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EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHRÖDINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION
2003,23B(4):453-460
athemi(c;a9citntia
.L
1~~JJ!~trl EXACT EXPLICIT SOLUTIONS OF THE .. NONLINEAR SCHRODINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION 1 Yao Ruoxia 1 ,2
(
.loJt*1~)
Li Zhibin 2
(
-t-.t~
)
1.Department of Computer Science, Weinan Teachers' College, Weinan 714000, China; 2.Department of Computer Science, East China Normal University, Shanghai 200062, China
Abstract A system comprised of the nonlinear Schrodinger equation coupled to the Boussinesq equation (S-B equations) which dealing with the stationary propagation of coupled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed. To examine its solitary wave solutions, a reduced set of ordinary differential equations are considered by a simple traveling wave transformation. It is then shown that several new solutions (either functional or parametrical) can be obtained systematically, in addition to rederiving all known -mes by means of our simple and direct algebra method with the help of the computer algebra system Maple. Key words Coupled Schrodinger-Boussinesq equations, traveling wave transformation, Riccati equations, solitary wave solution 2000 MR Subject Classification
1
35Q20, 35J05
Introduction
The bidirectional near-magnetosonic propagation of upper-hybrid waves of frequency Wo and wavenumber ko coupled to the magnetosonic waves in a homogeneous magnetized plasma is governedl'] by a Schrodinger-like equation, namely 8E i(-r(It
·8E
1
8 2E
->..E + wHoJ-LNE,
+ Vg -8 x ) + -2Do-2 = 8x
(1)
which is coupled to the (driven) Boussinesq equation: 2
8 N _ 8t 2
viI 8
2
N _ 8x 2
(32
84N _ 8x 4
2
0:2
8 (N 2 )
8x
2
2
= TJ2 8 2 (EE*). 8x
(2)
In (1) and (2), E(x, t) is the complex envelope of the upper-hybrid wave electric field normalized with respect to (167rn oTe ) 1/2 , N(x, t) is the perturbed number density (in the low-frequency response) normalized with respect to no,x and t are, respectively, the space and time variables, and the asterisk denotes complex conjugation. The following notations are used in (1) and 1 Received July 9, 2001. Project supported by the Natural Science Foundation of the Education Bureau of Shaanxi Province, China (01JK119), and the State Key Program of Basic Research of China (G1998030600).
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ACTA MATHEMATICA SCIENTIA
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= koDo,Do = 3w~eov;e(w~eo - 3w;J-1/WHo,>" = c.,o - WHo - Dok5/ 2,wk o = w~eo + 0~0'f.L = (iw~eo + O;o)/(w~eo + 0;0)' viI = Vl + C;,(3 = VAC/wpeO, a 2 = (3Vl + 2C;)/2 (2): Vg
and "l = WHoCs/Wpeo' The remaining notations are standard [1] and the subscript "0" denotes equilibrium quantities. These equations were previously studied by Rao[2] who obtained solitary wave solutions in sech x tanh and sech'' forms, as well as sech and sech'' forms. He used a power series expansion method to arrive at these solutions. In 1993, Wang et al.[3] examined these coupled equations using an ansatz making method and obtained a separate variety of solitary solutions in the form of sech? and sech'' type solutions. In 1999, Panigrahyltl studied them again by using a mixing exponential method and obtained some solutions in A + sech 2 and B + sech'' forms et al.(A, B are real constants). These solutions are not general and by nq means exhaust all possibilities. They are only some particular solutions within some specific parameters choices. More recently, Zhang et aI,[5] 'proposed a hyperbolic-function method which can deal with nonlinear differential equation effectively, but for coupled nonlinear equations a lot remains to be done. In this paper we intend to analyze these coupled equations by using a simple but widely applicable algebra method whose key idea is to construct such solutions to a given nonlinear equation as a combination of the solutions of the coupled Riccati equations. It is extended essentially from hyperbolic-function method, which can be extensively used for ordinary as well as partial differential equations. In fact, in addition to rederiving all known solutions in a systematic way, several new and more general solutions can be obtained by using our method. In order to examine solitary wave solutions to (1), (2), introducing a simple traveling wave transformation, we consider a reduced set of ordinary equations (ODEs) obtained from (1), (2). We construct analytically solutions to the ODEs thus derived. Then we apply the results to the original nonlinear differential equations (1), (2), and discuss the physical interest of some solutions.
2
Traveling-wave Transformation and A Direct Method To begin with, let us consider the following simple transformation
= EW exp[i(X(x) + T(t))], N(x, t) = N(~), E(x, t)
{
(3)
where ~ = x - Mt is the traveling wave variable and the parameter M determines the speed of the stationary envelope wave. The functions X(x) and T(t) are introduced to account for the shifts in the frequency and wavenumber of the carrier wave due to non-linearities. Substituting (3) into (1) and from the imaginary parts we get
E d2X dx 2
+
[2 dX 2(Vg - M)] dx + Do
then X(x) is determined by X(x)
= M- v:gx. Do
dE
d~
=0
,
(4)
(5)
No.4
Yao & Li: EXACT EXPLICIT SOLUTIONS OF NONLINEAR SCHRODINGER EQUATION 455
Correspondingly, the stationary envelope
E(~)
can be obtained from the real parts:
(6)
i '
, 111 2 _ V 2 where>. = 2b - 2>' + Do 9 and b = that means, T(t) = bt. Using (3) in (2) and integrating twice with respect to ~, one obtains
(7) where A, B are integration constants. Before giving details we rescale (6) and (7) for convenience by using the transformations
to arrive at e
{
d2E
de
= >.E + hEN,
d 2N
2 2 12 =pN +qN +rE +A~+B,
(8)
d~
where E, N and ~ are the same as the former, and all the other remaining quantities are free parameters. For solitary wave solutions to (8), we impose on E and N such boundary conditions as N ---* G2 , E ---* (9) { E', E", N', Nil ---* 0, as I~I ---* 00,
c..
where Gl , G2 are arbitrary constants. It is to be noted that A = 0 in (8), but the integration constant B is not essential to be zero in order to obtain solitary wave solutions under the localized boundary conditions (9). Thus, the system of equations to be discussed has seven free parameters. By inspection, it follows that the equations are invariant under the transformations (i) ~ ---* -~, (ii) E ---* -E, and (iii) ~ ---* ~ + G, where G is a constant. In order to obtain exact solitary wave solutions to (8), we consider the coupled Riccati equations u' = -kuv, v' = k(l - v 2 ) .
(10)
It can be easily verified that (10) has the following general solutions u(~)
= ±sech[k(~ + ~o)], v(~) = tanh[k(~ + ~o)],
(11)
where k is an arbitrary nonzero constant and ~o is an arbitrary one. Solutions (11) satisfy the relation V2(~) = 1 - u2(~). The next crucial step is that the solution we are looking for is expressed in the form
n
N
=L i=O
Ci Ui
n
+L i=l
divU
i- l
(12) ,
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ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
where ai, bi, c., di are constants to be determined. The parameters m, n will be found by balancing the linear terms of highest order with the nonlinear terms in (6) and (7). It is easy to show that n = 2, and m = 0,1,2 if balancing E" with EN, and Nil with E 2 , N 2 . So we may choose E {
°
= ao + al U + a2u2 + i. v + b2vu,
N = Co + CI U + C2U 2 + dl V
(13)
+ d2vu,
or d2 i 0. where C2 i Substituting (13) into (8), using (10) repeatedly, we can express all derivatives of U and v in term of series in U and v, collecting all terms with the same order of U and v, then setting the coefficients of each order of u and v to zero, it yields a set of algebraic equations which contain eighteen nonlinear algebraic equations, for parameters e, I, A, h,p, q, r, Bin 11 unknowns ai,ci, bj , dj , (i = 0, 1,2,j = 1,2) and k:
- Abl -
hb l
Co - haod l = 0,
= 0, 2qCodl = 0,
-Aao - haoCo - hbldl -2raobl - pd, -
- B - ra5 - rbi - pco - qC5 - qdi = 0, -Ab 2 + f3k 2b2 - hb2Co - hblCI - haldl - haad 2 = 0, -2ek 2bl - hb2CI - hblC2 - ha2dl - ha ld2 = 0, -6ek 2b2 - hb2C2 - ha2d2 = 0, -Aal
+ f3k2al - halCo - haOCI - hb2dl - hbld2 = 0,
-2ek2al - ha2cI - baicz -Aa2
+ 4ek2a2 -
ha2Co -
+ hb2dl + bbida = 0, halCI - haOC2 + hbldl
(14)
- hb2d2 = 0,
-6ek2a2 - ha2c2 + hb2d2 = 0,
-2ra lb l - 2ra ob2 - 2qcldl - pd 2 + k 2fd 2 - 2qcod2 = 0, -2(ra2bl + ra lb 2 + k 2fd l + qC2dl + qcld2) = 0, -2(ra2b2 + 3k 2fd 2 + qC2d2) = 0, -2raOal - 2rblb2 - PCI + k 2 fCI - 2qCoCI - 2qdld2 = 0, -2(rala2 - rb lb2 + k 2 fCI + qCIC2 - qdld2) = 0, -rai - 2raOa2 + rbi - rb~ - qcr - PC2 + 4k 2fC2 - 2qcaC2 -ra~ + rb~ - 6k 2fC2 - qc~
+ qdr -
qd~ = 0,
+ qd~ = 0.
The only task in the following is to find all the constants and unknown coefficients such that (13) actually satisfy (8).
3
Exact Analytical Solutions
With the aid of symbolic computation system Maple, and in the interactive mode of Maple, we have found five non-trivial solutions of system (8), for which, if we choose some specific parameter values, such as B = 0, we can cover all the previous known solitary wave solutions
No.4
Yao & Li: EXACT EXPLICIT SOLUTIONS OF NONLINEAR SCHRODINGER EQUATION 457
found by other more sophisticated methods. More importantly, all solutions found by our method are more general types of solutions, which, to our knowledge, have not been reported in the literature. Let us list them as follows. For f = two types of solutions are obtained:
fk,
EI W = {
N I (0
±3~ J6
(2k
2eq
- 6 + 3ph)e r>..q
sech[k(~ - ~o)],
k 2e - >.. k 2e 2 = h - 2 sech [k(~ - ~o)],
(15)
h
where k 2 = _ph - 2>..q ±2Z!p2 - 4Bq and ~o is an integration constant. In order that the solutions are localized, it is necessary that the parameters satisfy the inequalities:
ph - 2>..q ± hJp2 - 4Bq 0 (2k 2eq - 6>..q + 3ph)e =------=----'--=------=- < , > O. eq r E2(~) =
{
N2(~)
Xph. - h 2B _ >..2 q h 2r tanh[k(~ - ~o)],
± >..
2k 2 e
= -h - T
(16)
2
sech [k(~ - ~o)],
13ph - 6>..q ± J3J4>..2 q2 eq - 4>..phq + 3p2h2 - Sh 2qB , lik h k2 = g were l ethe e rirevi previous case, th e various parameters should satisfy the inequalities >..ph - h 2B - >.. 2 q > 0 3ph - 6>..q ± J3/r4>"-:-;2~q~2---4:-:'>..-ph:-q-+-3p2---;:-:"h-;:"2-_-S:-:-h-;:"2qB=- > 0 r ' eq . For B = (>"f - pe)(hpf - >"qf - pqe) another two type solutions are' (hf - 2qe)2 , . E (C) 3 <"
{
= ±~2 (hf (hp - 2>..q)e _ 2qe)h
J
hf - qe h 2[k(C _ C )] er sec <" <,,0,
pe - >"f 3 e(2)..q - hp) 2 N3(~) = hf _ 2qe + "2 (hf _ 2qe)h sech [k(~ - ~o)],
1 h'7- 2>..q . h hi - qe k 2 = 41L h were _ 2eq' WIt er
0 2>..q - h 0 >, 2qe - hy > .
(17)
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- 2>..] . h hp - 2>..] 0 qe - hI 0 h k 2 = 21 hp were 2qe _ h' WIt 2qe - h >, re >. Obviously, while the solutions E(~) given by (16) and (19) are antisymmetric with respect to ~ = ~o, whereas the solutions N(~) always have a symmetric structure. Also we can see, on the contrary, that the structure of the rest three sets of solutions E(~) and N(~) all have a symmetric structure with respect to ~ = ~o. Although the above five sets of solutions possess the known functional forms, they are general with wide parameter ranges. In the following, we consider the specific case for which B = 0 in (8), which is often encountered and also worthwhile in the study of envelope waves in plasmas. In this case, (8) reduces to d2E e --2 = >..E + hEN, d~
(20)
2N
d 2 2 { 1 2 =pN +qN +rE . d~
For (20), Rao[2] studied them and obtained some solitary wave solutions under the following boundary conditions: dE dN (21) as I~I -+ 00 E,N, d~' df -+ 0 and Takao[6] did under such conditions as N
dE dN
-+ C 3,E, d[' df -+ 0
as I~I-+
00,
(22)
where C3 is an arbitrary constant. But by inspection, we find that, to obtain solitary wave solutions, E, N are not necessary to be zero, so we set the boundary conditions as (9). Under such boundary conditions and for the case B = 0, from (15),(17),(18) and (19) we can obtain two sets of solutions respectively, but from (16) only one set of solutions is obtained. Thus, nine non-trivial solutions are delivered in all, in which, in addition to covering all those found by [2,4,7]' the others are new and first reported here. For the sake of conciseness we only list two new and more general sets of them. (i). For qe - 31h = 0,
E21(~) = >..(ePe~r3>"f) tanh[k(~ - ~o)], {
where
N 21W =
>..
2k 2 e
-it - T
2
(23)
sech [k(~ - ~o)],
k 2 = -6>"1 + ep ± V12>..2 J2 8el
-
4>..pel
+ p2e2
(24)
and for localized solutions the parameters should satisfy the following inequalities ep - 6>"1 0 hr 0 >.. 0 < ' -e 1 > ' -e < ' e1
or
ep - 3Af 0 hr 0 >.. 0 < ' -e 1 < ' -e < · e1
Solution (23) is new and more general than that ofPanigrahy et al[4l. It is valid for qe-3lh = 0, whereas the corresponding one in [4] is only valid for qe - 31h = 0 and pe - >"1 = O. In fact, if
Yao & Li: EXACT EXPLICIT SOLUTIONS OF NONLINEAR SCHRODINGER EQUATION 459
No.4
we further set the restriction pe - >"1 = 0, we can get two sets of solutions from (23), in which one is obtained by Panigrahyl''I, however, the other is completely new and reads
E(E)
= ±~J-2~~ tanh[~J -y(E + Eo)],
{ N(E) = '1 ') (1
D
2~ [-2 + sech2[~A(E + Eo)]].
2 >..q(3qe - hf) h(5qe - 2hf) ,
-
ror p -
E5I(E) =
±6k~e. Jqe ~r hi sech[k(E -
3e { N (0 = 2k 2 (h 5I where k
=
(25)
I
- q) -
Eo)]tanh[k(E - Eo)],
6k 2 e 2 -h- sech [k(E - ~o)],
(26)
V
2hf >..~ 5qe' with 2hf >..~ 5qe < 0, qe;rhi > O. This solution is new and obtained
from (19). In the following section, we apply the above results to the original coupled SchrodingerBoussinesq equations.
4
Traveling Wave Solutions to Coupled S-B Equations Equations (1) and (2) are in the standard form of (8). Using the results in Section 3, their
exact localized solutions satisfying the boundary conditions (9) can easily be obtained. For D oa 2 + 6f-lWHo(P = 0, (1) and (2) have 2 sets of solutions
-2k 2D o(32
+ 6>"(32 - DoQ . sech[k(x - Mt)] exp[I(rx + 8t)], f-lWHo k 2Do - >.. k 2Do 2 { N(x, t) = 2 - --sech [k(x - Mt)], f-lWH o f-lWHo E(x, t)
= ±-k 'T/
(27)
2 _ 6>"(32 - DoQ ± JD o(DoQ2 - 24f-lWHg(32B). 6D , o(32
h k were
= ± V2
->"DoQ + 3>..2(32 + ;f-lWHoDoB tanh[k(x _ Mt)] exp[i(rx + 8t)], DOf-lWHo'T/ 2 1 x k Do { N(x, t) = - - - - - --sech 2 [k(x - Mt)], 2 f-lWH o f-lWHo E(x, t)
2
(28)
12>..2(34 - 4>"D o(32 + D5Q 2 - 16f-lwHoBDo(32 B D o(3 Except the above 2 sets of solutions, (1) and (2) admit another 3 sets of solutions for which
where k 2 = 1
DoQ - 6>"(32 ±
all parameters are free:
2
±~ (>..a + f-lWHoQ)J -D02(P + f-lWHo(32) sech 2[k(x
_ Mt)] exp[i(rx + 8t)], 'T/f-lWHoP { N(x, t) = _ >"(32 - DoQ _ ~ Do(>..2 a2 + f-lWHoQ) sech 2[k(x _ Mt)]. 2P 4 f-lWHoP
E(x, t) =
4
460
ACTA MATHEMATICA SCIENTIA
(29)
where k 2 = _ >. + ~pHoQ;
.
E(x, t) = ±3
k2VD (a. 2D 0
0
+ 2J.Lw -
H (J2) 0
J.LWHo'TJ
tanh[k(x - Mt)]sech[k(x - Mt)] exp[i(TX + 6t)],
k 2D o 2[k(x { N(x, t) = -3--sech - Mt)], • J.LWHo 2_ _ J.LWHoQ + >'0'.2 h k were 2( 2D (32) . a. 0 + J.LWHo For the sake of conciseness, we denote
the above five solutions,
5
T
Vol.23 Ser.B
= M
Do V
g
(30)
0'.2
Do + J.LWHo(32 as P and M 2 - V~ as Q. In all of
and 6, M, B are arbitrary constants.
Summary
By using a simple transformation we derive the coupled non-linear ordinary differential equations from coupled Schr6dinger-Boussinesq equations, then we obtain five important and general solitary wave solutions, for which, if choosing specific parameters, it not only produces the same solutions as originally obtained by [2,4] and [7] but also can pick up what we believe to be new solutions missed by other authors. For all solutions we verify them by putting them back into (8). Hence, the approach used here shows some novel aspects. Firstly, it is an effective and systematic method in solving nonlinear differential equations. Secondly, it can serve a more general recipe for finding traveling wave solutions to the great majority equations. Finally, it only needs simple differential computation and solving a set of lower algebraic equations because the order of partial differential equations is sufficiently reduced and Riccati equations are used. More importantly, it is a computerizable method, which allows us to perform complicated and tedious algebraic calculation on computer. References 1 Rao N N. Near-magnetosonic envelope upper-hybrid waves. J Plasma Phys, 1988, 39: 385-405 2 Rao N N. Exact solutions of coupled scalar field equations. J Phys A: Math Gen, 1989, 22: 4813-4825 3 Wang X Y, Xu B C, Taylor P L. Exact soliton solutions for a class of coupled field equations. Phys Lett, 1993, A173: 30-32 4 Panigrahy M. Dash P C. Soliton solutions of a coupled field using the mixing exponential method. Phys Lett, 1999, A261: 284-288 5 Zhang G X, Li Z B, Duan Y S. Exact solitary wave solutions to nonlinear wave equation. Science in China, 2000, 30A: 1103-1108 (in Chinese) 6 Yoshinaga T, Kakutani T. Solitary and E-shock waves in a resonant system between long and short waves. J Phys Soc Japan, 1994, 63: 445-459 7 Yoko Hase, Junkichi Satsuma. An n-soliton solution for the nonlinear Schrodinger equation coupled to the Boussinesq equation. J Phys Soc Japan, 1988, 57: 679-682
athemi(c;a9citntia
.L
1~~JJ!~trl EXACT EXPLICIT SOLUTIONS OF THE .. NONLINEAR SCHRODINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION 1 Yao Ruoxia 1 ,2
(
.loJt*1~)
Li Zhibin 2
(
-t-.t~
)
1.Department of Computer Science, Weinan Teachers' College, Weinan 714000, China; 2.Department of Computer Science, East China Normal University, Shanghai 200062, China
Abstract A system comprised of the nonlinear Schrodinger equation coupled to the Boussinesq equation (S-B equations) which dealing with the stationary propagation of coupled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed. To examine its solitary wave solutions, a reduced set of ordinary differential equations are considered by a simple traveling wave transformation. It is then shown that several new solutions (either functional or parametrical) can be obtained systematically, in addition to rederiving all known -mes by means of our simple and direct algebra method with the help of the computer algebra system Maple. Key words Coupled Schrodinger-Boussinesq equations, traveling wave transformation, Riccati equations, solitary wave solution 2000 MR Subject Classification
1
35Q20, 35J05
Introduction
The bidirectional near-magnetosonic propagation of upper-hybrid waves of frequency Wo and wavenumber ko coupled to the magnetosonic waves in a homogeneous magnetized plasma is governedl'] by a Schrodinger-like equation, namely 8E i(-r(It
·8E
1
8 2E
->..E + wHoJ-LNE,
+ Vg -8 x ) + -2Do-2 = 8x
(1)
which is coupled to the (driven) Boussinesq equation: 2
8 N _ 8t 2
viI 8
2
N _ 8x 2
(32
84N _ 8x 4
2
0:2
8 (N 2 )
8x
2
2
= TJ2 8 2 (EE*). 8x
(2)
In (1) and (2), E(x, t) is the complex envelope of the upper-hybrid wave electric field normalized with respect to (167rn oTe ) 1/2 , N(x, t) is the perturbed number density (in the low-frequency response) normalized with respect to no,x and t are, respectively, the space and time variables, and the asterisk denotes complex conjugation. The following notations are used in (1) and 1 Received July 9, 2001. Project supported by the Natural Science Foundation of the Education Bureau of Shaanxi Province, China (01JK119), and the State Key Program of Basic Research of China (G1998030600).
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ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
= koDo,Do = 3w~eov;e(w~eo - 3w;J-1/WHo,>" = c.,o - WHo - Dok5/ 2,wk o = w~eo + 0~0'f.L = (iw~eo + O;o)/(w~eo + 0;0)' viI = Vl + C;,(3 = VAC/wpeO, a 2 = (3Vl + 2C;)/2 (2): Vg
and "l = WHoCs/Wpeo' The remaining notations are standard [1] and the subscript "0" denotes equilibrium quantities. These equations were previously studied by Rao[2] who obtained solitary wave solutions in sech x tanh and sech'' forms, as well as sech and sech'' forms. He used a power series expansion method to arrive at these solutions. In 1993, Wang et al.[3] examined these coupled equations using an ansatz making method and obtained a separate variety of solitary solutions in the form of sech? and sech'' type solutions. In 1999, Panigrahyltl studied them again by using a mixing exponential method and obtained some solutions in A + sech 2 and B + sech'' forms et al.(A, B are real constants). These solutions are not general and by nq means exhaust all possibilities. They are only some particular solutions within some specific parameters choices. More recently, Zhang et aI,[5] 'proposed a hyperbolic-function method which can deal with nonlinear differential equation effectively, but for coupled nonlinear equations a lot remains to be done. In this paper we intend to analyze these coupled equations by using a simple but widely applicable algebra method whose key idea is to construct such solutions to a given nonlinear equation as a combination of the solutions of the coupled Riccati equations. It is extended essentially from hyperbolic-function method, which can be extensively used for ordinary as well as partial differential equations. In fact, in addition to rederiving all known solutions in a systematic way, several new and more general solutions can be obtained by using our method. In order to examine solitary wave solutions to (1), (2), introducing a simple traveling wave transformation, we consider a reduced set of ordinary equations (ODEs) obtained from (1), (2). We construct analytically solutions to the ODEs thus derived. Then we apply the results to the original nonlinear differential equations (1), (2), and discuss the physical interest of some solutions.
2
Traveling-wave Transformation and A Direct Method To begin with, let us consider the following simple transformation
= EW exp[i(X(x) + T(t))], N(x, t) = N(~), E(x, t)
{
(3)
where ~ = x - Mt is the traveling wave variable and the parameter M determines the speed of the stationary envelope wave. The functions X(x) and T(t) are introduced to account for the shifts in the frequency and wavenumber of the carrier wave due to non-linearities. Substituting (3) into (1) and from the imaginary parts we get
E d2X dx 2
+
[2 dX 2(Vg - M)] dx + Do
then X(x) is determined by X(x)
= M- v:gx. Do
dE
d~
=0
,
(4)
(5)
No.4
Yao & Li: EXACT EXPLICIT SOLUTIONS OF NONLINEAR SCHRODINGER EQUATION 455
Correspondingly, the stationary envelope
E(~)
can be obtained from the real parts:
(6)
i '
, 111 2 _ V 2 where>. = 2b - 2>' + Do 9 and b = that means, T(t) = bt. Using (3) in (2) and integrating twice with respect to ~, one obtains
(7) where A, B are integration constants. Before giving details we rescale (6) and (7) for convenience by using the transformations
to arrive at e
{
d2E
de
= >.E + hEN,
d 2N
2 2 12 =pN +qN +rE +A~+B,
(8)
d~
where E, N and ~ are the same as the former, and all the other remaining quantities are free parameters. For solitary wave solutions to (8), we impose on E and N such boundary conditions as N ---* G2 , E ---* (9) { E', E", N', Nil ---* 0, as I~I ---* 00,
c..
where Gl , G2 are arbitrary constants. It is to be noted that A = 0 in (8), but the integration constant B is not essential to be zero in order to obtain solitary wave solutions under the localized boundary conditions (9). Thus, the system of equations to be discussed has seven free parameters. By inspection, it follows that the equations are invariant under the transformations (i) ~ ---* -~, (ii) E ---* -E, and (iii) ~ ---* ~ + G, where G is a constant. In order to obtain exact solitary wave solutions to (8), we consider the coupled Riccati equations u' = -kuv, v' = k(l - v 2 ) .
(10)
It can be easily verified that (10) has the following general solutions u(~)
= ±sech[k(~ + ~o)], v(~) = tanh[k(~ + ~o)],
(11)
where k is an arbitrary nonzero constant and ~o is an arbitrary one. Solutions (11) satisfy the relation V2(~) = 1 - u2(~). The next crucial step is that the solution we are looking for is expressed in the form
n
N
=L i=O
Ci Ui
n
+L i=l
divU
i- l
(12) ,
456
ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
where ai, bi, c., di are constants to be determined. The parameters m, n will be found by balancing the linear terms of highest order with the nonlinear terms in (6) and (7). It is easy to show that n = 2, and m = 0,1,2 if balancing E" with EN, and Nil with E 2 , N 2 . So we may choose E {
°
= ao + al U + a2u2 + i. v + b2vu,
N = Co + CI U + C2U 2 + dl V
(13)
+ d2vu,
or d2 i 0. where C2 i Substituting (13) into (8), using (10) repeatedly, we can express all derivatives of U and v in term of series in U and v, collecting all terms with the same order of U and v, then setting the coefficients of each order of u and v to zero, it yields a set of algebraic equations which contain eighteen nonlinear algebraic equations, for parameters e, I, A, h,p, q, r, Bin 11 unknowns ai,ci, bj , dj , (i = 0, 1,2,j = 1,2) and k:
- Abl -
hb l
Co - haod l = 0,
= 0, 2qCodl = 0,
-Aao - haoCo - hbldl -2raobl - pd, -
- B - ra5 - rbi - pco - qC5 - qdi = 0, -Ab 2 + f3k 2b2 - hb2Co - hblCI - haldl - haad 2 = 0, -2ek 2bl - hb2CI - hblC2 - ha2dl - ha ld2 = 0, -6ek 2b2 - hb2C2 - ha2d2 = 0, -Aal
+ f3k2al - halCo - haOCI - hb2dl - hbld2 = 0,
-2ek2al - ha2cI - baicz -Aa2
+ 4ek2a2 -
ha2Co -
+ hb2dl + bbida = 0, halCI - haOC2 + hbldl
(14)
- hb2d2 = 0,
-6ek2a2 - ha2c2 + hb2d2 = 0,
-2ra lb l - 2ra ob2 - 2qcldl - pd 2 + k 2fd 2 - 2qcod2 = 0, -2(ra2bl + ra lb 2 + k 2fd l + qC2dl + qcld2) = 0, -2(ra2b2 + 3k 2fd 2 + qC2d2) = 0, -2raOal - 2rblb2 - PCI + k 2 fCI - 2qCoCI - 2qdld2 = 0, -2(rala2 - rb lb2 + k 2 fCI + qCIC2 - qdld2) = 0, -rai - 2raOa2 + rbi - rb~ - qcr - PC2 + 4k 2fC2 - 2qcaC2 -ra~ + rb~ - 6k 2fC2 - qc~
+ qdr -
qd~ = 0,
+ qd~ = 0.
The only task in the following is to find all the constants and unknown coefficients such that (13) actually satisfy (8).
3
Exact Analytical Solutions
With the aid of symbolic computation system Maple, and in the interactive mode of Maple, we have found five non-trivial solutions of system (8), for which, if we choose some specific parameter values, such as B = 0, we can cover all the previous known solitary wave solutions
No.4
Yao & Li: EXACT EXPLICIT SOLUTIONS OF NONLINEAR SCHRODINGER EQUATION 457
found by other more sophisticated methods. More importantly, all solutions found by our method are more general types of solutions, which, to our knowledge, have not been reported in the literature. Let us list them as follows. For f = two types of solutions are obtained:
fk,
EI W = {
N I (0
±3~ J6
(2k
2eq
- 6 + 3ph)e r>..q
sech[k(~ - ~o)],
k 2e - >.. k 2e 2 = h - 2 sech [k(~ - ~o)],
(15)
h
where k 2 = _ph - 2>..q ±2Z!p2 - 4Bq and ~o is an integration constant. In order that the solutions are localized, it is necessary that the parameters satisfy the inequalities:
ph - 2>..q ± hJp2 - 4Bq 0 (2k 2eq - 6>..q + 3ph)e =------=----'--=------=- < , > O. eq r E2(~) =
{
N2(~)
Xph. - h 2B _ >..2 q h 2r tanh[k(~ - ~o)],
± >..
2k 2 e
= -h - T
(16)
2
sech [k(~ - ~o)],
13ph - 6>..q ± J3J4>..2 q2 eq - 4>..phq + 3p2h2 - Sh 2qB , lik h k2 = g were l ethe e rirevi previous case, th e various parameters should satisfy the inequalities >..ph - h 2B - >.. 2 q > 0 3ph - 6>..q ± J3/r4>"-:-;2~q~2---4:-:'>..-ph:-q-+-3p2---;:-:"h-;:"2-_-S:-:-h-;:"2qB=- > 0 r ' eq . For B = (>"f - pe)(hpf - >"qf - pqe) another two type solutions are' (hf - 2qe)2 , . E (C) 3 <"
{
= ±~2 (hf (hp - 2>..q)e _ 2qe)h
J
hf - qe h 2[k(C _ C )] er sec <" <,,0,
pe - >"f 3 e(2)..q - hp) 2 N3(~) = hf _ 2qe + "2 (hf _ 2qe)h sech [k(~ - ~o)],
1 h'7- 2>..q . h hi - qe k 2 = 41L h were _ 2eq' WIt er
0 2>..q - h 0 >, 2qe - hy > .
(17)
458
ACTA MATHEMATICA SCIENTIA
Vol.23 Ser.B
- 2>..] . h hp - 2>..] 0 qe - hI 0 h k 2 = 21 hp were 2qe _ h' WIt 2qe - h >, re >. Obviously, while the solutions E(~) given by (16) and (19) are antisymmetric with respect to ~ = ~o, whereas the solutions N(~) always have a symmetric structure. Also we can see, on the contrary, that the structure of the rest three sets of solutions E(~) and N(~) all have a symmetric structure with respect to ~ = ~o. Although the above five sets of solutions possess the known functional forms, they are general with wide parameter ranges. In the following, we consider the specific case for which B = 0 in (8), which is often encountered and also worthwhile in the study of envelope waves in plasmas. In this case, (8) reduces to d2E e --2 = >..E + hEN, d~
(20)
2N
d 2 2 { 1 2 =pN +qN +rE . d~
For (20), Rao[2] studied them and obtained some solitary wave solutions under the following boundary conditions: dE dN (21) as I~I -+ 00 E,N, d~' df -+ 0 and Takao[6] did under such conditions as N
dE dN
-+ C 3,E, d[' df -+ 0
as I~I-+
00,
(22)
where C3 is an arbitrary constant. But by inspection, we find that, to obtain solitary wave solutions, E, N are not necessary to be zero, so we set the boundary conditions as (9). Under such boundary conditions and for the case B = 0, from (15),(17),(18) and (19) we can obtain two sets of solutions respectively, but from (16) only one set of solutions is obtained. Thus, nine non-trivial solutions are delivered in all, in which, in addition to covering all those found by [2,4,7]' the others are new and first reported here. For the sake of conciseness we only list two new and more general sets of them. (i). For qe - 31h = 0,
E21(~) = >..(ePe~r3>"f) tanh[k(~ - ~o)], {
where
N 21W =
>..
2k 2 e
-it - T
2
(23)
sech [k(~ - ~o)],
k 2 = -6>"1 + ep ± V12>..2 J2 8el
-
4>..pel
+ p2e2
(24)
and for localized solutions the parameters should satisfy the following inequalities ep - 6>"1 0 hr 0 >.. 0 < ' -e 1 > ' -e < ' e1
or
ep - 3Af 0 hr 0 >.. 0 < ' -e 1 < ' -e < · e1
Solution (23) is new and more general than that ofPanigrahy et al[4l. It is valid for qe-3lh = 0, whereas the corresponding one in [4] is only valid for qe - 31h = 0 and pe - >"1 = O. In fact, if
Yao & Li: EXACT EXPLICIT SOLUTIONS OF NONLINEAR SCHRODINGER EQUATION 459
No.4
we further set the restriction pe - >"1 = 0, we can get two sets of solutions from (23), in which one is obtained by Panigrahyl''I, however, the other is completely new and reads
E(E)
= ±~J-2~~ tanh[~J -y(E + Eo)],
{ N(E) = '1 ') (1
D
2~ [-2 + sech2[~A(E + Eo)]].
2 >..q(3qe - hf) h(5qe - 2hf) ,
-
ror p -
E5I(E) =
±6k~e. Jqe ~r hi sech[k(E -
3e { N (0 = 2k 2 (h 5I where k
=
(25)
I
- q) -
Eo)]tanh[k(E - Eo)],
6k 2 e 2 -h- sech [k(E - ~o)],
(26)
V
2hf >..~ 5qe' with 2hf >..~ 5qe < 0, qe;rhi > O. This solution is new and obtained
from (19). In the following section, we apply the above results to the original coupled SchrodingerBoussinesq equations.
4
Traveling Wave Solutions to Coupled S-B Equations Equations (1) and (2) are in the standard form of (8). Using the results in Section 3, their
exact localized solutions satisfying the boundary conditions (9) can easily be obtained. For D oa 2 + 6f-lWHo(P = 0, (1) and (2) have 2 sets of solutions
-2k 2D o(32
+ 6>"(32 - DoQ . sech[k(x - Mt)] exp[I(rx + 8t)], f-lWHo k 2Do - >.. k 2Do 2 { N(x, t) = 2 - --sech [k(x - Mt)], f-lWH o f-lWHo E(x, t)
= ±-k 'T/
(27)
2 _ 6>"(32 - DoQ ± JD o(DoQ2 - 24f-lWHg(32B). 6D , o(32
h k were
= ± V2
->"DoQ + 3>..2(32 + ;f-lWHoDoB tanh[k(x _ Mt)] exp[i(rx + 8t)], DOf-lWHo'T/ 2 1 x k Do { N(x, t) = - - - - - --sech 2 [k(x - Mt)], 2 f-lWH o f-lWHo E(x, t)
2
(28)
12>..2(34 - 4>"D o(32 + D5Q 2 - 16f-lwHoBDo(32 B D o(3 Except the above 2 sets of solutions, (1) and (2) admit another 3 sets of solutions for which
where k 2 = 1
DoQ - 6>"(32 ±
all parameters are free:
2
±~ (>..a + f-lWHoQ)J -D02(P + f-lWHo(32) sech 2[k(x
_ Mt)] exp[i(rx + 8t)], 'T/f-lWHoP { N(x, t) = _ >"(32 - DoQ _ ~ Do(>..2 a2 + f-lWHoQ) sech 2[k(x _ Mt)]. 2P 4 f-lWHoP
E(x, t) =
4
460
ACTA MATHEMATICA SCIENTIA
(29)
where k 2 = _ >. + ~pHoQ;
.
E(x, t) = ±3
k2VD (a. 2D 0
0
+ 2J.Lw -
H (J2) 0
J.LWHo'TJ
tanh[k(x - Mt)]sech[k(x - Mt)] exp[i(TX + 6t)],
k 2D o 2[k(x { N(x, t) = -3--sech - Mt)], • J.LWHo 2_ _ J.LWHoQ + >'0'.2 h k were 2( 2D (32) . a. 0 + J.LWHo For the sake of conciseness, we denote
the above five solutions,
5
T
Vol.23 Ser.B
= M
Do V
g
(30)
0'.2
Do + J.LWHo(32 as P and M 2 - V~ as Q. In all of
and 6, M, B are arbitrary constants.
Summary
By using a simple transformation we derive the coupled non-linear ordinary differential equations from coupled Schr6dinger-Boussinesq equations, then we obtain five important and general solitary wave solutions, for which, if choosing specific parameters, it not only produces the same solutions as originally obtained by [2,4] and [7] but also can pick up what we believe to be new solutions missed by other authors. For all solutions we verify them by putting them back into (8). Hence, the approach used here shows some novel aspects. Firstly, it is an effective and systematic method in solving nonlinear differential equations. Secondly, it can serve a more general recipe for finding traveling wave solutions to the great majority equations. Finally, it only needs simple differential computation and solving a set of lower algebraic equations because the order of partial differential equations is sufficiently reduced and Riccati equations are used. More importantly, it is a computerizable method, which allows us to perform complicated and tedious algebraic calculation on computer. References 1 Rao N N. Near-magnetosonic envelope upper-hybrid waves. J Plasma Phys, 1988, 39: 385-405 2 Rao N N. Exact solutions of coupled scalar field equations. J Phys A: Math Gen, 1989, 22: 4813-4825 3 Wang X Y, Xu B C, Taylor P L. Exact soliton solutions for a class of coupled field equations. Phys Lett, 1993, A173: 30-32 4 Panigrahy M. Dash P C. Soliton solutions of a coupled field using the mixing exponential method. Phys Lett, 1999, A261: 284-288 5 Zhang G X, Li Z B, Duan Y S. Exact solitary wave solutions to nonlinear wave equation. Science in China, 2000, 30A: 1103-1108 (in Chinese) 6 Yoshinaga T, Kakutani T. Solitary and E-shock waves in a resonant system between long and short waves. J Phys Soc Japan, 1994, 63: 445-459 7 Yoko Hase, Junkichi Satsuma. An n-soliton solution for the nonlinear Schrodinger equation coupled to the Boussinesq equation. J Phys Soc Japan, 1988, 57: 679-682