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Homoclinic breather-wave solutions and doubly periodic wave solutions for coupled KdV equations 总被引:1,自引:0,他引:1
In this paper, we first find out a proper variable transformation by the ideas of Painleve expansion. Then we apply the extended homoclinic test approach to obtain two-cycle breathing places wave solutions of Eq. (1) which describe interactions between two physical waves, and these special solutions can be applied to explain the structure of certain physical phenomena. Thus this method can be applied to the study of other similar nonlinear coupled system. 相似文献
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Abdul‐Majid Wazwaz 《Mathematical Methods in the Applied Sciences》2017,40(6):2277-2283
In this work, we study the two‐mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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J. Douglas Wright Arnd Scheel 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(4):535-570
We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between
long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers.
We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as
such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.” 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2014,19(9):3074-3085
The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied. 相似文献
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In this paper an algebraic method is devised to uniformly construct a series of complete new exact solutions for general nonlinear equations. For illustration, we apply the modified proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions. 相似文献
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In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(13):5065-5073
The existence and nonexistence of periodic solutions are discussed for fractional differential equations by varying the lower limits of Caputo derivatives. The developed approach is illustrated on several examples. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(7):2769-2794
In this paper, we will establish the bounded solutions, periodic solutions, quasiperiodic solutions, almost periodic solutions, and almost automorphic solutions for linearly coupled complex cubic‐quintic Ginzburg‐Landau equations, under suitable conditions. The main difficulty is the nonlinear terms in the equations that are not Lipschitz‐continuity, traditional methods cannot deal with the difficulty in our problem. We overcome this difficulty by the Galerkin approach, energy estimate method, and refined inequality technique. 相似文献
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Wei Gengping~ Shen Jianhua~ 《高校应用数学学报(英文版)》2006,21(3):320-326
This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained. 相似文献
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A hierarchy of the inverse KdV equation is discussed. Through the bilinear form of Lax pairs, we prove a generalized Darboux-Crum theorem of the hierarchy. The Bäcklund transformation and the generalized Wronskian solutions are presented. The soliton solutions, explicit rational solutions are obtained then. 相似文献
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New exact solutions for a generalized variable-coefficient KdV equation were obtained using the generalized expansion method [R. Sabry, M.A. Zahran, E.G. Fan, Phys. Lett. A 326 (2004) 93]. The obtained solutions include solitary wave solutions besides Jacobi and Weierstrass doubly periodic wave solutions. 相似文献
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A. A. Panov 《Mathematical Notes》1998,64(5):622-628
We estimate the number of periodic solutions for special classes ofnth-order ordinary differential equations with variable coefficients.
Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 720–727, November, 1998.
The author thanks Yu. S. Il'yashenko for setting the problems, permanent advice, and overall support. The author is also thankful
to D. A. Panov for numerous discussions.
This research was supported by the CRDF Foundation under grant MR1-220, by the INTAS Foundation under grant No. 93-05-07,
and by the Russian Foundation for Basic Research under grant No. 95-01-01258. 相似文献
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O. Cârją 《Journal of Optimization Theory and Applications》1996,89(3):637-657
The present paper is concerned with the study of a Hamilton-Jacobi-Bellman equation in finite-dimensional spaces, from both the point of view of l.s.c. viscosity solutions and the point of view of l.s.c. contingent solutions. The results have been used in the study of the uniqueness problem for the Bellman equation associated to a time-optimal control problem (Ref. 1).This paper was completed while the author was visiting the University of California at Los Angeles as a Fulbright Scholar. 相似文献
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Baojian Hong 《Applied mathematics and computation》2010,217(2):472-479
In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics. 相似文献
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This paper is devoted to Young measure solutions of a class of forward-backward diffusion equations. Inspired by the idea from a recent work of Demoulini, we first discuss the regular case by introducing the Young measure solutions and prove the existence for such solutions, and then approximate the extreme case by the approach of regularization and establish the existence of Young measure solutions in the class of functions with bounded variation. 相似文献
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In this paper we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case. Moreover, we study the periodicity of solutions for such systems. Finally, some numerical examples are presented. 相似文献
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New exact traveling wave solutions are derived for two coupled nonlinear water wave equations by using a delicate way of rank analysis two-step ansatz method. 相似文献
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Almost periodic solutions for nonlinear duffing equations 总被引:6,自引:0,他引:6
Zeng Weiyao 《数学学报(英文版)》1997,13(3):373-380
The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation.
By combining the theory of exponential dichotomies with Liapunov functions, we obtain an intersting result on the existence
of almost periodic solutions.
This work is supported by NSF of China, No.19401013 相似文献