these periodic solutions is verified independently by direct differentiation with computer algebra software. ©2002 The American Physical Society.published_or_final_versio