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The modified reductive perturbation method as applied to Boussinesq equation: strongly dispersive case
Applied Mathematics and Computation 164 (2005) 1–9 www.elsevier.com/locate/amc
The modified reductive perturbation method as applied to Boussinesq equation: strongly dispersive case Hilmi Demiray Faculty of Arts and Sciences, Department of Mathematics, Isik University, Bu¨yu¨kdere Caddesi, 34398 Maslak, Istanbul, Turkey
Abstract In this work, we extended the application of ‘‘the modified reductive perturbation method’’ to Boussinesq equation for strongly dispersive case and tried to obtain the contribution of higher order terms in the perturbation expansion. It is shown that the first order term in the perturbation expansion is governed by the non-linear Schro¨dinger equation and the second order term is governed by the linearized Schro¨dinger equation with a non-homogeneous term. In the long-wave limit, a travelling wave type of solution to these equations is also given. Ó 2004 Published by Elsevier Inc.
1. Introduction In collisionless cold plasma, in fluid-filled elastic tubes and in shallow-water waves, due to nonlinearity of the governing equations, for weakly dispersive case one obtains the Korteweg–de Vries (KdV) equation for the lowest order term in the perturbation expansion, the solution of which may be described E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2004 Published by Elsevier Inc. doi:10.1016/j.amc.2004.06.076
2
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
by solitons (Davidson [1]). To study the higher order terms in the perturbation expansion, the reductive perturbation method has been introduced by use of the stretched time and space variables (Taniuti [2]). However, in such approach some secular terms appear which can be eliminated by introducing some slow scale variables (Sugimoto and Kakutani [3]) or by a renormalization procedure of the velocity of the KdV soliton (Kodama and Taniuti [4]). Nevertheless, this approach remains somewhat artificial, since there is no reasonable connection between the smallness parameters of the stretched variables and the one used in the perturbation expansion of the field variables. The choice of the former parameter is based on the linear wave analysis of the concerned problem and the wave number or the frequency is taken as the perturbation parameter (Washimi and Taniuti [5]). On the other hand, at the lowest order, the amplitude and the width of the wave are expressed in terms of the unknown perturbed velocity, which is also used as the smallness parameter. This causes some ambiguity over the correction terms. Another attempt to remove such secularities made by Kraenkel and Manna [6] for long water waves by use of the multiple time scale expansion but could not obtain explicitly the correction terms to the speed of the wave. In order to remove these ambiguities, Malfliet and Wieers [7] presented a dressed solitary wave approach, which is based on the assumption that the field variables admit localized travelling wave solution. Then, for the longwave limit, they expanded the field variables and the wave speed into a power series of the wave number, which is assumed to be the only smallness parameter, and obtained the explicit solution for various order terms in the expansion. However, this approach can only be used when one studies progressive wave solution to the original nonlinear equations and it does not give any idea about the form of evolution equations governing the various order terms in the perturbation expansion. In our previous paper [8], we have presented a method called ‘‘the modified reductive perturbation method’’ to examine the contributions of higher order terms in the perturbation expansion and applied it to weakly dispersive ion-acoustic plasma waves and solitary waves in a fluid filled elastic tube [9]. In these works, we have shown that the lowest order term in the perturbation expansion is governed by the nonlinear Korteweg–de Vries equation, whereas the higher order terms in the expansion are governed by the degenerate Korteweg–de Vries equation with non-homogeneous term. By employing hyperbolic tangent method a progressive wave type of solution was sought and the possible secularities were removed by selecting the scaling parameter in a special way. The basic idea in this method was the inclusion of higher order dispersive effects through the introduction of the scaling parameter c, to balance the higher order nonlinearities with dispersion. The negligence of higher order dispersive effects in the classical reductive perturbation method leads to the imbalance of nonlinearity and dispersion, which resulted in some secular terms in the solution of evolution equations. As a matter of fact, the renormal-
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
3
ization method presented by Kodama and Taniuti [4] is different but rather involved formulation of the same idea. In this work, we extended the application of ‘‘the modified reductive perturbation method’’ to Boussinesq equation for strongly dispersive case and tried to obtain the contribution of higher order terms in the perturbation expansion. It is shown that the first order term in the perturbation expansion is governed by the non-linear Schro¨dinger equation and the second order term is governed by the linearized Schro¨dinger equation with a non-homogeneous term. In the long-wave limit, a travelling wave type of solution to these equations is also given. The present method is seen to be fairly simple and practical as compared to the renormalization method of Kodama and Taniuti [4] and the multiple scale expansion method of Kraenkel and Manna [6].
2. Modified reductive perturbation formalism for Boussinesq equation The one-dimensional form of the Boussinesq equation may be given by o 2 u o2 u o4 u o2 2 þ 3 ðu Þ ¼ 0: ot2 ox2 ox4 ox2
ð1Þ
For our future purposes, it is convenient to introduce following stretched coordinates n ¼ ðx ktÞ;
s ¼ 2 gt;
ð2Þ
where is a small parameter measuring the width of wave packet, k and g are two scale parameters to be determined from the solution. We shall assume that the field variables are functions of slow variables (n, s) as well as the fast variables (x, t). Thus, the following operators are valid o o o o ! k þ 2 g ; ot ot on os o o o ! þ : ox ox on
ð3Þ
We shall further assume that the field variable u and the scale parameter g can be expanded into asymptotic series as u ¼ u1 þ 2 u2 þ 3 u3 þ 4 u4 þ . . . ; g ¼ 1 þ g1 þ 2 g2 þ 3 g3 þ 4 g4 þ
ð4Þ
Introducing the expansions (3) and (4) into the field equation (1) and setting the coefficients of like powers of equal to zero, we obtain the following sets of differential equations:
4
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
O() order equation o 2 u1 o2 u1 o4 u1 2 þ 4 ¼ 0: ot2 ox ox
ð5Þ
O(2) order equation o 2 u2 o2 u2 o4 u2 o4 u1 o2 u1 o2 u1 o2 2 2k 2 3 þ þ 4 ðu Þ ¼ 0: ot2 ox2 ox4 ox3 on oton oxon ox2 1
ð6Þ
O(3) order equation o2 u3 o2 u3 o4 u3 o2 u2 o2 u2 o4 u2 o4 u1 2 þ 4 þ 6 þ 2k ot2 ox2 ox4 oton oxon ox3 on on2 ox2 2 2 2 2 o o o u1 o u1 ðu2 Þ þ ðk2 1Þ 2 þ 2 ¼ 0: 6 2 ðu1 u2 Þ 6 ox oxon 1 otos on
ð7Þ
O(4) order equation 2
o2 u4 o2 u4 o4 u4 o2 u3 o2 u3 o4 u3 o2 ðu2 Þ 2 þ 4 3 þ 2k ot2 ox2 ox4 oton oxon ox3 on ox2 2 2 2 4 o ðu1 u3 Þ o u2 o u2 o u2 o2 2 þ 6 ðu1 u2 Þ 6 þ ðk 1Þ þ 2 12 ox2 otos oxon on2 ox2 on2 o2 u1 o2 u1 o4 u1 o2 2k þ 4 3 3 2 ðu21 Þ ¼ 0: þ 2g1 otos onos on ox on
ð8Þ
2.1. Solution of the field equations For the solution of O() order equation, we shall propose u1 ¼ U ei/ þ c:c:;
ð9Þ
where U(n, s) is an unknown function whose governing equation will be obtained later, / = xt kx is the phasor and c.c. stands for the complex conjugate of the corresponding quantity. Introducing (9) into Eq. (5), and in order to have a non-zero solution for U(n, s) we must have the following dispersion relation x2 ðk 2 þ k 4 Þ ¼ 0;
ð10Þ
where x is the angular frequency and k is the wave number. In order to obtain the solution for O(2) order equation we introduce (9) into Eq. (6) and have o 2 u2 o2 u2 o4 u2 oU i/ e þ 12k 2 U 2 e2i/ þ c:c: ¼ 0: 2 þ 4 þ 2ið2k 3 kx þ kÞ on ot2 ox ox ð11Þ
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
5
For this order of equation we shall seek a solution of the form ð1Þ
ð2Þ
u2 ¼ U 2 ei/ þ U 2 e2i/ þ c:c: ð12Þ Introducing (12) into Eq. (11) and setting the coefficients of like powers of ei/ equal to zero, we obtain oU ¼ 0; ð13Þ ð2k 3 þ k kxÞ on ð2Þ
ð16k 4 þ 4k 2 4x2 ÞU 2 þ 12k 2 U 2 ¼ 0: In order to have a non-zero solution for U we must have 2k 3 þ k kx ¼ 0;
k¼
k þ 2k 3 dx ¼ dk x
ðgroup velocityÞ:
ð14Þ
ð15Þ
The solution of (14) yields U2 : ð16Þ k2 ð1Þ Here U 2 remains to be arbitrary, provided that the dispersion relation (10) is satisfied, and its governing equation will be obtained later. To obtain the solution for O(3) order equation, we introduce the solutions (9) and (12) into Eq. (7), to have 2 o2 u3 o2 u3 o4 u3 oU 2 2 o U 2 ð2Þ þ 6k þ þ ðk 1 6k Þ þ 2ix U U ei/ 2 os ot2 ox2 ox4 on2 " # ð2Þ oU 2i/ ð1Þ 3 oU 2 2 þ 24k UU 2 þ 24ikU þ 4iðk þ k þ 8k Þ e on on ð2Þ
U2 ¼
ð2Þ
þ 54k 2 U 2 U e3i/ þ c:c: ¼ 0:
ð17Þ
The form of this equation suggests us to seek the following type of solution ð1Þ
ð2Þ
ð3Þ
u3 ¼ U 3 ei/ þ U 3 e2i/ þ U 3 e3i/ þ c:c:
ð18Þ
Introducing (18) into Eq. (17) and setting the coefficients of like powers of ei/ equal to zero we obtain the following sets of differential equations 2ix
oU o2 U ð2Þ þ ðk2 6k 2 1Þ 2 þ 6k 2 U 2 U ¼ 0; os on
ð19Þ ð2Þ
ð2Þ
ð4k 2 þ 16k 4 4x2 ÞU 3 þ 4iðkx þ k þ 8k 3 Þ þ 24ikU
oU ¼ 0; on
oU 2 ð1Þ þ 24k 2 U 2 U on ð20Þ
ð3Þ
ð2Þ
ð9k 2 þ 81k 4 9x2 ÞU 3 þ 54k 2 U 2 U ¼ 0:
ð21Þ
6
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9 ð2Þ
Using the expression of U 2 in (19) and (20) we obtain the conventional nonlinear Schro¨dinger equation i
oU o2 U 2 þ l1 2 þ l2 jU j U ¼ 0; os on
ð22Þ
where jUj2 = UU*, U* being the complex conjugate of U and the coefficients l1 and l2 are defined by l1 ¼
1 2 ðk 6k 2 1Þ; 2x
l2 ¼
3 : x
ð23Þ
The solution of the remaining equation yields 2i oU 2 ð1Þ 2 UU 2 ; U 3 on 3k k
ð2Þ
U3 ¼
ð3Þ
U3 ¼
3 3 U : 4k 4
ð24Þ
For the solution of O(4) order equation, we need only the equation associated with the coefficient of ei/. From the Eq. (8) this equation can be written as ð1Þ
ð1Þ
o2 U 2 oU þ 2ix 2 2 os on 2 3 o oU oU oU ð2Þ 2k 4ik 3 ¼ 0: þ 6ik ðU 2 U Þ þ 2ig1 x on os onos on ð2Þ
ð1Þ
3k 2 U 2 U 2
ð2Þ
þ 6k 2 U 3 U þ ðk2 6k 2 1Þ
ð2Þ
ð25Þ
ð2Þ
Introducing the expressions of U 2 and U 3 into Eq. (25) we have ð1Þ
i
ð1Þ
oU 2 o2 U 2 l ð1Þ 2 ð1Þ þ l1 2 U 2 U 2 2l2 jU j U 2 ¼ iSðU Þ; 2 os 2 on
ð26Þ
where the non-homogeneous term S(U) is defined by 2
SðU Þ ¼ l3 U
ojU j o3 U oU 2 oU þ l5 3 g1 : þ l4 jU j on os on on
ð27Þ
Here, the coefficients l3, l4 and l5 are defined by
1 k 1 3k 2 ; l4 ¼ 2; kx x kx x 2k k l5 ¼ þ 2 ðk2 6k 2 1Þ: x 2x l3 ¼ 3
ð28Þ
2.1.1. Solution in the small wave number region Since the coefficients of the coupled set of Eqs. (22) and (26) are too complicated, we shall investigate the nonlinear wave modulation for small wave num-
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
7
bers. The asymptotic form of the coefficients appearing in Eqs. (23) and (28) take the following form l1 ¼
1 ; 2k
3 l2 ¼ ; k
l3 ¼ 3;
l4 ¼
2 ; k2
1 l5 ¼ : 2
ð29Þ
2.1.2. Progressive wave solution In this subsection we shall present a progressive wave solution to Eqs. (22) and (26). For that purpose, we propose a solution of the following form U ¼ f ðfÞ exp½iðKn XsÞ ; ð1Þ
U 2 ¼ hðfÞ exp½iðKn XsÞ ; f ¼ n 2l1 Ks;
ð30Þ
where f(f) is a real function, h(f) is a complex function, K and X are some constants. Introducing (30) into Eq. (22) we have l1 f 00 þ ðX l1 K 2 Þf þ l2 f 3 ¼ 0:
ð31Þ
Here a prime denotes the differentiation of the corresponding quantity with respect to f. Since the coefficients l1 and l2 satisfy the inequality l1l2 < 0, the solution for f(f) may be given by f ¼ A tanh bf;
ð32Þ
where A is the amplitude of the shock wave, X and b are defined by b2 ¼
l2 2 A; 2l1
X ¼ l 1 K 2 l 2 A2 :
ð33Þ
Introducing the solution given in (30) and (32) into Eq. (26), the governing equation for h(f) may be given as follows l1 ðh00 þ 2b2 hÞ þ l1 b2 tanh2 bfðh þ 4hÞ ¼ ½Xg1 A þ l5 ð6KAb2 þ K 3 AÞ tanh bf ð6l5 KAb2 þ l4 KA3 Þtanh3 bf þ if½2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab þ ½l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab tanh2 bf þ ½l3 A3 b l4 A3 b 6l5 Ab3 tanh4 bfg:
ð34Þ
The form of Eq. (34) suggests us that the function h(f) has the following form hðfÞ ¼ h1 ðfÞ þ ih2 ðfÞ:
ð35Þ
8
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
Introducing (35) into Eq. (34), the real and imaginary parts of it yields the following differential equations l1 ½h001 þ ð2b2 þ 5b2 tanh2 bfÞh1 ¼ ½Xg1 A þ l5 ð6KAb2 þ K 3 AÞ tanh bf ð6l5 KAb2 þ l4 KA3 Þtanh3 bf:
ð36Þ
l1 ½h002 þ ð2b2 þ 3b2 tanh2 bfÞh2 ¼ ½2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab þ ½l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab tanh2 bf þ ½l3 A3 b l4 A3 b 6l5 Ab3 tanh4 bf:
ð37Þ
For the solution of Eq. (36) we shall propose the following function h1 ¼ b tanh bf;
ð38Þ
where b is a constant to be determined from the solution. Introducing (38) into (36) and noting that b2 = 3A2, for this case, we have b¼
ðl4 þ 18l5 ÞKA ; 21l1
g1 ¼
ðK 3 þ 18KA2 Þl5 : X
ð39Þ
Here, we note that, in order to balance the term related to tanh bf (in Eq. (36), we set the coefficient of this term equal to zero and determined the g1. The method presented here makes it possible to determine the scale parameter g1 as part of the solution. The solution for h2 may be expressed as ð40Þ h2 ¼ a0 þ a1 tanh2 bf; where a0 and a1 are the two constants to be determined from the solution. Introducing (40) into Eq. (37) we obtain 2l1 b2 ða0 þ a1 Þ þ l1 b2 ð3a0 6a1 Þtanh2 bf þ 9a1 b2 l1 tanh4 bf ¼ ½2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab þ ½l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab tanh2 bf þ ½l3 A3 b l4 A3 b 6l5 Ab3 tanh4 bf: ð41Þ
From this equation we obtain the following algebraic equations 2l1 b2 ða0 þ a1 Þ ¼ 2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab; 3l1 b2 ða0 2a1 Þ ¼ l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab; 9b2 l1 a1 ¼ l3 A3 b l4 A3 b 6l5 Ab3 : ð42Þ
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
From the solution of these equations we obtain pffiffiffi ! pffiffiffi pffiffiffi ! pffiffiffi 2 3 2 3 2 3 3 2 a0 ¼ þ A ; a1 ¼ A2 : 9l1 27k 2 l1 10l1 27k 2 l1
9
ð43Þ
Hence, in terms of the real physical quantities the solution becomes eiu
u ¼ fA tanh bf þ 2 b tanh bf þ i2 ½a0 þ a1 tanh2 bf g ;
ð44Þ
where f ¼ ðx kt 2l1 Kt 2 2l1 Kg1 tÞ; u ¼ ½Kðx ktÞ Xt 2 Xg1 t :
ð45Þ
Here, it is seen that the second order correction terms to the speed of enveloping and harmonic waves are, respectively, given by 2l1Kg1 and Xg1. As pointed out before, with this method, it is possible to obtain progressive wave solution to the evolution equations of any order term in the perturbation expansion without causing the secularities. However, it is not possible to obtain similar results through the use of the conventional reductive perturbation method [4,6].
Acknowledgment This work was supported by the Turkish Academy of Sciences.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
N. Davidson, Methods in Nonlinear Plasma Theory, Academic Press, New York, 1972. T. Taniuti, Progress in Theoretical Physics Supplement 55 (1974) 1. N. Sugimoto, T. Kakutani, J. Phys. Soc. Japan 43 (1977) 1469. Y. Kodama, T. Taniuti, J. Phys. Soc. Japan 45 (1978) 298. H. Washimi, T. Taniuti, Phys. Rev. Lett. 17 (1966) 996. R.A. Kraenkel, M.A. Manna, J. Math. Phys. 36 (1995) 307. M. Malfliet, E. Wieers, J. Plasma Phys. 56 (1996) 444. H. Demiray, J. Phys. Soc. Japan 68 (1999) 1833. H. Demiray, Z. Agnew. Math. Phys. (ZAMP) 51 (2000) 75–91.
The modified reductive perturbation method as applied to Boussinesq equation: strongly dispersive case Hilmi Demiray Faculty of Arts and Sciences, Department of Mathematics, Isik University, Bu¨yu¨kdere Caddesi, 34398 Maslak, Istanbul, Turkey
Abstract In this work, we extended the application of ‘‘the modified reductive perturbation method’’ to Boussinesq equation for strongly dispersive case and tried to obtain the contribution of higher order terms in the perturbation expansion. It is shown that the first order term in the perturbation expansion is governed by the non-linear Schro¨dinger equation and the second order term is governed by the linearized Schro¨dinger equation with a non-homogeneous term. In the long-wave limit, a travelling wave type of solution to these equations is also given. Ó 2004 Published by Elsevier Inc.
1. Introduction In collisionless cold plasma, in fluid-filled elastic tubes and in shallow-water waves, due to nonlinearity of the governing equations, for weakly dispersive case one obtains the Korteweg–de Vries (KdV) equation for the lowest order term in the perturbation expansion, the solution of which may be described E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2004 Published by Elsevier Inc. doi:10.1016/j.amc.2004.06.076
2
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
by solitons (Davidson [1]). To study the higher order terms in the perturbation expansion, the reductive perturbation method has been introduced by use of the stretched time and space variables (Taniuti [2]). However, in such approach some secular terms appear which can be eliminated by introducing some slow scale variables (Sugimoto and Kakutani [3]) or by a renormalization procedure of the velocity of the KdV soliton (Kodama and Taniuti [4]). Nevertheless, this approach remains somewhat artificial, since there is no reasonable connection between the smallness parameters of the stretched variables and the one used in the perturbation expansion of the field variables. The choice of the former parameter is based on the linear wave analysis of the concerned problem and the wave number or the frequency is taken as the perturbation parameter (Washimi and Taniuti [5]). On the other hand, at the lowest order, the amplitude and the width of the wave are expressed in terms of the unknown perturbed velocity, which is also used as the smallness parameter. This causes some ambiguity over the correction terms. Another attempt to remove such secularities made by Kraenkel and Manna [6] for long water waves by use of the multiple time scale expansion but could not obtain explicitly the correction terms to the speed of the wave. In order to remove these ambiguities, Malfliet and Wieers [7] presented a dressed solitary wave approach, which is based on the assumption that the field variables admit localized travelling wave solution. Then, for the longwave limit, they expanded the field variables and the wave speed into a power series of the wave number, which is assumed to be the only smallness parameter, and obtained the explicit solution for various order terms in the expansion. However, this approach can only be used when one studies progressive wave solution to the original nonlinear equations and it does not give any idea about the form of evolution equations governing the various order terms in the perturbation expansion. In our previous paper [8], we have presented a method called ‘‘the modified reductive perturbation method’’ to examine the contributions of higher order terms in the perturbation expansion and applied it to weakly dispersive ion-acoustic plasma waves and solitary waves in a fluid filled elastic tube [9]. In these works, we have shown that the lowest order term in the perturbation expansion is governed by the nonlinear Korteweg–de Vries equation, whereas the higher order terms in the expansion are governed by the degenerate Korteweg–de Vries equation with non-homogeneous term. By employing hyperbolic tangent method a progressive wave type of solution was sought and the possible secularities were removed by selecting the scaling parameter in a special way. The basic idea in this method was the inclusion of higher order dispersive effects through the introduction of the scaling parameter c, to balance the higher order nonlinearities with dispersion. The negligence of higher order dispersive effects in the classical reductive perturbation method leads to the imbalance of nonlinearity and dispersion, which resulted in some secular terms in the solution of evolution equations. As a matter of fact, the renormal-
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
3
ization method presented by Kodama and Taniuti [4] is different but rather involved formulation of the same idea. In this work, we extended the application of ‘‘the modified reductive perturbation method’’ to Boussinesq equation for strongly dispersive case and tried to obtain the contribution of higher order terms in the perturbation expansion. It is shown that the first order term in the perturbation expansion is governed by the non-linear Schro¨dinger equation and the second order term is governed by the linearized Schro¨dinger equation with a non-homogeneous term. In the long-wave limit, a travelling wave type of solution to these equations is also given. The present method is seen to be fairly simple and practical as compared to the renormalization method of Kodama and Taniuti [4] and the multiple scale expansion method of Kraenkel and Manna [6].
2. Modified reductive perturbation formalism for Boussinesq equation The one-dimensional form of the Boussinesq equation may be given by o 2 u o2 u o4 u o2 2 þ 3 ðu Þ ¼ 0: ot2 ox2 ox4 ox2
ð1Þ
For our future purposes, it is convenient to introduce following stretched coordinates n ¼ ðx ktÞ;
s ¼ 2 gt;
ð2Þ
where is a small parameter measuring the width of wave packet, k and g are two scale parameters to be determined from the solution. We shall assume that the field variables are functions of slow variables (n, s) as well as the fast variables (x, t). Thus, the following operators are valid o o o o ! k þ 2 g ; ot ot on os o o o ! þ : ox ox on
ð3Þ
We shall further assume that the field variable u and the scale parameter g can be expanded into asymptotic series as u ¼ u1 þ 2 u2 þ 3 u3 þ 4 u4 þ . . . ; g ¼ 1 þ g1 þ 2 g2 þ 3 g3 þ 4 g4 þ
ð4Þ
Introducing the expansions (3) and (4) into the field equation (1) and setting the coefficients of like powers of equal to zero, we obtain the following sets of differential equations:
4
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
O() order equation o 2 u1 o2 u1 o4 u1 2 þ 4 ¼ 0: ot2 ox ox
ð5Þ
O(2) order equation o 2 u2 o2 u2 o4 u2 o4 u1 o2 u1 o2 u1 o2 2 2k 2 3 þ þ 4 ðu Þ ¼ 0: ot2 ox2 ox4 ox3 on oton oxon ox2 1
ð6Þ
O(3) order equation o2 u3 o2 u3 o4 u3 o2 u2 o2 u2 o4 u2 o4 u1 2 þ 4 þ 6 þ 2k ot2 ox2 ox4 oton oxon ox3 on on2 ox2 2 2 2 2 o o o u1 o u1 ðu2 Þ þ ðk2 1Þ 2 þ 2 ¼ 0: 6 2 ðu1 u2 Þ 6 ox oxon 1 otos on
ð7Þ
O(4) order equation 2
o2 u4 o2 u4 o4 u4 o2 u3 o2 u3 o4 u3 o2 ðu2 Þ 2 þ 4 3 þ 2k ot2 ox2 ox4 oton oxon ox3 on ox2 2 2 2 4 o ðu1 u3 Þ o u2 o u2 o u2 o2 2 þ 6 ðu1 u2 Þ 6 þ ðk 1Þ þ 2 12 ox2 otos oxon on2 ox2 on2 o2 u1 o2 u1 o4 u1 o2 2k þ 4 3 3 2 ðu21 Þ ¼ 0: þ 2g1 otos onos on ox on
ð8Þ
2.1. Solution of the field equations For the solution of O() order equation, we shall propose u1 ¼ U ei/ þ c:c:;
ð9Þ
where U(n, s) is an unknown function whose governing equation will be obtained later, / = xt kx is the phasor and c.c. stands for the complex conjugate of the corresponding quantity. Introducing (9) into Eq. (5), and in order to have a non-zero solution for U(n, s) we must have the following dispersion relation x2 ðk 2 þ k 4 Þ ¼ 0;
ð10Þ
where x is the angular frequency and k is the wave number. In order to obtain the solution for O(2) order equation we introduce (9) into Eq. (6) and have o 2 u2 o2 u2 o4 u2 oU i/ e þ 12k 2 U 2 e2i/ þ c:c: ¼ 0: 2 þ 4 þ 2ið2k 3 kx þ kÞ on ot2 ox ox ð11Þ
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
5
For this order of equation we shall seek a solution of the form ð1Þ
ð2Þ
u2 ¼ U 2 ei/ þ U 2 e2i/ þ c:c: ð12Þ Introducing (12) into Eq. (11) and setting the coefficients of like powers of ei/ equal to zero, we obtain oU ¼ 0; ð13Þ ð2k 3 þ k kxÞ on ð2Þ
ð16k 4 þ 4k 2 4x2 ÞU 2 þ 12k 2 U 2 ¼ 0: In order to have a non-zero solution for U we must have 2k 3 þ k kx ¼ 0;
k¼
k þ 2k 3 dx ¼ dk x
ðgroup velocityÞ:
ð14Þ
ð15Þ
The solution of (14) yields U2 : ð16Þ k2 ð1Þ Here U 2 remains to be arbitrary, provided that the dispersion relation (10) is satisfied, and its governing equation will be obtained later. To obtain the solution for O(3) order equation, we introduce the solutions (9) and (12) into Eq. (7), to have 2 o2 u3 o2 u3 o4 u3 oU 2 2 o U 2 ð2Þ þ 6k þ þ ðk 1 6k Þ þ 2ix U U ei/ 2 os ot2 ox2 ox4 on2 " # ð2Þ oU 2i/ ð1Þ 3 oU 2 2 þ 24k UU 2 þ 24ikU þ 4iðk þ k þ 8k Þ e on on ð2Þ
U2 ¼
ð2Þ
þ 54k 2 U 2 U e3i/ þ c:c: ¼ 0:
ð17Þ
The form of this equation suggests us to seek the following type of solution ð1Þ
ð2Þ
ð3Þ
u3 ¼ U 3 ei/ þ U 3 e2i/ þ U 3 e3i/ þ c:c:
ð18Þ
Introducing (18) into Eq. (17) and setting the coefficients of like powers of ei/ equal to zero we obtain the following sets of differential equations 2ix
oU o2 U ð2Þ þ ðk2 6k 2 1Þ 2 þ 6k 2 U 2 U ¼ 0; os on
ð19Þ ð2Þ
ð2Þ
ð4k 2 þ 16k 4 4x2 ÞU 3 þ 4iðkx þ k þ 8k 3 Þ þ 24ikU
oU ¼ 0; on
oU 2 ð1Þ þ 24k 2 U 2 U on ð20Þ
ð3Þ
ð2Þ
ð9k 2 þ 81k 4 9x2 ÞU 3 þ 54k 2 U 2 U ¼ 0:
ð21Þ
6
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9 ð2Þ
Using the expression of U 2 in (19) and (20) we obtain the conventional nonlinear Schro¨dinger equation i
oU o2 U 2 þ l1 2 þ l2 jU j U ¼ 0; os on
ð22Þ
where jUj2 = UU*, U* being the complex conjugate of U and the coefficients l1 and l2 are defined by l1 ¼
1 2 ðk 6k 2 1Þ; 2x
l2 ¼
3 : x
ð23Þ
The solution of the remaining equation yields 2i oU 2 ð1Þ 2 UU 2 ; U 3 on 3k k
ð2Þ
U3 ¼
ð3Þ
U3 ¼
3 3 U : 4k 4
ð24Þ
For the solution of O(4) order equation, we need only the equation associated with the coefficient of ei/. From the Eq. (8) this equation can be written as ð1Þ
ð1Þ
o2 U 2 oU þ 2ix 2 2 os on 2 3 o oU oU oU ð2Þ 2k 4ik 3 ¼ 0: þ 6ik ðU 2 U Þ þ 2ig1 x on os onos on ð2Þ
ð1Þ
3k 2 U 2 U 2
ð2Þ
þ 6k 2 U 3 U þ ðk2 6k 2 1Þ
ð2Þ
ð25Þ
ð2Þ
Introducing the expressions of U 2 and U 3 into Eq. (25) we have ð1Þ
i
ð1Þ
oU 2 o2 U 2 l ð1Þ 2 ð1Þ þ l1 2 U 2 U 2 2l2 jU j U 2 ¼ iSðU Þ; 2 os 2 on
ð26Þ
where the non-homogeneous term S(U) is defined by 2
SðU Þ ¼ l3 U
ojU j o3 U oU 2 oU þ l5 3 g1 : þ l4 jU j on os on on
ð27Þ
Here, the coefficients l3, l4 and l5 are defined by
1 k 1 3k 2 ; l4 ¼ 2; kx x kx x 2k k l5 ¼ þ 2 ðk2 6k 2 1Þ: x 2x l3 ¼ 3
ð28Þ
2.1.1. Solution in the small wave number region Since the coefficients of the coupled set of Eqs. (22) and (26) are too complicated, we shall investigate the nonlinear wave modulation for small wave num-
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
7
bers. The asymptotic form of the coefficients appearing in Eqs. (23) and (28) take the following form l1 ¼
1 ; 2k
3 l2 ¼ ; k
l3 ¼ 3;
l4 ¼
2 ; k2
1 l5 ¼ : 2
ð29Þ
2.1.2. Progressive wave solution In this subsection we shall present a progressive wave solution to Eqs. (22) and (26). For that purpose, we propose a solution of the following form U ¼ f ðfÞ exp½iðKn XsÞ ; ð1Þ
U 2 ¼ hðfÞ exp½iðKn XsÞ ; f ¼ n 2l1 Ks;
ð30Þ
where f(f) is a real function, h(f) is a complex function, K and X are some constants. Introducing (30) into Eq. (22) we have l1 f 00 þ ðX l1 K 2 Þf þ l2 f 3 ¼ 0:
ð31Þ
Here a prime denotes the differentiation of the corresponding quantity with respect to f. Since the coefficients l1 and l2 satisfy the inequality l1l2 < 0, the solution for f(f) may be given by f ¼ A tanh bf;
ð32Þ
where A is the amplitude of the shock wave, X and b are defined by b2 ¼
l2 2 A; 2l1
X ¼ l 1 K 2 l 2 A2 :
ð33Þ
Introducing the solution given in (30) and (32) into Eq. (26), the governing equation for h(f) may be given as follows l1 ðh00 þ 2b2 hÞ þ l1 b2 tanh2 bfðh þ 4hÞ ¼ ½Xg1 A þ l5 ð6KAb2 þ K 3 AÞ tanh bf ð6l5 KAb2 þ l4 KA3 Þtanh3 bf þ if½2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab þ ½l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab tanh2 bf þ ½l3 A3 b l4 A3 b 6l5 Ab3 tanh4 bfg:
ð34Þ
The form of Eq. (34) suggests us that the function h(f) has the following form hðfÞ ¼ h1 ðfÞ þ ih2 ðfÞ:
ð35Þ
8
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
Introducing (35) into Eq. (34), the real and imaginary parts of it yields the following differential equations l1 ½h001 þ ð2b2 þ 5b2 tanh2 bfÞh1 ¼ ½Xg1 A þ l5 ð6KAb2 þ K 3 AÞ tanh bf ð6l5 KAb2 þ l4 KA3 Þtanh3 bf:
ð36Þ
l1 ½h002 þ ð2b2 þ 3b2 tanh2 bfÞh2 ¼ ½2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab þ ½l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab tanh2 bf þ ½l3 A3 b l4 A3 b 6l5 Ab3 tanh4 bf:
ð37Þ
For the solution of Eq. (36) we shall propose the following function h1 ¼ b tanh bf;
ð38Þ
where b is a constant to be determined from the solution. Introducing (38) into (36) and noting that b2 = 3A2, for this case, we have b¼
ðl4 þ 18l5 ÞKA ; 21l1
g1 ¼
ðK 3 þ 18KA2 Þl5 : X
ð39Þ
Here, we note that, in order to balance the term related to tanh bf (in Eq. (36), we set the coefficient of this term equal to zero and determined the g1. The method presented here makes it possible to determine the scale parameter g1 as part of the solution. The solution for h2 may be expressed as ð40Þ h2 ¼ a0 þ a1 tanh2 bf; where a0 and a1 are the two constants to be determined from the solution. Introducing (40) into Eq. (37) we obtain 2l1 b2 ða0 þ a1 Þ þ l1 b2 ð3a0 6a1 Þtanh2 bf þ 9a1 b2 l1 tanh4 bf ¼ ½2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab þ ½l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab tanh2 bf þ ½l3 A3 b l4 A3 b 6l5 Ab3 tanh4 bf: ð41Þ
From this equation we obtain the following algebraic equations 2l1 b2 ða0 þ a1 Þ ¼ 2l5 Ab3 3l5 K 2 Ab þ 2l1 Kg1 Ab; 3l1 b2 ða0 2a1 Þ ¼ l3 A3 b þ l4 A3 b þ 8l5 Ab3 þ 3l5 K 2 Ab 2l1 Kg1 Ab; 9b2 l1 a1 ¼ l3 A3 b l4 A3 b 6l5 Ab3 : ð42Þ
H. Demiray / Appl. Math. Comput. 164 (2005) 1–9
From the solution of these equations we obtain pffiffiffi ! pffiffiffi pffiffiffi ! pffiffiffi 2 3 2 3 2 3 3 2 a0 ¼ þ A ; a1 ¼ A2 : 9l1 27k 2 l1 10l1 27k 2 l1
9
ð43Þ
Hence, in terms of the real physical quantities the solution becomes eiu
u ¼ fA tanh bf þ 2 b tanh bf þ i2 ½a0 þ a1 tanh2 bf g ;
ð44Þ
where f ¼ ðx kt 2l1 Kt 2 2l1 Kg1 tÞ; u ¼ ½Kðx ktÞ Xt 2 Xg1 t :
ð45Þ
Here, it is seen that the second order correction terms to the speed of enveloping and harmonic waves are, respectively, given by 2l1Kg1 and Xg1. As pointed out before, with this method, it is possible to obtain progressive wave solution to the evolution equations of any order term in the perturbation expansion without causing the secularities. However, it is not possible to obtain similar results through the use of the conventional reductive perturbation method [4,6].
Acknowledgment This work was supported by the Turkish Academy of Sciences.
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