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Elliptic function propagating wave for Maccari system
Abstract
Based on the exact solution by multi-linear variable separation approach and introducing Jacobi elliptic functions in the variable separation functions, two types of doubly periodic propagating wave patterns for the Maccari system are derived. When the moduli of elliptic functions are set different, these periodic waves show different features with interesting properties. Especially, in the limiting case, the elliptic function waves may degenerate to dromion and peakon excitations. Graphical investigation of the interaction of elliptic function waves shows it to be inelastic.
Different general solutions are obtained by selecting different seed solutions to the multilinear variable-separation approach in the (2+1)-dimensional Maccari system. By different variable separated solutions with arbitrary functions selected as Jacobian elliptic functions, periodic wave with chaotic behavior and the localized fractal structure, and the interaction between (elliptic) periodic wave and peakon are discussed respectively.
A general solution, including three arbitrary functions, is obtained for a (2+1)-dimensional modified dispersive water-wave (MDWW) equation by means of the WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and the degenerated single folded solitary waves are investigated graphically and found to be completely elastic.
A Jacobi elliptic function expansion method, which is more general than the hyperbolic tangent function expansion method, is proposed to construct the exact periodic solutions of nonlinear wave equations. It is shown that the periodic solutions obtained by this method include some shock wave solutions and solitary wave solutions.
In a previous paper (Lou S-y 1995 J. Phys. A: Math. Gen. 28 7227), a generalized dromion structure was revealed for the (2+1)-dimensional KdV equation, which was first derived by Boiti et al (Boiti M, Leon J J P, Manna M and Pempinelli F 1986 Inverse Problems 2 271) using the idea of the weak Lax pair. In this paper, using a Bäcklund transformation and the variable separation approach, we find there exist much more abundant localized structures for the (2+1)-dimensional KdV equation. The abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution. Some special types of dromion solution, lumps, breathers, instantons and the ring type of soliton, are discussed by selecting the arbitrary functions appropriately. The dromion solutions can be driven by sets of straight-line and curved-line ghost solitons. The dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves. The breathers may breathe both in amplitude and in shape.
The variable separation approach is used to find exact solutions of
the (2+1)-dimensional long-wave-short-wave resonance interaction
equation. The abundance of the coherent soliton structures of this
model is introduced by the entrance of an arbitrary function
of the seed solutions. For some special selections of the arbitrary
function, it is shown that the coherent soliton structures may be
dromions, solitoffs, etc.
A variable separation procedure for the Davey - Stewartson (DS) equation is proposed by using a prior ansatz to its bilinear form. The reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables. The trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a Boussinesq-type equation system. Whenever a pair of solutions of the reduced fields are obtained, a corresponding solution of the DS equation can be obtained algebraically. The single dromion solution and some kinds of positon solutions are obtained explicitly.
The extended F-expansion method or mapping method is used to construct exact solutions for the coupled Klein-Gordon Schrödinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.
Two exact, doubly periodic, propagating wave patterns of the Davey-Stewartson system are computed analytically by a special separation of variables procedure. For the first solution there is a cluster of smaller peaks within each period. The second one consists of a rectangular array of 'plates' joined together by sharp edges, and is thus a kind of 'peakons' for this system of (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations. A long wave limit will yield exponentially localized waves different from the conventional dromion. The stability properties and nonlinear dynamics must await further investigations.
A general solution including two arbitrary functions is first obtained
for the generalized Nizhnik Novikov Veselov equation by means of WTC
truncation method. A class of doubly periodic wave solutions, which are
expressed as rational functions of the Jacobi elliptic functions with
different moduli, result from the general solution. Limit cases are
considered and some new solitary structures are revealed. The
interaction properties of periodic waves are numerically studied and
found to be nonelastic. Under long wave limit, a two-dromion solution
with the new solution structure is obtained and interaction between the
two dromions is completely elastic.
With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansatz, in which periodic solutions of nonlinear partial differential equations that can be expressed as a finite Laurent series of some of 12 Jacobi elliptic functions, is more powerful than exiting Jacobi elliptic function methods and is very powerful to uniformly construct more new exact periodic solutions in terms of rational formal Jacobi elliptic function solution of nonlinear partial differential equations. As an application of the method, we choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.
A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev–Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev–Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs.
In this note, we prove that the recently proposed new higher-dimensional nonlinear partial differential equation admits the Painlevé property. We briefly discuss the integrability properties of the equation. © 1997 American Institute of Physics.
In this paper, the discrete mKdV lattice is solved by using the modified Jacobian elliptic function expansion method. As a consequence, abundant families of Jacobian elliptic function solutions are obtained. When the modulus m→1, these periodic solutions degenerate to the corresponding solitary wave solutions, including bell-type and kink-type excitations.
The periodic wave solutions for the Zakharov system of
nonlinear wave equations and a long–short-wave interaction system
are obtained by using the F-expansion method, which can be regarded
as an overall generalization of Jacobi elliptic function expansion
proposed recently. In the limit cases, the solitary wave solutions for
the systems are also obtained.
A variable separation approach is proposed and extended to the (1+1)-dimensional physics system. The variable separation solution of (1+1)-dimensional Ito system is obtained. Some special types of solutions such as non-propagating solitary wave solution, propagating solitary wave solution and looped soliton solution are found by selecting the arbitrary function appropriately.
An extended F-expansion method for finding periodic wave
solutions of nonlinear evolution equations in mathematical physics is
presented, which
can be thought of as a concentration of extended Jacobi elliptic function
expansion method proposed more recently. By using the homogeneous balance
principle and the extended F-expansion, more periodic wave solutions
expressed by Jacobi elliptic functions for the coupled KdV equations are
derived. In the limit cases, the solitary wave solutions and the other type
of travelling wave solutions for the system are also obtained.
A general solution including three arbitrary functions is obtained for the (2 + 1)-dimensional high-order Broer–Kaup equation by means of WTC truncation method. From the general solution, doubly periodic wave solutions in terms of the Jacobian elliptic functions with different modulus and folded solitary wave solutions determined by appropriate multiple valued functions are obtained. Some interesting novel features and interaction properties of these exact solutions and coherent localized structures are revealed.
The periodic wave solutions to a coupled KdV equations with variable coefficients are obtained by using F -expansion method which can be thought of as an over-all generalization of Jacobi elliptic function expansion method. In the limit cases, the solitary wave solutions are obtained as well. 2003 Elsevier Science B.V. All rights reserved.
With the aid of computerized symbolic computation, the extended Jacobian elliptic function expansion method and its algorithm are presented by using some relations among ten Jacobian elliptic functions and are very powerful to construct more new exact doubly-periodic solutions of nonlinear differential equations in mathematical physics. The new (2+1)-dimensional complex nonlinear evolution equations is chosen to illustrate our algorithm such that sixteen families of new doubly-periodic solutions are obtained. When the modulus m→1 or 0, these doubly-periodic solutions degenerate as solitonic solutions including bright solitons, dark solitons, new solitons as well as trigonometric function solutions.
The Jacobi elliptic function method with symbolic computation is extended to special-type nonlinear equations for constructing their doubly periodic wave solutions. Such equations cannot be directly dealt with by the method and require some kinds of pre-possessing techniques. It is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions. The different Jacobi function expansions may lead to new Jacobi doubly periodic wave solutions, triangular periodic solutions and soliton solutions. In addition, as an illustrative sample, the properties for the Jacobi doubly periodic wave solutions of the coupled Schrödinger–KdV equation are shown with some figures.
A variable separation approach is used to obtain exact solutions of high-dimensional nonlinear physical models. Taking the Nizhnik–Novikov–Veselov (NNV) equation as a simple example, we show that a high-dimensional nonlinear physical model may have quite rich localized coherent structures. For the NNV equation, the richness of the localized structures caused by the entrance of two variable-separated arbitrary functions. For some special selections of the arbitrary functions, it is shown that the localized structures of the NNV equation may be dromions, lumps, breathers, instantons and ring solitons, etc.
We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the extended F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously a number of periodic wave solutions expressed by various Jacobi elliptic functions for the generalized Zakharov equations. In the limit cases, the solitary wave solutions are obtained as well.