89 research outputs found
Group-invariant solutions of a nonlinear acoustics model
Based on a recent classification of subalgebras of the symmetry algebra of
the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation
into ordinary differential equations are obtained. Large classes of
group-invariant solutions of the equation are also determined, and some
properties of the reduced equations and exact solutions are discussed.Comment: 14 page
Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations
We study the propagation of two-dimensional finite-amplitude shear waves in a
nonlinear pre-strained incompressible solid, and derive several asymptotic
amplitude equations in a simple, consistent, and rigorous manner. The scalar
Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations
of motion for all elastic generalized neo-Hookean solids (with strain energy
depending only on the first principal invariant of Cauchy-Green strain).
However, we show that the Z equation cannot be a scalar equation for the
propagation of two-dimensional shear waves in general elastic materials (with
strain energy depending on the first and second principal invariants of
strain). Then we introduce dispersive and dissipative terms to deduce the
scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and
Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid
mechanics.Comment: 15 page
Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics
Analytical solutions to nonlinear differential equations arising in physical problems
Nonlinear partial differential equations are difficult to solve, with many of the approximate solutions in the literature being numerical in nature. In this work, we apply the Homotopy Analysis Method to give approximate analytical solutions to nonlinear ordinary and partial differential equations. The main goal is to apply different linear operators, which can be chosen, to solve nonlinear problems. In the first three chapters, we study ordinary differential equations (ODEs) with one or two linear operators. As we progress, we apply the method to partial differential equations (PDEs) and use several linear operators. The results are all purely analytical, meaning these are approximate solutions that we can evaluate at points and take their derivatives. Another main focus is error analysis, where we test how good our approximations are. The method will always produce approximations, but we use residual errors on the domain of the problem to find a measure of error. In the last two chapters, we apply similarity transforms to PDEs to transform them into ODEs. We then use the Homotopy Analysis Method on one, but are able to find exact solutions to both equations
Infinite - dimensional symmetries of two - dimensional generalized Burgers equations
The conditions for a class of generalized Burgers equations which a priori involve nine arbitrary functions of one or two variables to allow an infinite-dimensional symmetry algebra are determined. Although this algebra can involve up to two arbitrary functions of time, it does not allow a Virasoro subalgebra. This result reconfirms a long-standing fact that variable coefficient generalizations of a nonintegrable equation should be expected to remain as such
Algebraic Approaches to Partial Differential Equations
Partial differential equations are fundamental tools in mathematics,sciences
and engineering. This book is mainly an exposition of the various algebraic
techniques of solving partial differential equations for exact solutions
developed by the author in recent years, with emphasis on physical equations
such as: the Calogero-Sutherland model of quantum many-body system in
one-dimension, the Maxwell equations, the free Dirac equations, the generalized
acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and
Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave
equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the
equation of geopotential forecast, the nonlinear Schrodinger equation and
coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson
equations of three-dimensional packets of surface waves, the equation of the
dynamic convection in a sea, the Boussinesq equations in geophysics, the
incompressible Navier-Stokes equations and the classical boundary layer
equations.
In linear partial differential equations, we focus on finding all the
polynomial solutions and solving the initial-value problems. Intuitive
derivations of Lie symmetry of nonlinear partial differential equations are
given. These symmetry transformations generate sophisticated solutions with
more parameters from relatively simple ones. They are also used to simplify our
process of finding exact solutions. We have extensively used moving frames,
asymmetric conditions, stable ranges of nonlinear terms, special functions and
linearizations in our approaches to nonlinear partial differential equations.
The exact solutions we obtained usually contain multiple parameter functions
and most of them are not of traveling-wave type.Comment: This is part of the monograph to be published by Springe
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