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1.
A. O. Smirnov 《Theoretical and Mathematical Physics》1994,100(2):937-947
Two ansatzes of the Krichever curves for solutions of the KdV equation that are elliptic int are considered. Examples are given.State Academy of Aviation Instrument Manufacture, St. Petersburg. Translated from Teoreticheskaya i Mathematicheskaya Fizika, Vol. 100, No. 2, pp. 183–198, August, 1994. 相似文献
2.
An automorphismf of an abelian varietyX is called fixed point free if it admits no fixed points other than the origin and this is of multiplicity one. It is well known that the elliptic curve withj-invariant 0 is the only elliptic curve admitting a fixed point free automorphism. In this note, this result is extended to abelian varieties of higher dimensions and some connected commutative algebraic groups.Supported by DFG-contract La 318/4 and EC-contract SC1-0398-C(A). 相似文献
3.
A. O. Smirnov 《Mathematical Notes》1995,58(1):735-743
We study spectral surfaces associated with elliptic two-gap solutions to the nonlinear Schrödinger equation (NLS), the Korteweg-de Vries equation (KdV), and the sine-Gordon equation (SG). It is shown that elliptic solutions to the NLS and SG equations, as well as solutions to the KdV equation elliptic with respect tot, can be assigned to any hyperelliptic surface of genus 2 that forms a covering over an elliptic surface. 相似文献
4.
Wenjun Yuan Fanning Meng Jianming Lin Yonghong Wu 《Mathematical Methods in the Applied Sciences》2016,39(8):2083-2092
Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
5.
This paper is concerned with several aspects of travelling wave solutions for a (N+1) dimensional potential KdV equation. The Weierstrass elliptic function solutions, the Jaccobi elliptic function solutions, solitary wave solutions, periodic wave solutions to the equation are acquired under certain circumstances. It is shown that the coefficients of derivative terms in the equation cause the qualitative changes of physical structures of the solutions. 相似文献
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Damping of periodic waves in the classically important nonlinear wave systems—nonlinear Schrödinger, Korteweg–deVries (KdV), and modified KdV—is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m=m(t) . The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m= 0 to soliton waves when m= 1 . 相似文献
8.
Summary In this paper we rigorously show the existence and smoothness inε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently
small values of the perturbation parameterε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability
theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically.
Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions. 相似文献
9.
《Chaos, solitons, and fractals》2005,23(1):93-100
Some doubly periodic (Jacobi elliptic function) solutions of the coupled Schrödinger–Boussinesq (KdV) equations are presented in closed form. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solution to construct doubly periodic solutions of the coupled equations. When the module m→1, these solutions degenerate to the exact solitary wave solutions of the coupled equations. 相似文献
10.
Real three-phase solutions of the sine-Laplace equation are constructed. All smooth and singular real doubly periodic solutions
are found. The corresponding three-dimensional theta functions are reduced to the elliptic Jacobi functions. Some classes
of solutions with symmetries giving possibilities for physical applications are determined. Bibliography: 18 titles.
Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1996, pp. 199–216. 相似文献
11.
A Generalized (G'/G)-Expansion Method to Find the Traveling Wave Solutions of Nonlinear Evolution Equations
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In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized (G'/G)-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained. 相似文献
12.
Heng Huat Chan 《Journal of Number Theory》2008,128(3):680-699
We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient η4(7τ)/η4(τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series. 相似文献
13.
张辉群 《数学物理学报(A辑)》2005,25(7):996-1003
借助齐次平衡原则,提出了一种新的构造非线性发展方程的Jacobi椭圆函数精确解的方法. 并利用之得到了KdV方程,Boussinesq方程,KGS方程组的新形式
Jacobi椭圆函数解. 相似文献
14.
Anatoli F. Ivanov Bernhard Lani-Wayda 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(2):205-233
We derive sufficient conditions for the stability and instability of periodic solutions
of Kaplan–Yorke type to the equation
where f is even in the first and odd in the second argument. The criteria are based on the monotonicity of the coefficient in a transformed
version of the variational equation. For the special case of cubic f, we show that this monotonicity property is satisfied if and only if the set
is contained in a region E defined by a quadratic form (bounded by an an ellipse or a hyperbola). The coefficients of this quadratic form are expressible
in terms of the Taylor coefficients of f. Further, the parameter α in the equation and the amplitude z of the periodic solution are related by an elliptic integral. Using the relation between this integral and the arithmeticgeometric
mean, we obtain upper and lower estimates on this relation, and on the inverse function. Combining these estimates with the
inequality that defines the region E, we obtain stability criteria explicit in terms of the Taylor coefficients of f. These criteria go well beyond local stability analysis, as examples show.
This research was supported by the Alexander von Humboldt Foundation (Germany)
Received: March 14, 2005; revised: August 16, 2005 相似文献
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Hongyou Wu 《Annals of Global Analysis and Geometry》1993,11(1):49-64
The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR
3 with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR
3 is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR
3 and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR
3 and having geometricso(3,C)-scattering systems. 相似文献
17.
A new method of constructing elliptic finite-gap solutions of the stationary Korteweg-de Vries (KdV) hierarchy, based on a theorem due to PICARD , is illustrated in the concrete case of the Lamé-Ince potentials -s(s + 1) ρ(z), sε (ρ(.) the elliptic Weierstrass function). Analogous results are derived in the context of the stationary modified Korteweg-de Vries (mKd V) hierarchy for the first time. 相似文献
18.
《Chaos, solitons, and fractals》2007,31(1):70-74
In this note, many new exact solutions of the generalized KdV equation, such as rational solutions, periodic solutions like Jacobian elliptic and triangular functions, soliton-like solutions, are constructed by symbolic computation and the extended mapping method, with the auxiliary ordinary equation replaced by a more general one. 相似文献
19.
Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pn. For an algebraic subvariety S?Pn, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case. 相似文献
20.
张辉群 《数学物理学报(A辑)》2005,25(Z1):996-1003
借助齐次平衡原则,提出了一种新的构造非线性发展方程的Jacobi椭圆函数精确解的方法.并利用之得到了KdV方程,Boussinesq方程,KGS方程组的新形式Jacobi椭圆函数解. 相似文献