New Non-traveling Solitary Wave Solutions for a Second-order Korteweg-de Vries Equation
Woo-Pyo Hong and Young-Dae Junga
Department of Physics, Catholic University of Taegu-Hyosung, Hayang, Kyongsan, Kyungbuk 712-702, South Korea
aDepartment of Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea Reprint requests to Prof. W.-P. H.; E-mail: wphong@cuth.cataegu.ac.kr
Z. Naturforsch. 54 a, 375–378 (1999); received March 12, 1999
Modeling the propagation of two different wave modes simultaneously, the second-order KdV equation is of current interest. Applying a tanh-typed method with symbolic computation, we have found certain new analytic soliton-typed solutions which go beyond the the previously obtained traveling wave solutions.
Key words: Nonlinear Evolution Equations; Second-order KdV Equation; Solitonic Solutions;
Symbolic Computation.
We investigate the second-order KdV equation pro- posed by Korsunsky [1], which is assumed to gov- ern propagation in the same direction of two wave modes with the same dispersion relation, but with different phase velocities, nonlinearity and dispersion parameters:
u
xx+ (c1+c2)uxt+c1c2uxx
+
h
(1+2)∂
∂t+ (1c2+2c1) ∂
∂x
i
uu
x
+
h
(1+2)∂
∂t+ (1c2+2c1) ∂
∂x
i
uu
xxxx= 0; (1)
whereu(x;t) is a field function,ciare the phase ve- locities,ithe parameters of nonlinearity, andithe dispersion parameters for the first (i= 1) and second (i = 2) mode. This equation exhibits two important features: (i) if one of the modes is absent, the other obeys the ordinary KdV equation, and (ii) on applica- tion of the perturbation techniqu,e this equation leads to the uncoupled KdV equations for each mode on a corresponding temporal and spatial scale [1]. We can show that in the absence of the other wave the evolution of each mode is described by its own KdV equation
u
t+ci u
x+i uu
x+i u
xxx= 0 (2)
0932–0784 / 99 / 0600–0375 $ 06.00c Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingenwww.znaturforsch.com with the traveling solitary wave solutions
u(x;t) =Asech2f[x;(ci+1
31A)t]=Lig;
where 12L2i =Ai
=
i :
(3)
To simplify the analysis, we transform (1), using the transformations
= (1+2);1=2(x;c0t); T = (1+2);1=2t;
c0= 1
2(c1+c2); U(;T) = (1+2)u(x;t); (4)
into
U
TT
;s
2
U
(5)
+
∂
∂T +s ∂
∂
UU
+
∂
∂T +s ∂
∂
UU
= 0; where
s=1
2(c1;c2); = 2;1
2+1
; = 2;1
2+1
(6) withs>0,jj1, andjj1.
Two families of traveling wave solutions have been found in [1] for (5):
376 W.-P. Hong and Y.-D. Jung · Solutions for a Second-order Korteweg-de Vries Equation
U
I(x;t) =U0+ 3U0sech2
h
s
U0(1) 4(1)
(1+2);1=2(x;c0t);t
i
; (7)
whereU0is a constant wave amplitude and=scorresponds to the two modes represented in (1) and
U
II(x;t) =Asech2
h
s
A(s;) 12(s;)
(1+2);1=2(x;c0t);t
i
; (8)
where the two roots of2=s2+ 1=3A(s;) correspond to the two solitary wave modes, andAis the wave amplitude.
In this work we apply the tanh-typed method [2 - 4] with symbolic computation to (5) to find some non-traveling solitary wave solutions. As an Ansatz we assume that the physical fieldU(;T) has the form
U(;T) =
N
X
n=0
A
n(T)tanhn[G(T)+H(T)]; (9) whereN is the integer determined via the balance of the highest-order contributions from both the linear and nonlinear terms of (5) asN = 2, whileAN(t),G(T), andH(T) are the non-trivial differentiable functions to be determined.
With the symbolic computation package Maple we substitute the Ansatz (9), together with the above conditions, into (5) and collect the coefficients of like powers of tanh:
(tanh6) : 10A2(T)G(T)[A2(T)sG(T) +A2(T)G(T)T
+A2(T)H(T)T + 12sG(T)3 (10) + 12G(T)2G(T)T+ 12G(T)2H(T)T]
(tanh5) : ;2A2(T)2G(T)T
;24A2(T)TT
G(T)3+ 12A1(T)G(T)T
A2(T)G(T) (11) + 24A1(T)G(T)3G(T)T
+ 24sA1(T)G(T)4;72A2(T)G(T)2G(T)T + 24A1(T)G(T)3H(T)T
+ 12sA1(T)G(T)2A2(T) + 12A1(T)H(T)TA2(T)G(T);4A2(T)TTA2(T)G(T) (tanh4) : ;6A1(T)T(G(T))3+ 6A2(T)H(T)T
2
;6s2A2(T)(G(T))2+ 3A1(T)2G(T)T
G(T) (12)
; 3A1(T)T
A2(T)G(T) + 6A0(T)A2(T)G(T)H(T)T
;240A2(T)(G(T))3H(T)T
; 240A2(T)G(T)3G(T)T ;18A1(T)G(T)2G(T)T + 6A2(T)G(T)T22 + 6A0(T)A2(T)G(T)G(T)T ;240sA2(T)G(T)4;16A2(T)2H(T)TG(T) + 12A2(T)G(T)T
H(T)T
;3A2(T)TT
A1(T)G(T);16s(A2(T))2(G(T))2 + 3A1(T)2H(T)T
G(T) + 3s(A1(T))2G(T)2;16 (A2(T))2G(T)T
G(T)
; 3A1(T)A2(T)G(T)T + 6sA0(T)A2(T)(G(T))2 (tanh3) : ;18A1(T)H(T)T
A2(T)G(T);40A1(T)G(T)3H(T)T
;40A1(T)G(T)3G(T)T
(13)
; 18A1(T)G(T)T
A2(T)G(T);4A2(T)T
H(T)T + 120A2(T)G(T)2G(T)T
; 2A2(T)G(T)TT ;A1(T)2G(T)T ;2A2(T)H(T)TT + 2A1(T)H(T)T2 + 2A1(T)G(T)T
2
2
;2A0(T)A2(T)G(T)T
;4A2(T)T G(T)T
;2A1(T)T
A1(T)G(T)
W.-P. Hong and Y.-D. Jung · Solutions for a Second-order Korteweg-de Vries Equation 377 + 4A2(T)T
A2(T)G(T);2A0(T)A2(T)T
G(T);2A0(T)T
A2(T)G(T) + 40A2(T)T G(T)3
; 40sA1(T)G(T)4;18sA1(T)G(T)2A2(T);2s2A1(T)G(T)2 + 2A0(T)A1(T)G(T)G(T)T
+ 2sA0(T)A1(T)G(T)2+ 2A0(T)A1(T)G(T)H(T)T
+ 4A1(T)G(T)T H(T)T
+ 2A2(T)2G(T)T
(tanh2) :A2(T)TT ;4A1(T)2H(T)TG(T) + 6A2(T)2H(T)TG(T) + 136A2(T)G(T)3H(T)T (14) + 3A1(T)A2(T)G(T)T
;8A2(T)G(T)T
2
2+ 24A1(T)G(T)2G(T)T
;4A1(T)2G(T)T
G(T) + 3A1(T)T
A2(T)G(T);16A2(T)G(T)T H(T)T
;8A0(T)A2(T)G(T)H(T)T
; 8A0(T)A2(T)G(T)G(T)T + 136A2(T)G(T)3G(T)T + 136sA2(T)G(T)4
; 4sA1(T)2G(T)2;8sA0(T)A2(T)G(T)2+ 6A2(T)2G(T)TG(T)+ 8s2A2(T)G(T)2
; A
1(T)G(T)TT
;A
0(T)A1(T)G(T)T + 6sA2(T)2G(T)2+ 8A1(T)T G(T)3
; 8A2(T)H(T)T
2
;A0(T)T
A1(T)G(T) + 3A2(T)T
A1(T)G(T);2A1(T)T G(T)T
; A0(T)A1(T)TG(T);2A1(T)TH(T)T ;A1(T)H(T)TT (tanh1) :A1(T)TT + 6A1(T)H(T)T
A2(T)G(T) + 16A1(T)G(T)3H(T)T + 16A1(T)G(T)3G(T)T
(15) + 6A1(T)G(T)T
A2(T)G(T)+ 4A2(T)T H(T)T
;48A2(T)G(T)2G(T)T + 2A2(T)G(T)TT
+ A1(T)2G(T)T + 2A2(T)H(T)TT ;2A1(T)H(T)T2;2A1(T)G(T)T22 + 2A0(T)A2(T)G(T)T + 4A2(T)T
G(T)T
+ 2A1(T)T
A1(T)G(T) + 2A0(T)A2(T)T G(T) + 2A0(T)T
A2(T)G(T);16A2(T)T
G(T)3+ 16sA1(T)G(T)4+ 6sA1(T)G(T)2A2(T) + 2s2A1(T)G(T)2;2A0(T)A1(T)G(T)G(T)T ;2sA0(T)A1(T)G(T)2
; 2A0(T)A1(T)G(T)H(T)T
;4A1(T)G(T)T H(T)T
(tanh0) :A0(T)TT
;2A1(T)T
G(T)3+ 2A1(T)T
H(T)T + 2A2(T)H(T)T
2+A1(T)H(T)TT (16) + A0(T)A1(T)G(T)T + 2A1(T)T
G(T)T
+ 2A2(T)G(T)T
2
2
;2s2A2(T)G(T)2 + A1(T)2H(T)T
G(T) +A0(T)T
A1(T)G(T) +A0(T)A1(T)T
G(T) +A1(T)G(T)TT
; 16A2(T)G(T)3H(T)T
;6A1(T)G(T)2G(T)T + 4A2(T)G(T)T H(T)T
; 16sA2(T)G(T)4+sA1(T)2G(T)2+ 2sA0(T)A2(T)G(T)2+A1(T)2G(T)T
G(T) + 2A0(T)A2(T)G(T)G(T)T + 2A0(T)A2(T)G(T)H(T)T ;16A2(T)G(T)3G(T)T ; where the subscriptT denotes time derivative.
Our goal is to find the conditions forAN(T);G(T), andH(T) which simultaneously let the above terms become zero. After dealing with some complicated symbolic calculations using Maple, we obtained a new family of non-traveling solitary-wave solutions
as
U
new(;T) =A0(T) +A1(T)tanh1[G(T)+H(T)]
+A2(T)tanh2[G(T)+H(T)]; (17) where
378 W.-P. Hong and Y.-D. Jung · Solutions for a Second-order Korteweg-de Vries Equation
Fig. 1. Beyond traveling solitary-wave solution Unew(x;t) with the parameters 2 = 10;1 = 0:01;2 = 10,
1= 0:01, c1 = 1;c2 = 0:01;C1 = 0:1;C2 = 0:01, andC3 = 0:01, satisfying the solitary wave property that
U
new(x;t) tends to zerojxjas approaches infinity.
G(T) =G= nonzero constant; (18)
H(T) =C1T2+C2T +C3; (19) whereCiare arbitary constants;
A2(T) =;12G2; (20)
A1(T) = 0; (21)
A0(T) R(T)
S(T)
; (22)
R(T);24G(2C1T+C2)4+ (192G4;48G2s)
(2C1T +C2)3+ 576G5s(2C1T +C2)2 + (576s2G6+ 48s3G4)(2C1T+C2) + 24s4G5+ 192s3G7;
S(T)3G3s(2C1T +C2)2+ 3s2G4(2C1T+C2)
G
2(2C1T +C2)3+s3G3;
and the following auxiliary conditions are required forUnew(;T) to be a solution of (5):
[1] S. V. Korsunsky, Phys. Lett. A 185, 174 (1994).
[2] B. Tian, K. Zhao and Y. T. Gao, Int. J. Engng. Sci.
(Lett.) 35, 1081 (1997).
[3] Y. T. Gao and B. Tian, Acta Mechanica 128, 137 (1998).
[4] E. Parkes and B. Duffy, Computer Phys. Comm. 98, 288 (1996).
Fig. 2. A typical sech2-typed solitary wave solutionUII(x;t) withA= 1;2= 10;1= 0:01;2= 10;1= 0:01;c1= 1, andc2= 0:01.
== 1 or==;1; (23)
which imply that2 1 and 2 1 for =
= 1 or1 2 and1 2 for = = ;1.
Physically, these conditions indicate that two solitary wave modes propagate in the medium, where one mode’s nonlinearity and dispersion parameters are much bigger than the other one’s.
Finally we present two figures with some selected parameters. We set2= 10;1= 0:01;2= 10;1= 0:01;c1 = 1;c2 = 0:01;C1 = 0:1;C2 = 0:01, and
C3 = 0:01 for the new solitary-wave solutions (17) and plot Unew(x;t) in Figure 1. The new solutions satisfy solitary wave property that Unew(x;t) tends to zero asjxjapproaches infinity. For comparison in Fig. 2 we plot the traveling wave solution ofUII(x;t) (8) withA = 1;2 = 10;1 = 0:01;2 = 10;1 = 0:01;c1= 1, andc2 = 0:01.
To sum up, the tanh method and symbolic computa- tions lead to the new analytic solitary-wave solutions (17), different from the previously obtained results [1]
for the second order KdV equation.
Acknowledgements
This research was supported by the Korean Re- search Foundation through the Basic Science Re- search Institute Program (1998-015-D00128).
