Abstract The stability of a new solution of the equations of one-dimensional gas dynamics is investigated. This solution is a generalization of the solutions of Sedov /1, 2/ to the case of a viscous, thermally conducting ideal gas with an... more
Abstract The stability of a new solution of the equations of one-dimensional gas dynamics is investigated. This solution is a generalization of the solutions of Sedov /1, 2/ to the case of a viscous, thermally conducting ideal gas with an exponential dependence of the coefficient of viscosity and thermal conductivity on temperature. The linearized equations for small perturbations (the effects of thermal conductivity are not allowed for in the equations for the perturbations), which contain functions of time and the radial coordinate in the coefficients, can be solved by separation of the variables. The conditions under which instability arises are determined from an analysis of the time parts of the solutions. The stability of the solutions /1/ has been considered in /3–5/.
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info:eu-repo/semantics/nonPublishe
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ABSTRACT
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ABSTRACT A method for construction of new integrable PDEs, whose properties are related to an asymptotic perturbation expansion with the leading-order term given by an integrable equation, is developed. A new integrable equation is... more
ABSTRACT A method for construction of new integrable PDEs, whose properties are related to an asymptotic perturbation expansion with the leading-order term given by an integrable equation, is developed. A new integrable equation is constructed by applying the properly defined Lie–Bäcklund group of transformations to the leading-order equation. The integrable equations related to the Korteweg–de Vries (KdV) equation with higher-order corrections are used to investigate the limits of applicability of the so-called asymptotic integrability concept. It is found that the solutions of the higher-order KdV equations obtained by a near identity transform from the normal form solitary waves cannot, in principle, describe some intrinsic features of the high-order KdV solitons.
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Abstract The stability of a new solution of the equations of one-dimensional gas dynamics is investigated. This solution is a generalization of the solutions of Sedov /1, 2/ to the case of a viscous, thermally conducting ideal gas with an... more
Abstract The stability of a new solution of the equations of one-dimensional gas dynamics is investigated. This solution is a generalization of the solutions of Sedov /1, 2/ to the case of a viscous, thermally conducting ideal gas with an exponential dependence of the coefficient of viscosity and thermal conductivity on temperature. The linearized equations for small perturbations (the effects of thermal conductivity are not allowed for in the equations for the perturbations), which contain functions of time and the radial coordinate in the coefficients, can be solved by separation of the variables. The conditions under which instability arises are determined from an analysis of the time parts of the solutions. The stability of the solutions /1/ has been considered in /3–5/.
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A qualitatively new approach to the transformation of boundary layer equations into ordinary differential equations is suggested. The solution obtained describe flows induced by motions of solid surface.
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The one-dimensional motions of a perfect gas are considered in cases of spherical, cylindrical and plane symmetry, when the velocity is proportional to the distance to the center of symmetry. The solutions obtained are an extension of the... more
The one-dimensional motions of a perfect gas are considered in cases of spherical, cylindrical and plane symmetry, when the velocity is proportional to the distance to the center of symmetry. The solutions obtained are an extension of the known solutions of Sedov [1, 2] to the case of a viscous heat-conducting gas with a power-law temperature dependence of the coefficient of viscosity and thermal conductivity.
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Continuous groups of transformations acting on the expanded space of variables, which in-cludes the equation parameters in addition to independent and dependent variables, areconsidered. It is shown that the use of the expanded... more
Continuous groups of transformations acting on the expanded space of variables, which in-cludes the equation parameters in addition to independent and dependent variables, areconsidered. It is shown that the use of the expanded transformations enables one to enrichthe concept of similarity reductions of PDEs. The expanded similarity reductions of diffe-rential equations may be used as a tool for finding changes of variables, which convert theoriginal PDE into another (presumably simpler) PDE. A new view on the common simi-larity reductions as the singular expanded group transformations may be used for definingreductions of a PDE to a specific target ODE.
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The so-called 'direct' approach to separation of variables in linear PDEs is applied to the hydrodynamic stability problem. Calculations are made for the complete linear stability equations in cylindrical coordinates. Several... more
The so-called 'direct' approach to separation of variables in linear PDEs is applied to the hydrodynamic stability problem. Calculations are made for the complete linear stability equations in cylindrical coordinates. Several classes of the exact solutions of the Navier-Stokes equations describing spatially developing and unsteady flows, for which the linear stability problems can be rigorously reduced to eigenvalue problems of ordinary differential equations, are defined. Those exactly solvable nonparallel and unsteady flow stability problems can be used for testing approximate approaches and the methods based on direct numerical simulations of the (linearized) Navier-Stokes equations. The exact solutions of the viscous incompressible Navier-Stokes equations determined as the basic states, for which the linear stability problem is exactly separable, may be themselves of interest from theoretical and engineering points of view.
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Problems that arise when the perturbative analysis of the Boussinesq equations for the flow in a shallow-water layer over a flat horizontal bottom is extended beyond the KdV approximation are studied. It is shown that the solution... more
Problems that arise when the perturbative analysis of the Boussinesq equations for the flow in a shallow-water layer over a flat horizontal bottom is extended beyond the KdV approximation are studied. It is shown that the solution unavoidably contains a third-order non-local term. In the single-soliton case, this term is localized along the soliton trajectory. This ensures that the unidirectional solution is mass conserving at least through third order. In the multiple-soliton case, the non-local term represents inelastic soliton interactions; it does not vanish along a strip in the x-t plane, which emanates form the soliton-collision region. Due to the freedom inherent in the perturbation analysis, this term may be incorporated either in the solution for the surface elevation, or in the solution for the horizontal velocity component. With the first choice, through third order, the unidirectional solution is mass conserving, whereas with the second – it is not.
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ABSTRACT
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ABSTRACT A new approach to the use of the Lie group technique for partial and ordinary differential equations dependent on a small parameter is developed. In addition to determining approximate solutions to the perturbed equation, the... more
ABSTRACT A new approach to the use of the Lie group technique for partial and ordinary differential equations dependent on a small parameter is developed. In addition to determining approximate solutions to the perturbed equation, the approach allows constructing integrable equations that have solutions with (partially) prescribed features. Examples of application of the approach to partial differential equations are given.
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Some effects in the soliton dynamics governed by higher-order Korteweg-de Vries (KdV) type equations are discussed. This is done based on the exact explicit solutions of the equations derived in the paper. It is shown that some higher... more
Some effects in the soliton dynamics governed by higher-order Korteweg-de Vries (KdV) type equations are discussed. This is done based on the exact explicit solutions of the equations derived in the paper. It is shown that some higher order KdV equations possessing multisoliton solutions also admit steady state solutions in terms of algebraic functions describing localized patterns. Solutions including both those static patterns and propagating KdV-like solitons are combinations of algebraic and hyperbolic functions. It is shown that the localized structures behave like static solitons upon collisions with regular moving solitons, with their shape remaining unchanged after the collision and only the position shifted. These phenomena are not revealed in common multisoliton solutions derived using inverse scattering or Hirota's method. The solutions of the higher-order KdV type equations were obtained using a method devised for obtaining soliton solutions of nonlinear evolution equations. This method can be combined with Hirota's method with a modified representation of the solution which allows the results to be extended to multisoliton solutions. The prospects for applying the methods to soliton equations not of KdV type are discussed.
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Some new explicit solutions of the unsteady two-dimensional Navier-Stokes equations describing nonsteady stagnation-point flows with surface suction or injection are presented. The solutions have been obtained using a new approach for... more
Some new explicit solutions of the unsteady two-dimensional Navier-Stokes equations describing nonsteady stagnation-point flows with surface suction or injection are presented. The solutions have been obtained using a new approach for finding explicit similarity solutions of partial differential equations. As distinct from the common Birkhoff’s similarity transformation, which permits only one form of an unsteady potential flow field and only one form of time dependence for suction (or blowing) velocity, the transforms obtained permit consideration of a variety of special solutions differing in the forms of time dependence.
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This review preceding a description of a new recycling model (Part II of this paper) discusses one aspect of the regional precipitation recycling studies, namely, the mathematical modeling of the recycling process. Several recycling... more
This review preceding a description of a new recycling model (Part II of this paper) discusses one aspect of the regional precipitation recycling studies, namely, the mathematical modeling of the recycling process. Several recycling models developed in recent decades are discussed within a unified framework of equations of the conservation of atmospheric water vapor mass. Most of the recycling models
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The problem of a laminar jet, inmiscible in a surrounding fluid, is analyzed using a boundary layer approximation. It is assumed that both fluids are incompressible, their interface is smooth, and the jet does not break up. A self-similar... more
The problem of a laminar jet, inmiscible in a surrounding fluid, is analyzed using a boundary layer approximation. It is assumed that both fluids are incompressible, their interface is smooth, and the jet does not break up. A self-similar solution (in Mises variables) to the plane problems is obtained for the special case where the viscosities of the fluids are inversely proportional to their densities.
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Some new effects in the soliton dynamics governed by higher order Korteweg-de Vries (KdV) equations are discussed based on the exact explicit solutions of the equations on the positive half-line. The solutions describe the process of... more
Some new effects in the soliton dynamics governed by higher order Korteweg-de Vries (KdV) equations are discussed based on the exact explicit solutions of the equations on the positive half-line. The solutions describe the process of generation of a soliton that occurs without boundary forcing and on the steady state background: the boundary conditions remain constant and the initial distribution is a steady state solution of the problem. The time moment when the soliton generation starts is not determined by the parameters present in the problem formulation, the additional parameters imbedded into the solution are needed to determine that moment. The equations found capable of describing those effects are the integrable Sawada-Kotera equation and the KdV-Kaup-Kupershmidt (KdV-KK) equation which, albeit not proven to be integrable, possesses multi-soliton solutions.
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ABSTRACT a b s t r a c t A simple, algorithmic approach is proposed for the construction of the most general family of equations of a given scaling weight, possessing, at least, the same single-soliton solution as a given, lower scaling... more
ABSTRACT a b s t r a c t A simple, algorithmic approach is proposed for the construction of the most general family of equations of a given scaling weight, possessing, at least, the same single-soliton solution as a given, lower scaling weight equation. The construction exploits special polynomials– differential polynomials in the solution, u, of an evolution equation, which vanish identi-cally when u is a single-soliton solution. Applying the approach to different types of evo-lution equations yields new results concerning the most general families of evolution equations in a given scaling weight, which possess solitary wave solutions. The same method can be applied in the identification of families of evolution equations of mixed scaling weight (and, in general, of any structure), which admit single-soliton solutions of a desired form. Ó 2012 Elsevier B.V. All rights reserved.
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The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for... more
The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter α and the long wavelength (or shallowness) parameter β, is developed.