Applied Mathematics, 2013, 4, 1595-1598
Published Online December 2013 (http://www.scirp.org/journal/am)
Open Access AM
Traveling Wave Solutions and Kind Wave Excitations for
the (2 + 1)-Dimensional Dissipative
Xiajie Liu, Chunliang Mei, Songhua Ma
College of Science, Lishui University, Lishui, China
Received October 1, 2013; revised November 1, 2013; accepted November 8, 2013
Copyright © 2013 Xiajie Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this work, with the help of the symbolic computation system Maple and the Riccati mapping approach and a linear
variable separation approach, a new family of traveling wave solutions of the (2 + 1)-dimensional dissipative Zabolot-
skaya-Khokhlov equation (DZK) is derived. Based on the derived solitary wave solution, some novel kind wave excita-
tions are investigated.
Keywords: Mapping Approach; Dissipative Zabolotskaya-Khokhlov Equation; Traveling Wave Solution; Kind Wave
In nonlinear science, soliton theory plays an essential
role and has been applied in almost all the natural sci-
ences, especially in all the physical branches such as
fluid physics, condensed matter, biophysics, plasma phy-
sics, nonlinear optics, quantum field theory, and particle
physics, etc. [1-5]. How to find exact solutions of non-
linear partial differential equations (PDEs) plays an im-
portant role in the research of nonlinear physical phe-
nomena. So it is always an interesting topic to search for
meaningful solutions for PDEs. In order to find some
new exact solutions, a wealth of effective methods have
been set up, for instance, the bilinear method, the stan-
dard Painlevé truncated expansion, the method of “coa-
lescence of eigenvalue” or “wavenumbers”, the homo-
genous balance method, the homotopy-perturbation me-
thod, the hyperbolic function method, the Jacobian ellip-
tic method, the (G’/G)-expansion method, the variable
separation method, and the mapping equation method
[6-15], etc. Among these methods, the mapping equation
approach is one of the most effectively straightforward
algebraic methods to construct exact solutions of NPDE
[16-19]. In this paper, via the Riccati mapping equation
we find some new exact solutions of the (2 + 1)-dimen-
sional dissipative Zabolotskaya-Khokhlov equation (DZK).
Based on the derived solution, we obtain some kind wave
excitations of the equation.
2. New Traveling Wave Solutions of the
The (2 + 1)-dimensional dissipative Zabolotskaya-Khok-
hlov equation is
In Ref. , some new exact solutions and time soli-
tons have been discussed by (G’/G)-expansion method.
As is well known, to search for the solitary wave solu-
tions for a nonlinear physical model, we can apply dif-
ferent approaches. One of the most efficient methods of
finding soliton excitations of a physical model is the
so-called mapping approach. The basic ideal of the algo-
rithm is as follows. For a given nonlinear partial differ-
ential equation (NPDE) with the independent variables
xtxx x and the dependent variable u, in
Puu uu (2)
where P is in general a polynomial function of its argu-
ments, and the subscripts denote the partial derivatives,
the solution can be assumed to be in the form
X. J. LIU ET AL.
where σ is a constant and the prime denotes the different-
tiation with respect to q. To determine U explicitly, one
may substitute (3) and (4) into the given NPDE and col-
lect coefficients of polynomials of Φ, then eliminate each
coefficient to derive a set of partial differential equations
and q, and solve the system of partial differential
equations to obtain ,
and q. Finally, as (4) is known
to possess the solutions
q and (5) into (3), one obtains the ex-
act solutions to the given NPDE. Now we apply the
mapping approach to (1). By the balancing procedure,
the ansatz (3) becomes
l, m, n are arbitrary constants. Substituting
(6), (7) and (4) into (1) and collecting coefficients of
, then setting each coefficient to zero,
Based on the solutions of (4), one thus obtains follow-
ing exact solutions of Equation (1):
3. Kind Wave Excitations of DZK Equation
In the following discussion, we merel
wave excitations of DZK equation. According to the so-
y analyze kind
when we set the parameters
at time , we can
obtain a kind wave excitation of the physical quantity
U presented in Figure 1.
Furthermore, According to the solution 2,U when we
set the parame ers 20, 20, 20lmn
es a) 6t at tim
, b) 3t , c) 0t
, e) 6t
, we can obtain the time evolution of a
kind wave presented in Figure 2. From Figure 2, one
tive Zabolotskaya-Khokhlov equation. Based on the de-
ds that the kind wave moves in the same dirion and
the amplitude, velocity, and wave shape of the kind wave
do not undergo any chanth tim
4. Summary and Discussion
In summary, with the help of a Riccati mapping method
and a linear variable separation method, we find some
new exact solutions of the (2 + 1)-dim
Figure 1. Plot of the kind wave structure for the physical
Open Access AM
X. J. LIU ET AL. 1597
Figure 2. Plot of the time evolution of a kind wave for the
rived solution , we obtained the kind wave solution
and studied the evolution of a kind wave, which are
different from the ones of the previous work. Because of
wide applications of the DZK equation in physics, more
properties are worthy to be studied such as its Lax pair,
symmetry reduction, bilinear form, and Darboux trans-
formation, etc. All these properties are worthy of study-
The authors would like to thank Professor Senyue Lou
for his fruitful and helpful suggestions. This work has
been supported by the Natural Science Foundation of
Zhejiang Province (Grant No. Y6100257).
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