106,573 research outputs found
I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent
In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.
We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.
We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.
Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.
Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.
In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.</p
Meromorphic solutions of nonlinear ordinary differential equations
Exact solutions of some popular nonlinear ordinary differential equations are
analyzed taking their Laurent series into account. Using the Laurent series for
solutions of nonlinear ordinary differential equations we discuss the nature of
many methods for finding exact solutions. We show that most of these methods
are conceptually identical to one another and they allow us to have only the
same solutions of nonlinear ordinary differential equations
On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation
The Conte-Musette method has been modified for the search of only elliptic
solutions to systems of differential equations. A key idea of this a priory
restriction is to simplify calculations by means of the use of a few Laurent
series solutions instead of one and the use of the residue theorem. The
application of our approach to the quintic complex one-dimensional
Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave
form. We also find restrictions on coefficients, which are necessary conditions
for the existence of elliptic solutions for the CGLE5. Using the investigation
of the CGLE5 as an example, we demonstrate that to find elliptic solutions the
analysis of a system of differential equations is more preferable than the
analysis of the equivalent single differential equation.Comment: LaTeX, 21 page
Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method
The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions
Seven common errors in finding exact solutions of nonlinear differential equations
We analyze the common errors of the recent papers in which the solitary wave
solutions of nonlinear differential equations are presented. Seven common
errors are formulated and classified. These errors are illustrated by using
multiple examples of the common errors from the recent publications. We show
that many popular methods in finding of the exact solutions are equivalent each
other. We demonstrate that some authors look for the solitary wave solutions of
nonlinear ordinary differential equations and do not take into account the well
- known general solutions of these equations. We illustrate several cases when
authors present some functions for describing solutions but do not use
arbitrary constants. As this fact takes place the redundant solutions of
differential equations are found. A few examples of incorrect solutions by some
authors are presented. Several other errors in finding the exact solutions of
nonlinear differential equations are also discussed.Comment: 42 page
Construction of Doubly Periodic Solutions via the Poincare-Lindstedt Method in the case of Massless Phi^4 Theory
Doubly periodic (periodic both in time and in space) solutions for the
Lagrange-Euler equation of the (1+1)-dimensional scalar Phi^4 theory are
considered. The nonlinear term is assumed to be small, and the
Poincare-Lindstedt method is used to find asymptotic solutions in the standing
wave form. The principal resonance problem, which arises for zero mass, is
solved if the leading-order term is taken in the form of a Jacobi elliptic
function. It have been proved that the choice of elliptic cosine with fixed
value of module k (k=0.451075598811) as the leading-order term puts the
principal resonance to zero and allows us constructed (with accuracy to third
order of small parameter) the asymptotic solution in the standing wave form. To
obtain this leading-order term the computer algebra system REDUCE have been
used. We have appended the REDUCE program to this paper.Comment: 16 pages, LaTeX 2.09. This paper have been published in the
Electronic Proceedings of the Fourth International IMACS Conference on
Applications of Computer Algebra (ACA'98) {Prague (Czech Republic)} at
http://math.unm.edu/ACA/1998/sessions/dynamical/verno
Finding the one-loop soliton solution of the short-pulse equation by means of the homotopy analysis method
The homotopy analysis method is applied to the short-pulse equation in order to find an analytic approximation to the known exact solitary upright-loop solution. It is demonstrated that the approximate solution agrees well with the exact solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves
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