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1.
The rational solutions of the Korteweg-de Vries equation are obtained as limits of the soliton solutions in wronskian form and these solutions are verified by direct substitution using a novel determinantal identity. 相似文献
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Abdul-Majid Wazwaz 《Waves in Random and Complex Media》2018,28(3):533-543
A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions. 相似文献
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In this Letter, a few new types of interaction solutions to the KdV equation are obtained through a constructed Wronskian form expansion method. The method takes advantage of the forms and structures of Wronskian solutions to the KdV equation, and the functions used in the Wronskian determinants don't satisfy the systems of linear partial differential equations. 相似文献
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E.J. Parkes 《Physics letters. A》2010,374(42):4321-4323
It is shown that the N-loop soliton solution to the short-pulse equation may be decomposed exactly into N separate soliton elements by using a Moloney-Hodnett type decomposition. For the case N=2, the decomposition is used to calculate the phase shift of each soliton caused by its interaction with the other one. Corrections are made to some previous results in the literature. 相似文献
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In this Letter, Wronskian solutions for the complex KdV equation are obtained by Hirota's bilinear method. Moreover, starting from the bilinear Bäcklund transformation, multi-soliton solutions are presented for the same equation. At the same time, it is also shown that these two kinds of solutions are equivalent. 相似文献
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Michael Nivala 《Physica D: Nonlinear Phenomena》2010,239(13):1147-1158
The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. The KdV equation is known to have large families of periodic solutions that are parameterized by hyperelliptic Riemann surfaces. They are generalizations of the famous multi-soliton solutions. We show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations: perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution. 相似文献
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Yair Zarmi 《Physica D: Nonlinear Phenomena》2008,237(23):2987-3007
The solution of the perturbed KdV equation (PKDVE), when the zero-order approximation is a multiple-soliton wave, is constructed as a sum of two components: elastic and inelastic. The elastic component preserves the elastic nature of soliton collisions. Its perturbation series is identical in structure to the series-solution of the PKDVE when the zero-order approximation is a single soliton. The inelastic component exists only in the multiple-soliton case, and emerges from the first order and onwards. Depending on initial data or boundary conditions, it may contain, in every order, a plethora of inelastic processes. Examples are given of sign-exchange soliton-anti-soliton scattering, soliton-anti-soliton creation or annihilation, soliton decay or merging, and inelastic soliton deflection. The analysis has been carried out through third order in the expansion parameter, exploiting the freedom in the expansion to its fullest extent. Both elastic and inelastic components do not modify soliton parameters beyond their values in the zero-order approximation. When the PKDVE is not asymptotically integrable, the new expansion scheme transforms it into a system of two equations: The Normal Form for ordinary KdV solitons, and an auxiliary equation describing the contribution of obstacles to asymptotic integrability to the inelastic component. Through the orders studied, the solution of the latter is a conserved quantity, which contains the dispersive wave that has been observed in previous works. 相似文献
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Wen-Xiu Ma 《理论物理通讯》2021,73(6):65001
A linear superposition is studied for Wronskian rational solutions to the Kd V equation, which include rogue wave solutions. It is proved that it is equivalent to a polynomial identity that an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders is again a solution to the bilinear Kd V equation. It is also conjectured that there is no other rational solutions among general linear superpositions of Wronskian rational solutions. 相似文献
11.
Zhenya Yan 《Physics letters. A》2008,372(7):969-977
In this Letter, the modified Korteweg-de Vries (mKdV) equations with the focusing (+) and defocusing (−) branches are investigated, respectively. Many new types of binary travelling-wave periodic solutions are obtained for the mKdV equation in terms of Jacobi elliptic functions such as sn(ξ,m)cn(ξ,m)dn(ξ,m) and their extensions. Moreover, we analyze asymptotic properties of some solutions. In addition, with the aid of the Miura transformation, we also give the corresponding binary travelling-wave periodic solutions of KdV equation. 相似文献
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In this paper, by using a transformation and an application of Fan subequation, we study a class of generalized Korteweg–de
Vries (KdV) equation with generalized evolution. As a result, more types of exact solutions to the generalized KdV equation
with generalized evolution are obtained, which include more general single-hump solitons, multihump solitons, kink solutions
and Jacobian elliptic function solutions with double periods. 相似文献
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《Journal of Nonlinear Mathematical Physics》2013,20(4):527-533
Abstract In this paper we use the Painlevé analysis and study a special case of a water wave equation of the KdV type. More specifically, we use the Pickering algorithm [9] and obtain a new kind of solutions, which constitute of both algebraic and trigonometric (or hyperbolic) functions. 相似文献
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《中国物理 B》2015,(8)
The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie's first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method. 相似文献
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F. Lambert 《Zeitschrift fur Physik C Particles and Fields》1980,5(2):147-150
The closed form solutions of the Kortewegde Vries (KdV) and modified MKdV equations which are obtainable by inverse scattering or Hirota's method are found to be Pade approximants to the formal series which results from iterating particular solutions of the linearized equation. 相似文献
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Changbum Chun 《Physics letters. A》2008,372(16):2760-2766
In this Letter the Exp-function method is applied to obtain new generalized solitonary solutions and periodic solutions of the fifth-order KdV equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising in mathematical physics. 相似文献
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《Waves in Random and Complex Media》2013,23(1):151-160
We consider a generalized fifth-order KdV equation with time-dependent coefficients exhibiting higher-degree nonlinear terms. This nonlinear evolution equation describes the interaction between a water wave and a floating ice cover and gravity-capillary waves. By means of the subsidiary ordinary differential equation method, some new exact soliton solutions are derived. Among these solutions, we can find the well known bright and dark solitons with sech and tanh function shapes, and other soliton-like solutions. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous KdV system supporting high-order dispersive and nonlinear effects. 相似文献