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Exact solutions to mKdV equation with variable coefficients
Applied Mathematics and Computation 216 (2010) 2792–2798
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Exact solutions to mKdV equation with variable coefficients Alvaro H. Salas * Universidad de Caldas, Manizales, Colombia Department of Mathematics, Universidad Nacional de Colombia, Manizales, Colombia
a r t i c l e
i n f o
a b s t r a c t In this paper a special mKdV with variable coefficients is considered. A transformation of variables is first applied in order to obtain a mKdV equation with constant coefficients. Some its one-, two- and three-soliton as well as breather-type soliton solutions are derived by using Hirorta’s bilinear approach. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: KdV mKdV Hirota’s method Bilinear form Multisoliton solution
1. Introduction Modified Kortweg-de Vries (mKdV) equation manifests in diverse areas of physics [1–6]. For example, it appears in the context of, electromagnetic waves in size-quantized films, van Alfén waves in collisionless plasma [7], phonons in anharmonic lattice [8], interfacial waves in two layer liquid with gradually varying depth [9], transmission lines in Schottky barrier [10], ion acoustic solitons [11,12], elastic media [13], and traffic flow problems [14,15]. It is an integrable dynamical system with an infinite number of conserved quantities; the solutions of this equation are well studied [16,17]. Recently nonlinear equations with variable coefficients have attracted considerable attention in the literature. Nonlinear Schro¯dinger equation (NLSE) with variable nonlinearity and dispersion is relevant to both optical fibers and Bose–Einstein condensates [18]. Nonlinear Schro¯dinger equation with source, having distributed coefficients like variable dispersion, variable Kerr nonlinearity and gain or loss, is applicable to asymmetric twin-core optical fibers [19,20]. It has been shown that, solitons can be compressed and their dynamics effectively controlled through these variable parameters. The Kortweg-de Vries (KdV) equation with variable coefficients has been studied recently in the context of ocean waves, where the spatio-temporal variability of the coefficients are due to the changes in the water depth and other physical conditions. The fact that, mKdV equation is relevant to hydrodynamics and a variety of physical phenomena, it is natural to expect the possibility of temporal variations in the equation parameters occurring in the same. Furthermore, for propagating solitons, the first integral of the mKdV equation yields NLSE with a source, making it imperative to investigate the effect of the temporal variation of the distributed parameters on the solitary wave solutions of this dynamical system. The mKdV with variable coefficients we shall consider has the form
ws þ pðsÞw2 wn þ cpðsÞwnnn ¼ 0;
where w ¼ wðs; nÞ and c ¼ constant > 0:
ð1:1Þ
Our next purpose is to apply a suitable transformation of the variables n and s to reduce Eq. (1.1) to an equation with constant coefficients. To this end, let
t ¼ tðs; nÞ;
x ¼ xðs; nÞ and wðs; nÞ ¼ v ðtðs; nÞ; xðs; nÞÞ ¼ v ðt; xÞ;
where x(s, n) and t(s, n) are some functions to be determined. Inserting (1.2) into (1.1) gives following equation: * Address: Department of Mathematics, Universidad Nacional de Colombia, Manizales, Colombia. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.129
ð1:2Þ
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A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
ðcpðsÞt nnn þ ts Þv t þ pðsÞxn v 2 v x þ cpðsÞx3n v xxx þ ðcpðsÞxnnn þ xs Þv x þ 3cpðsÞxn xnn v xx þ ð3cpðsÞt n xnn þ 3cpðsÞxn tnn Þv tx þ 3cpðsÞx2n t n v txx þ 3cpðsÞxn t 2n v ttx þ cpðsÞt3n v ttt þ pðsÞtn v 2 v t þ 3cpðsÞtn tnn v tt ¼ 0:
ð1:3Þ
Now, we choose functions x(s, n) and t(s, n) so that
1 1 ts ¼ pffiffiffiffiffiffi pðsÞ and xn ¼ pffiffiffiffiffiffi ; a ac ac
tn ¼ xnn ¼ xs ¼ 0;
where a ¼ constant > 0:
ð1:4Þ
Under restrictions (1.4) Eq. (1.3) takes the form
pðsÞ
v t þ av 2 v x þ v xxx
¼ 0:
ð1:5Þ
In view of (1.4) and (1.5), to solve (1.1), we first find solutions to following mKdV equation with constant coefficients
v t þ av 2 v x þ v xxx ¼ 0
ð1:6Þ
and then a solution to Eq. (1.1) may be found by the formula
wðs; nÞ ¼ v ðt; xÞ;
1 where t ¼ tðsÞ ¼ pffiffiffiffiffiffi a ac
Z
1 pðsÞdsx ¼ xðnÞ ¼ pffiffiffiffiffiffi n: ac
ð1:7Þ
To solve Eq. (1.5), we may solve its potential version
1 ut þ aðux Þ3 þ uxxx ¼ 0 with 3
v ¼ ux :
ð1:8Þ
2. Hirota’s method In this section we make use of Hirota’s approach [21] to solve nonlinear pde’s. For any two smooth functions f = f(t, x) and g = g(t, x) we define Hirota operators as
@m @n ½ f ðt þ s; x þ yÞgðt s; x yÞ for m; n ¼ 1; 2; 3; . . . ; @sm @yn s¼0;y¼0 @m m Dm ½f ðt þ s; xÞgðt s; xÞ for m ¼ 1; 2; 3; . . . ; t ðf gÞ ¼ Dt ðf ðt; xÞgðt; xÞÞ ¼ m @s s¼0 @n Dnx ðf gÞ ¼ Dnx ðf ðt; xÞgðt; xÞÞ ¼ n ½f ðt; x þ yÞgðt; x yÞ for n ¼ 1; 2; 3; . . . : @y y¼0 n m n Dm t Dx ðf gÞ ¼ Dt Dx ðf ðt; xÞgðt; xÞÞ ¼
ð2:1Þ ð2:2Þ ð2:3Þ
In particular,
Dtx ðf f Þ ¼ Dxt ðf f Þ ¼ D1t D1x ðf f Þ ¼ 2ðfftx ft fx Þ;
ð2:4Þ
Dx ðf gÞ ¼ D1x ðf gÞ ¼ gfx fg x ; Dt ðf gÞ ¼ D1t ðf gÞ ¼ fg t gft ; Dxx ðf f Þ ¼ D2x ðf f Þ ¼ 2 fx2 ffxx ; Dxx ðg gÞ ¼ D2x ðg gÞ ¼ 2 g 2x gg xx ; Dxxx ðf gÞ ¼ D3x ðf gÞ ¼ 3f x g xx 3f xx g x
ð2:5Þ ð2:6Þ ð2:7Þ ð2:8Þ fg xxx þ gfxxx :
ð2:9Þ
From (2.4)–(2.9) it follows that
2f t fx þ Dtx ðf f Þ gfx Dx ðf gÞ gft Dt ðf gÞ ; gx ¼ ; gt ¼ ; 2f f f 2f 2 þ Dxx ðf f Þ 2g 2 þ Dxx ðg gÞ 3f g 3f xx g x þ gfxxx Dxxx ðf gÞ fxx ¼ x ; g xx ¼ x ; g xxx ¼ x xx : 2f 2g f
ftx ¼ fxt ¼
ð2:10Þ ð2:11Þ
We seek solutions to Eq. (1.8) in the form
u ¼ uðt; xÞ ¼ arctan
gðt; xÞ : f ðt; xÞ
Substituting ansatz (2.12) into (1.8) gives following pde:
ð2:12Þ
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A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
ð3g xxx þ 3g t Þf 5 3ðfxxx þ ft Þf 4 g 9ðfxx g x þ fx g xx Þf 4 þ 6ðg xxx þ g t Þf 3 g 2 þ 18fx2 g x þ ag 3x 6g 3x f 3 4 þ 18ðfx fxx g x g xx Þf 3 g 6ðfxxx þ ft Þf 2 g 3 3 afx g 2x 18f x g 2x þ 6fx3 f 2 g 3ðfxxx þ ft Þg 5 þ 3ðg xxx þ g t Þfg 2 3 þ 9ðfxx g x þ fx g xx Þg 4 18f x g 2x þ afx3 6fx3 g 3 þ 18ðfx fxx g x g xx Þfg þ 3 afx2 g x 18fx2 g x þ 6g 3x fg ¼ 0:
ð2:13Þ
To put this equation in its bilinear form, we substitute (2.10) and (2.11) into (2.13) to obtain
2 3 ða 24ÞðDx ðf gÞÞ 9 f 2 þ g 2 Dx ðf gÞ½Dxx ðf f Þ þ Dxx ðg gÞ þ 3 f 2 þ g 2 ½Dt ðf gÞ þ Dxxx ðf gÞ ¼ 0:
ð2:14Þ
Eq. (2.14) shows that it is convenient to choose a = 24. Decoupling Eq. (2.14) gives following bilinear form to Eq. (1.8):
Dxx ðf f Þ þ Dxx ðg gÞ ¼ 0;
ð2:15Þ
Dt ðf gÞ þ Dxxx ðf gÞ ¼ 0:
Observe that we have derived the bilinear form of Eq. (1.8) algorithmically. Accordingly in view of (2.6)–(2.9), system (2.14) may be written equivalently as
(
fx2 fxx f þ g 2x g xx g ¼ 0;
ð2:16Þ
fg t gft þ 3f x g xx 3f xx g x fg xxx þ gfxxx ¼ 0: It is obvious that if the pair f = f(t, x) and g = g(t, x) is a solution to system (2.16), then function
u ¼ uðx; tÞ ¼ arctan
gðt; xÞ f ðt; xÞ
ð2:17Þ
is a solution to the potential mKdV equation (1.8) with a = 24 and then
v ¼ v ðt; xÞ ¼ @ x arctan
gðt; xÞ fg gfx ¼ 2x f ðt; xÞ f þ g2
ð2:18Þ
is a solution to mKdV equation (1.6). Consequently, a solution to our main Eq. (1.1) may be calculated by formula (1.7) with a = 24:
wðs; nÞ ¼ v
1 pffiffiffiffiffiffi 48 6c
1 pðsÞds; pffiffiffiffiffiffi n : 2 6c
Z
ð2:19Þ
In the next section we obtain exact solutions to Eq. (1.6) with a = 24:
v t þ 24v 2 v x þ v xxx ¼ 0:
ð2:20Þ
3. Solutions to mKdV equation 3.1. One-soliton solutions We seek solutions to system (2.16) in the form
f ¼ a0 þ a1 expðkx xtÞ and g ¼ 1 þ b1 expðkx xtÞ;
ð3:1Þ
for some constants a0, a1, b1, k – 0 and x – 0. From (2.16) and (3.1) we obtain following equations:
(
2
k ða0 a1 þ b1 Þ expðkx xtÞ ¼ 0; 2
ð3:2Þ
3
k ða0 b1 a1 Þðx k Þ expðkx xtÞ ¼ 0: Solving (3.2) gives
x ¼ k3 ; b1 ¼ a0 a1 :
ð3:3Þ
This in turns gives following solution to Eq. (2.20): 3
v 1 ðt; xÞ ¼ @ x arctan
1 a0 a1 expðkx k tÞ 3
a0 þ a1 expðkx k tÞ
!
3
¼
ka1 expðkx k tÞ 3
1 þ a21 expð2ðkx k tÞÞ
:
ð3:4Þ
In particular, if a = 1 we obtain the solution
vf1 ðt; xÞ ¼
1 3 k sechðkx k tÞ: 2
ð3:5Þ
A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
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3.2. Two-soliton solutions We seek solutions to system (2.16) in the form
f ¼ 1 þ a1 expðk1 x x1 tÞ þ a2 expðk2 x x2 tÞ þ a3 expðk1 x x1 t þ k2 x x2 tÞ; g ¼ 1 þ b1 expðk1 x x1 tÞ þ b2 expðk2 x x2 tÞ þ b3 expðk1 x x1 t þ k2 x x2 tÞ;
ð3:6Þ
for some constants a1, a2, a 3, b1, b2, b3, k1 – 0, k2 – 0, x1 – 0 and x2 – 0. Let
X ¼ expðk1 x x1 tÞ and Y ¼ expðk2 x x2 tÞ:
ð3:7Þ i j
Inserting (3.6) into (2.16) and equating to zero the different coefficients of X Y (i, j = 0, 1, 2, 3, . . . ) yields an algebraic system in the unknowns ai, bi, ki, xi. Solving it we get following solutions:
a1 ¼ b1 ;
a2 ¼ b2 ;
a3 ¼ b3 ¼ b1 b2
ðk1 k2 Þ2 ðk1 þ k2 Þ2
x1 ¼ k31 ; x2 ¼ k32 :
;
ð3:8Þ
From (2.18), (3.6) and (3.8) we see that a two-soliton solution to Eq. (2.20) is
1
2 3 3 3 3 1 k2 Þ 1 þ b1 exp k1 x k1 t þ b2 exp k2 x k2 t b1 b2 ðk ðk1 þ k2 Þx k1 þ k2 t 2 exp ðk þk Þ 1 2
A:
v 2 ðt; xÞ ¼ @ x arctan @ 2 3 3 3 3 1 k2 Þ t ðk þ k Þx k þ k 1 b1 exp k1 x k1 t b2 exp k2 x k2 t b1 b2 ðk 1 2 2 exp 1 2 ðk þk Þ 0
1
ð3:9Þ
2
Similarly, if we assume an anstaz of the form
f ¼ 1 þ a1 expðk1 x x1 tÞ þ a2 expðk2 x x2 tÞ þ a3 expðk1 x x1 t þ k2 x x2 tÞ; g ¼ b1 expðk1 x x1 tÞ þ b2 expðk2 x x2 tÞ þ b3 expðk1 x x1 t þ k2 x x2 tÞ;
ð3:10Þ
we obtain
a1 ¼ a2 ¼ b3 ¼ 0;
a3 ¼ b1 b2
ðk1 k2 Þ2 ðk1 þ k2 Þ2
;
x1 ¼ k31 ; x2 ¼ k32
ð3:11Þ
and the corresponding two-soliton solution to Eq. (2.20) is (see Fig. 1):
1 2 3 3 3 3 1 k2 Þ b1 exp k1 x k1 t þ b2 exp k2 x k2 t b1 b2 ðk k1 þ k2 t 2 exp ðk1 þ k2 Þx ðk þk Þ 1 2 A:
v 3 ðt; xÞ ¼ @ x arctan @ 2 3 3 1 k2 Þ ðk þ k Þx k þ k 1 b1 b2 ðk t 1 2 2 exp 1 2 ðk þk Þ 0
1
2
3.3. Three-soliton solutions We seek solutions to system (2.16) in the form
Fig. 1. Graphic of function
v3(x, t) for k1 = 3, k2 = 1, b1 = 3, b2 = 2, 0.1 6 t 6 0.5 and 4 6 x 6 6.
ð3:12Þ
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A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
8 > < f ¼ 1 þ a12 expðn1 þ n2 Þ þ a13 expðn1 þ n3 Þ þ a23 expðn2 þ n3 Þ; g ¼ b1 expðn1 Þ þ b2 expðn2 Þ þ b3 expðn3 Þ þ b123 expðn1 þ n2 þ n3 Þ; > : n1 ¼ k1 x x1 t; n2 ¼ k2 x x2 t; n3 ¼ k3 x x3 t;
ð3:13Þ
for some constants a12, a13, a23, b123, b2, b3, ki – 0 and xi – 0 (i = 1, 2, 3). Inserting (3.6) into (2.16) and equating to zero the different coefficients of ni1 nj2 nk3 (i, j, k = 0, 1, 2, 3, . . . ) yields an algebraic system. Solving it gives
a12 ¼ b1 b2
ðk1 k2 Þ2 ðk1 þ k2 Þ
b123 ¼ b1 b2 b3
2
;
a13 ¼ b1 b3
ðk1 k3 Þ2 2
ðk1 þ k3 Þ
ðk1 k2 Þ2 ðk1 k3 Þ2 ðk2 k3 Þ2 ðk1 þ k2 Þ2 ðk1 þ k3 Þ2 ðk2 þ k3 Þ2
;
;
a23 ¼ b2 b3
ðk2 k3 Þ2 ðk2 þ k3 Þ2
;
x1 ¼ k31 ; x2 ¼ k32 ; x3 ¼ k33 :
ð3:14Þ ð3:15Þ
From (3.13)–(3.15) we see that a three-soliton solution to Eq. (2.20) is
v 4 ðt; xÞ ¼ @ x arctan
b1 expðn1 Þ þ b2 expðn2 Þ þ b3 expðn3 Þ þ b123 expðn1 þ n2 þ n3 Þ : 1 þ a12 expðn1 þ n2 Þ þ a13 expðn1 þ n3 Þ þ a23 expðn2 þ n3 Þ
ð3:16Þ
3.4. Breather-type solutions We seek solutions to system (2.16) in the form
f ¼ a1 coshðk1 x x1 tÞ þ b1 cosðk2 x x2 tÞ; g ¼ c1 coshðk1 x x1 tÞ þ d1 cosðk2 x x2 tÞ;
ð3:17Þ
for some constants a1, b1, c1, d1, b2, k1 – 0, k2 – 0, x1 – 0 and x2 – 0. Let
X ¼ expðk1 x x1 tÞ and Y ¼ exp
pffiffiffiffiffiffiffi
1ðk2 x x2 tÞ :
ð3:18Þ
Inserting (3.17) into (2.16) and equating to zero the different coefficients of XiYj (i, j = 0, 1, 2, 3, . . . ) yields an algebraic system in the unknowns ai, bi, ki, xi. This system reads
Fig. 2. Graphic of function
v5(x, t) for k1 = 2, k2 = 3, a1 = 3, b1 = 1, 0.1 6 t 6 0.1 and 2.5 6 x 6 3.5.
A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
8 pffiffiffiffiffiffiffi 2 > > k 1k2 ða1 b1 þ c1 d1 Þ ¼ 0; > 1 > > <
2 2 2 2 k1 a21 þ c21 k2 b1 þ d1 ¼ 0; > >
> pffiffiffiffiffiffiffi > > : x1 k31 þ 3k22 k1 þ 1 x2 þ k32 3k21 k2 ðb1 c1 a1 d1 Þ ¼ 0:
2797
ð3:19Þ
Solving this system we obtain:
c1 ¼
b1 k2 ; k1
d1 ¼
a1 k1 ; k2
x1 ¼ k1 k21 3k22 ; x2 ¼ k2 k22 3k21 :
ð3:20Þ
From (2.18), (3.6) and (3.8) we see that a two-soliton solution to Eq. (2.20) is (see Fig. 2):
1 2 2 2 2 2 2 a1 k1 cos k2 x þ k2 k2 3k1 t b1 k2 cosh k1 x k1 k1 3k2 t 1
A: v 5 ðt; xÞ ¼ @ x arctan @ k1 k2 b1 cos k2 x þ k2 k2 3k2 t þ a1 cosh k1 x k1 k2 3k2 t 2 1 1 2 0
4. Concluding remarks Sometimes, we may reduce the problem of finding solutions to an nonlinear pde with variable coefficients to that of solving a similar equation with constant coefficients. This approach simplifies the tedious algebraic computations and allows us to use known results. On the other hand, Hirota’s method is a powerful tool for solving nonlinear pde’s in mathematical physics. Other results concerning pde’s with variable coefficients may be found in [22–24,34,35]. Additional results concerning the problem of finding analytic solutions to nonlinear pde’s are derived in [25–33]. We think that some of the results we presented in this work are new in the open literature. References [1] R.M. Miura, C.S. Gardner, M.D. Kruskal, Korteweg-de Vries equation and generalizations. II. existence of conservation laws and constants of motion, J. Math. Phys. 9 (1968) 1204. [2] M. Wadati, The exact solution of the modified Korteweg-de Vries equation, J. Phys. Soc. Jpn. 32 (1972) 1681. [3] M. Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Jpn. 34 (1973) 1289. [4] R. Hirota, Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (1972) 1456. [5] Z. Fu, S. Liu, S. Liu, New solutions to mKdV equation, Phys. Lett. A 326 (2004) 364–374. [6] S. Matsutani, Hyperelliptic solutions of modified Korteweg-de Vries equation of genus g: essentials of the Miura transformation, J. Phys. A: Math. Gen. 35 (2002) 4321. [7] A.H. Khater, O.H. El-Kakaawy, D.K. Callebaut, Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron–positron plasma, Phys. Scr. 58 (1998) 545. [8] H. Ono, Soliton fission in anharmonic lattices with reflectionless inhomogeneity, J. Phys. Soc. Jpn. 61 (1992) 4336–4343. [9] K.R. Helfrich, W.K. Melville, J.W. Miles, On interfacial solitary waves over slowly varying topography, J. Fluid Mech. 149 (1984) 305–317. [10] V. Ziegler, J. Dinkel, C. Setzer, K.E. Lonngren, On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line, Chaos, Solitons Fractals 12 (2001) 1719–1728. [11] K.E. Lonngren, Ion acoustic soliton experiments in a plasma, Opt. Quantum Electron. 30 (1998) 615–630. [12] S. Watanabe, Ion acoustic soliton in plasma with negative ion, J. Phys. Soc. Jpn. 53 (1984) 950–956. [13] S. Matsutani, H. Tsuru, Reflectionless quantum wire, J. Phys. Soc. Jpn. 60 (1991) 3640–3644. [14] T. Nagatani, TDGL and mKdV equations for jamming transition in the lattice models of traffic, Phys. A 264 (1999) 571–581. [15] T.S. Komatsu, S.I. Sasa, Kink soliton characterizing traffic congestion, Phys. Rev. E 52 (1995) 5574–5582. [16] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Waves, Academic Press, London, 1982. [17] P.G. Kevrekidis, A. Khare, A. Saxena, Breather lattice and its stabilization for the modified Korteweg-de Vries equation, Phys. Rev. E 68 (2003). [18] V.I. Kruglov, A.C. Peacock, J.D. Harvey, Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients, Phys. Rev. Lett. 90 (2003) 113902. [19] T. Soloman Raju, P.K. Panigrahi, K. Porsezian, Nonlinear compression of solitary waves in asymmetric twin-core fibers, Phys. Rev. E 71 (2005) 026608. [20] T. Soloman Raju, P.K. Panigrahi, K. Porsezian, Self-similar propagation and compression of chirped self-similar waves in asymmetric twin-core fibers with nonlinear gain, Phys. Rev. E 72 (2005) 046612. [21] H. Salas Alvaro, Symbolic computation of solutions for a forced Burgers equation, Applied Mathematics and Computation 216 (2010) 18–26. [22] H. Salas Alvaro, C.A. Gómez, Computing exact solutions for some fifth KdV equations with forcing term, Applied Mathematics and Computation 204 (2008) 257–260. [23] H. Salas Alvaro, C.A. Gómez, Exact solutions for a third-order KdV equation with variable coefficients and forcing term, Mathematical Problems in Engineering, Hindawi Pub. Corp., 2009, doi:10.1155/2009/737928. [24] Salas Alvaro, Some solutions for a type of generalized Sawada–Kotera equation, Applied Mathematics and Computation 196 (2008) 812–817. [25] H. Salas Alvaro, Exact solutions for the general fifth KdV equation by the exp function method, Applied Mathematics and Computation 205 (2008) 291– 297. [26] H. Salas Alvaro, C.A. Gómez, Application of the Cole–Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation (KdV7), Mathematical Problems in Engineering (2010), doi:10.1155/2010/194329. [27] H. Salas Alvaro, C.A. Gómez, L. Escobar, G. José, Exact solutions for the general fifth-order KdV equation by the extended tanh method, Journal of Mathematical Sciences: Advances and Applications 1 (2) (2008) 305–310. [28] H. Salas Alvaro, C.A. Gómez, A practical approach to solve coupled systems of nonlinear PDE’s, Journal of Mathematical Sciences: Advances and Applications 3 (1) (2009) 101–107. [29] H. Salas Alvaro, Computing exact solutions to a generalized Lax seventh-order forced KdV equation (KdV7), Applied Mathematics and Computation 216 (2010) 2333–2338. [30] H. Salas Alvaro, Exact solutions to coupled sine-Gordon equation, Nonlinear Analysis: Real World Applications, in press.
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[31] H. Salas Alvaro, New solutions to Korteweg-de Vries (kdV) equation by the Riccati equation expansion method, International Journal of Applied Mathematics (IJAM) 22 (2009) 1169–1177. [32] H. Salas Alvaro, Symbolic computation of exact solutions to KdV equation, Canadian Applied Mathematics Quarterly 16 (2008). [33] Biswas Anjan, 1-soliton solution of the Kðm; nÞ equation with generalized evolution and time-dependent damping and dispersion, Computers and Mathematics with Applications, in press. [34] Biswas Anjan, Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 3503–3506. [35] Hirota Ryogo, The Direct Method in Soliton Theory, Cambridge University Press, 2004.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Exact solutions to mKdV equation with variable coefficients Alvaro H. Salas * Universidad de Caldas, Manizales, Colombia Department of Mathematics, Universidad Nacional de Colombia, Manizales, Colombia
a r t i c l e
i n f o
a b s t r a c t In this paper a special mKdV with variable coefficients is considered. A transformation of variables is first applied in order to obtain a mKdV equation with constant coefficients. Some its one-, two- and three-soliton as well as breather-type soliton solutions are derived by using Hirorta’s bilinear approach. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: KdV mKdV Hirota’s method Bilinear form Multisoliton solution
1. Introduction Modified Kortweg-de Vries (mKdV) equation manifests in diverse areas of physics [1–6]. For example, it appears in the context of, electromagnetic waves in size-quantized films, van Alfén waves in collisionless plasma [7], phonons in anharmonic lattice [8], interfacial waves in two layer liquid with gradually varying depth [9], transmission lines in Schottky barrier [10], ion acoustic solitons [11,12], elastic media [13], and traffic flow problems [14,15]. It is an integrable dynamical system with an infinite number of conserved quantities; the solutions of this equation are well studied [16,17]. Recently nonlinear equations with variable coefficients have attracted considerable attention in the literature. Nonlinear Schro¯dinger equation (NLSE) with variable nonlinearity and dispersion is relevant to both optical fibers and Bose–Einstein condensates [18]. Nonlinear Schro¯dinger equation with source, having distributed coefficients like variable dispersion, variable Kerr nonlinearity and gain or loss, is applicable to asymmetric twin-core optical fibers [19,20]. It has been shown that, solitons can be compressed and their dynamics effectively controlled through these variable parameters. The Kortweg-de Vries (KdV) equation with variable coefficients has been studied recently in the context of ocean waves, where the spatio-temporal variability of the coefficients are due to the changes in the water depth and other physical conditions. The fact that, mKdV equation is relevant to hydrodynamics and a variety of physical phenomena, it is natural to expect the possibility of temporal variations in the equation parameters occurring in the same. Furthermore, for propagating solitons, the first integral of the mKdV equation yields NLSE with a source, making it imperative to investigate the effect of the temporal variation of the distributed parameters on the solitary wave solutions of this dynamical system. The mKdV with variable coefficients we shall consider has the form
ws þ pðsÞw2 wn þ cpðsÞwnnn ¼ 0;
where w ¼ wðs; nÞ and c ¼ constant > 0:
ð1:1Þ
Our next purpose is to apply a suitable transformation of the variables n and s to reduce Eq. (1.1) to an equation with constant coefficients. To this end, let
t ¼ tðs; nÞ;
x ¼ xðs; nÞ and wðs; nÞ ¼ v ðtðs; nÞ; xðs; nÞÞ ¼ v ðt; xÞ;
where x(s, n) and t(s, n) are some functions to be determined. Inserting (1.2) into (1.1) gives following equation: * Address: Department of Mathematics, Universidad Nacional de Colombia, Manizales, Colombia. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.129
ð1:2Þ
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ðcpðsÞt nnn þ ts Þv t þ pðsÞxn v 2 v x þ cpðsÞx3n v xxx þ ðcpðsÞxnnn þ xs Þv x þ 3cpðsÞxn xnn v xx þ ð3cpðsÞt n xnn þ 3cpðsÞxn tnn Þv tx þ 3cpðsÞx2n t n v txx þ 3cpðsÞxn t 2n v ttx þ cpðsÞt3n v ttt þ pðsÞtn v 2 v t þ 3cpðsÞtn tnn v tt ¼ 0:
ð1:3Þ
Now, we choose functions x(s, n) and t(s, n) so that
1 1 ts ¼ pffiffiffiffiffiffi pðsÞ and xn ¼ pffiffiffiffiffiffi ; a ac ac
tn ¼ xnn ¼ xs ¼ 0;
where a ¼ constant > 0:
ð1:4Þ
Under restrictions (1.4) Eq. (1.3) takes the form
pðsÞ
v t þ av 2 v x þ v xxx
¼ 0:
ð1:5Þ
In view of (1.4) and (1.5), to solve (1.1), we first find solutions to following mKdV equation with constant coefficients
v t þ av 2 v x þ v xxx ¼ 0
ð1:6Þ
and then a solution to Eq. (1.1) may be found by the formula
wðs; nÞ ¼ v ðt; xÞ;
1 where t ¼ tðsÞ ¼ pffiffiffiffiffiffi a ac
Z
1 pðsÞdsx ¼ xðnÞ ¼ pffiffiffiffiffiffi n: ac
ð1:7Þ
To solve Eq. (1.5), we may solve its potential version
1 ut þ aðux Þ3 þ uxxx ¼ 0 with 3
v ¼ ux :
ð1:8Þ
2. Hirota’s method In this section we make use of Hirota’s approach [21] to solve nonlinear pde’s. For any two smooth functions f = f(t, x) and g = g(t, x) we define Hirota operators as
@m @n ½ f ðt þ s; x þ yÞgðt s; x yÞ for m; n ¼ 1; 2; 3; . . . ; @sm @yn s¼0;y¼0 @m m Dm ½f ðt þ s; xÞgðt s; xÞ for m ¼ 1; 2; 3; . . . ; t ðf gÞ ¼ Dt ðf ðt; xÞgðt; xÞÞ ¼ m @s s¼0 @n Dnx ðf gÞ ¼ Dnx ðf ðt; xÞgðt; xÞÞ ¼ n ½f ðt; x þ yÞgðt; x yÞ for n ¼ 1; 2; 3; . . . : @y y¼0 n m n Dm t Dx ðf gÞ ¼ Dt Dx ðf ðt; xÞgðt; xÞÞ ¼
ð2:1Þ ð2:2Þ ð2:3Þ
In particular,
Dtx ðf f Þ ¼ Dxt ðf f Þ ¼ D1t D1x ðf f Þ ¼ 2ðfftx ft fx Þ;
ð2:4Þ
Dx ðf gÞ ¼ D1x ðf gÞ ¼ gfx fg x ; Dt ðf gÞ ¼ D1t ðf gÞ ¼ fg t gft ; Dxx ðf f Þ ¼ D2x ðf f Þ ¼ 2 fx2 ffxx ; Dxx ðg gÞ ¼ D2x ðg gÞ ¼ 2 g 2x gg xx ; Dxxx ðf gÞ ¼ D3x ðf gÞ ¼ 3f x g xx 3f xx g x
ð2:5Þ ð2:6Þ ð2:7Þ ð2:8Þ fg xxx þ gfxxx :
ð2:9Þ
From (2.4)–(2.9) it follows that
2f t fx þ Dtx ðf f Þ gfx Dx ðf gÞ gft Dt ðf gÞ ; gx ¼ ; gt ¼ ; 2f f f 2f 2 þ Dxx ðf f Þ 2g 2 þ Dxx ðg gÞ 3f g 3f xx g x þ gfxxx Dxxx ðf gÞ fxx ¼ x ; g xx ¼ x ; g xxx ¼ x xx : 2f 2g f
ftx ¼ fxt ¼
ð2:10Þ ð2:11Þ
We seek solutions to Eq. (1.8) in the form
u ¼ uðt; xÞ ¼ arctan
gðt; xÞ : f ðt; xÞ
Substituting ansatz (2.12) into (1.8) gives following pde:
ð2:12Þ
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ð3g xxx þ 3g t Þf 5 3ðfxxx þ ft Þf 4 g 9ðfxx g x þ fx g xx Þf 4 þ 6ðg xxx þ g t Þf 3 g 2 þ 18fx2 g x þ ag 3x 6g 3x f 3 4 þ 18ðfx fxx g x g xx Þf 3 g 6ðfxxx þ ft Þf 2 g 3 3 afx g 2x 18f x g 2x þ 6fx3 f 2 g 3ðfxxx þ ft Þg 5 þ 3ðg xxx þ g t Þfg 2 3 þ 9ðfxx g x þ fx g xx Þg 4 18f x g 2x þ afx3 6fx3 g 3 þ 18ðfx fxx g x g xx Þfg þ 3 afx2 g x 18fx2 g x þ 6g 3x fg ¼ 0:
ð2:13Þ
To put this equation in its bilinear form, we substitute (2.10) and (2.11) into (2.13) to obtain
2 3 ða 24ÞðDx ðf gÞÞ 9 f 2 þ g 2 Dx ðf gÞ½Dxx ðf f Þ þ Dxx ðg gÞ þ 3 f 2 þ g 2 ½Dt ðf gÞ þ Dxxx ðf gÞ ¼ 0:
ð2:14Þ
Eq. (2.14) shows that it is convenient to choose a = 24. Decoupling Eq. (2.14) gives following bilinear form to Eq. (1.8):
Dxx ðf f Þ þ Dxx ðg gÞ ¼ 0;
ð2:15Þ
Dt ðf gÞ þ Dxxx ðf gÞ ¼ 0:
Observe that we have derived the bilinear form of Eq. (1.8) algorithmically. Accordingly in view of (2.6)–(2.9), system (2.14) may be written equivalently as
(
fx2 fxx f þ g 2x g xx g ¼ 0;
ð2:16Þ
fg t gft þ 3f x g xx 3f xx g x fg xxx þ gfxxx ¼ 0: It is obvious that if the pair f = f(t, x) and g = g(t, x) is a solution to system (2.16), then function
u ¼ uðx; tÞ ¼ arctan
gðt; xÞ f ðt; xÞ
ð2:17Þ
is a solution to the potential mKdV equation (1.8) with a = 24 and then
v ¼ v ðt; xÞ ¼ @ x arctan
gðt; xÞ fg gfx ¼ 2x f ðt; xÞ f þ g2
ð2:18Þ
is a solution to mKdV equation (1.6). Consequently, a solution to our main Eq. (1.1) may be calculated by formula (1.7) with a = 24:
wðs; nÞ ¼ v
1 pffiffiffiffiffiffi 48 6c
1 pðsÞds; pffiffiffiffiffiffi n : 2 6c
Z
ð2:19Þ
In the next section we obtain exact solutions to Eq. (1.6) with a = 24:
v t þ 24v 2 v x þ v xxx ¼ 0:
ð2:20Þ
3. Solutions to mKdV equation 3.1. One-soliton solutions We seek solutions to system (2.16) in the form
f ¼ a0 þ a1 expðkx xtÞ and g ¼ 1 þ b1 expðkx xtÞ;
ð3:1Þ
for some constants a0, a1, b1, k – 0 and x – 0. From (2.16) and (3.1) we obtain following equations:
(
2
k ða0 a1 þ b1 Þ expðkx xtÞ ¼ 0; 2
ð3:2Þ
3
k ða0 b1 a1 Þðx k Þ expðkx xtÞ ¼ 0: Solving (3.2) gives
x ¼ k3 ; b1 ¼ a0 a1 :
ð3:3Þ
This in turns gives following solution to Eq. (2.20): 3
v 1 ðt; xÞ ¼ @ x arctan
1 a0 a1 expðkx k tÞ 3
a0 þ a1 expðkx k tÞ
!
3
¼
ka1 expðkx k tÞ 3
1 þ a21 expð2ðkx k tÞÞ
:
ð3:4Þ
In particular, if a = 1 we obtain the solution
vf1 ðt; xÞ ¼
1 3 k sechðkx k tÞ: 2
ð3:5Þ
A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
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3.2. Two-soliton solutions We seek solutions to system (2.16) in the form
f ¼ 1 þ a1 expðk1 x x1 tÞ þ a2 expðk2 x x2 tÞ þ a3 expðk1 x x1 t þ k2 x x2 tÞ; g ¼ 1 þ b1 expðk1 x x1 tÞ þ b2 expðk2 x x2 tÞ þ b3 expðk1 x x1 t þ k2 x x2 tÞ;
ð3:6Þ
for some constants a1, a2, a 3, b1, b2, b3, k1 – 0, k2 – 0, x1 – 0 and x2 – 0. Let
X ¼ expðk1 x x1 tÞ and Y ¼ expðk2 x x2 tÞ:
ð3:7Þ i j
Inserting (3.6) into (2.16) and equating to zero the different coefficients of X Y (i, j = 0, 1, 2, 3, . . . ) yields an algebraic system in the unknowns ai, bi, ki, xi. Solving it we get following solutions:
a1 ¼ b1 ;
a2 ¼ b2 ;
a3 ¼ b3 ¼ b1 b2
ðk1 k2 Þ2 ðk1 þ k2 Þ2
x1 ¼ k31 ; x2 ¼ k32 :
;
ð3:8Þ
From (2.18), (3.6) and (3.8) we see that a two-soliton solution to Eq. (2.20) is
1
2 3 3 3 3 1 k2 Þ 1 þ b1 exp k1 x k1 t þ b2 exp k2 x k2 t b1 b2 ðk ðk1 þ k2 Þx k1 þ k2 t 2 exp ðk þk Þ 1 2
A:
v 2 ðt; xÞ ¼ @ x arctan @ 2 3 3 3 3 1 k2 Þ t ðk þ k Þx k þ k 1 b1 exp k1 x k1 t b2 exp k2 x k2 t b1 b2 ðk 1 2 2 exp 1 2 ðk þk Þ 0
1
ð3:9Þ
2
Similarly, if we assume an anstaz of the form
f ¼ 1 þ a1 expðk1 x x1 tÞ þ a2 expðk2 x x2 tÞ þ a3 expðk1 x x1 t þ k2 x x2 tÞ; g ¼ b1 expðk1 x x1 tÞ þ b2 expðk2 x x2 tÞ þ b3 expðk1 x x1 t þ k2 x x2 tÞ;
ð3:10Þ
we obtain
a1 ¼ a2 ¼ b3 ¼ 0;
a3 ¼ b1 b2
ðk1 k2 Þ2 ðk1 þ k2 Þ2
;
x1 ¼ k31 ; x2 ¼ k32
ð3:11Þ
and the corresponding two-soliton solution to Eq. (2.20) is (see Fig. 1):
1 2 3 3 3 3 1 k2 Þ b1 exp k1 x k1 t þ b2 exp k2 x k2 t b1 b2 ðk k1 þ k2 t 2 exp ðk1 þ k2 Þx ðk þk Þ 1 2 A:
v 3 ðt; xÞ ¼ @ x arctan @ 2 3 3 1 k2 Þ ðk þ k Þx k þ k 1 b1 b2 ðk t 1 2 2 exp 1 2 ðk þk Þ 0
1
2
3.3. Three-soliton solutions We seek solutions to system (2.16) in the form
Fig. 1. Graphic of function
v3(x, t) for k1 = 3, k2 = 1, b1 = 3, b2 = 2, 0.1 6 t 6 0.5 and 4 6 x 6 6.
ð3:12Þ
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A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
8 > < f ¼ 1 þ a12 expðn1 þ n2 Þ þ a13 expðn1 þ n3 Þ þ a23 expðn2 þ n3 Þ; g ¼ b1 expðn1 Þ þ b2 expðn2 Þ þ b3 expðn3 Þ þ b123 expðn1 þ n2 þ n3 Þ; > : n1 ¼ k1 x x1 t; n2 ¼ k2 x x2 t; n3 ¼ k3 x x3 t;
ð3:13Þ
for some constants a12, a13, a23, b123, b2, b3, ki – 0 and xi – 0 (i = 1, 2, 3). Inserting (3.6) into (2.16) and equating to zero the different coefficients of ni1 nj2 nk3 (i, j, k = 0, 1, 2, 3, . . . ) yields an algebraic system. Solving it gives
a12 ¼ b1 b2
ðk1 k2 Þ2 ðk1 þ k2 Þ
b123 ¼ b1 b2 b3
2
;
a13 ¼ b1 b3
ðk1 k3 Þ2 2
ðk1 þ k3 Þ
ðk1 k2 Þ2 ðk1 k3 Þ2 ðk2 k3 Þ2 ðk1 þ k2 Þ2 ðk1 þ k3 Þ2 ðk2 þ k3 Þ2
;
;
a23 ¼ b2 b3
ðk2 k3 Þ2 ðk2 þ k3 Þ2
;
x1 ¼ k31 ; x2 ¼ k32 ; x3 ¼ k33 :
ð3:14Þ ð3:15Þ
From (3.13)–(3.15) we see that a three-soliton solution to Eq. (2.20) is
v 4 ðt; xÞ ¼ @ x arctan
b1 expðn1 Þ þ b2 expðn2 Þ þ b3 expðn3 Þ þ b123 expðn1 þ n2 þ n3 Þ : 1 þ a12 expðn1 þ n2 Þ þ a13 expðn1 þ n3 Þ þ a23 expðn2 þ n3 Þ
ð3:16Þ
3.4. Breather-type solutions We seek solutions to system (2.16) in the form
f ¼ a1 coshðk1 x x1 tÞ þ b1 cosðk2 x x2 tÞ; g ¼ c1 coshðk1 x x1 tÞ þ d1 cosðk2 x x2 tÞ;
ð3:17Þ
for some constants a1, b1, c1, d1, b2, k1 – 0, k2 – 0, x1 – 0 and x2 – 0. Let
X ¼ expðk1 x x1 tÞ and Y ¼ exp
pffiffiffiffiffiffiffi
1ðk2 x x2 tÞ :
ð3:18Þ
Inserting (3.17) into (2.16) and equating to zero the different coefficients of XiYj (i, j = 0, 1, 2, 3, . . . ) yields an algebraic system in the unknowns ai, bi, ki, xi. This system reads
Fig. 2. Graphic of function
v5(x, t) for k1 = 2, k2 = 3, a1 = 3, b1 = 1, 0.1 6 t 6 0.1 and 2.5 6 x 6 3.5.
A.H. Salas / Applied Mathematics and Computation 216 (2010) 2792–2798
8 pffiffiffiffiffiffiffi 2 > > k 1k2 ða1 b1 þ c1 d1 Þ ¼ 0; > 1 > > <
2 2 2 2 k1 a21 þ c21 k2 b1 þ d1 ¼ 0; > >
> pffiffiffiffiffiffiffi > > : x1 k31 þ 3k22 k1 þ 1 x2 þ k32 3k21 k2 ðb1 c1 a1 d1 Þ ¼ 0:
2797
ð3:19Þ
Solving this system we obtain:
c1 ¼
b1 k2 ; k1
d1 ¼
a1 k1 ; k2
x1 ¼ k1 k21 3k22 ; x2 ¼ k2 k22 3k21 :
ð3:20Þ
From (2.18), (3.6) and (3.8) we see that a two-soliton solution to Eq. (2.20) is (see Fig. 2):
1 2 2 2 2 2 2 a1 k1 cos k2 x þ k2 k2 3k1 t b1 k2 cosh k1 x k1 k1 3k2 t 1
A: v 5 ðt; xÞ ¼ @ x arctan @ k1 k2 b1 cos k2 x þ k2 k2 3k2 t þ a1 cosh k1 x k1 k2 3k2 t 2 1 1 2 0
4. Concluding remarks Sometimes, we may reduce the problem of finding solutions to an nonlinear pde with variable coefficients to that of solving a similar equation with constant coefficients. This approach simplifies the tedious algebraic computations and allows us to use known results. On the other hand, Hirota’s method is a powerful tool for solving nonlinear pde’s in mathematical physics. Other results concerning pde’s with variable coefficients may be found in [22–24,34,35]. Additional results concerning the problem of finding analytic solutions to nonlinear pde’s are derived in [25–33]. We think that some of the results we presented in this work are new in the open literature. References [1] R.M. Miura, C.S. Gardner, M.D. Kruskal, Korteweg-de Vries equation and generalizations. II. existence of conservation laws and constants of motion, J. Math. Phys. 9 (1968) 1204. [2] M. Wadati, The exact solution of the modified Korteweg-de Vries equation, J. Phys. Soc. Jpn. 32 (1972) 1681. 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