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Two new classes of exact solutions for the KdV equation via Bäcklund transformations
Two New Classes of Exact Solutions for the KdV Equation via Bgicklund Transformations-/A. H. KHATERS, Department
of Mathematics,
Faculty
0. H. EL-KALAAWY of Science,
Cairo
Liniversity.
Beni-Suef,
Egypt
and
M. A. HELAL Department
of Mathematics,
Faculty (Accepted
of Science, 3 April
Cairo
University.
Giza.
Egypt
1997)
Abstract-The
Korteweg-de Vries equation which includes nonlinear and dispersive terms quadratic in the wave amplitude is considered. The exact solutions can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system: that is. a linear eigenvaluc problem in the form of a system of first order partial differential equations. The method of characteristics is used and Backlund transformations (BTs) arc employed to generate two new solutions from the old one. 0 1997 Elscvier Science Ltd
1. INTRODUCTION
The problem of wave propagation may be considered as the one which stands at the frontier of nonlinear domains in both mathematical and physical sciences (e.g. fluid dynamics, plasma physics, solid-state physics, optics, etc.). Indeed, it was the first to be observed, analysed and termed as solitary wave or soliton [l, 21. During the last three decades or so an exciting and extremely active area of research has been devoted to constructing exact solutions for a wide class of nonlinear equations. This includes the most famous nonlinear equation, the Korteweg-de Vries (KdV) equation. Indeed, various branches of mathematics as well as physics have been developed, renewed or even begun to meet the needs of the nonlinear world. Certain important and novel subareas of research lead to the construction of exact solutions, such as the applications of the Jacobian elliptic function [3], the powerful inverse scattering transform (IST) [4], the Backlund transformations (BTs) [5], the Painleve analysis [&lo], the Lie group theoretic methods [ll], the direct algebraic method [12-161 and tangent hyperbolic method [17-201. In this paper, two classes of new exact analytical solution are generated by using the BT technique for the KdV equation qr + 6~7, + qvrr = 0. The outline
(1)
is as follows. First, we address the Ablowitz,
tcommunicated by Professor Wadati. fPresent address: Physics Department,
U.I.A.,
University 1901
of Antwerp.
Kaup, Newell, Segur (AKNS)
B-2610
Antwerp,
Belgium.
lYO2
A. H. KHATER ef ol
system and the BTs. Second, the derivation of the inverse method from the BTs is reported. Third, a class of exact solutions is generated from a known simple function solution. Fourth. another class of exact solutions is generated from a known travelling wave solution.
2. THE
AKNS
SYSTEM
AND
BACKLUND
TRANSFORMATIONS
Nonlinear partial differential equations integrable by the inverse scattering transform (IST) method form a wide class of soliton equations which possess many remarkable properties [4,21,22]. These soliton solutions can be obtained in different ways. The BT technique is one of the direct methods to generate a new solution of a nonhnear evolution equation (NLEE) from a known solution of that equation [5,23-261. Previously, Refs [27-311 had derived some BTs for the NLEEs of the AKNS class. These BTs explicitly express the new solutions in terms of the known solutions of the NLEEs and the corresponding wave functions which are solutions of the associated AKNS system. The AKNS system is a linear eigenvalue problem in the form of a system of first order partial differential equations. Therefore, the problem of obtaining new solutions by BTs is to obtain the wave function. The main aim of this section is not to derive BTs but rather to implement them in the construction of exact solutions for the KdV equation. It is known that many NLEEs can be derived from the following AKNS system eigenvalue problem, defined in the form a, = P@; P and Q are two 2 X 2 null-trace
a, = Q@,
where CD=
(2)
matrices: (3)
where n is a parameter, independent the integrability condition:
of x and t, while 9 and r are functions of x and t. From
P,-Q,+PQ-QP=O,
or in component
(4)
form: -A,+qC-rB=U,
(3
q, - B,v - 2qA + 2778 = 0,
(6)
r, - C, - 2qC + 2rA = 0.
(7)
A, B and C are functions of 7, q and r. A suitable choice of r, A, B and C in eqns (5)-(7) satisfy. For the KdV equation r = - 1.
Konno and Wadati [29] introduced
yields various NLEEs
which q must
the function &!?l ‘pz ’
and for any of the NLEEs
derived a BT with the following form: 4’ = 4 + W,rl),
where q is the old solution
and q’ is a new solution of the corresponding
NLEEs.
190.1
Exact solutions for KdV equations
To construct the KdV equation in the form
from the AKNS
system (2) and (5)-(7)
we take P and Q
(l(J) Q=[and, substituting
47f-227)q-q~ 4v2+ 2q
in eqns (5)-(7),
-4v2q -2w-Cl,, -2Y2 4g + 2774+ q, I
(11)
then eqn (7) gives
(12) 4, + WI, + 4rr.t = 0. In the discussion given later (P and Q) it will be found that the Riccati form of the inverse method is sometimes useful. Introducing the function I in eqn (8) eqns (2) and (3) are rewritten as
I-, =2T$+P+q, r,=2Ar+ecr*.
(13a) (13b)
The Backlund transformation originates from the transformation theory in differential geometry [32,33]. Its importance for the theory of nonlinear differential equations has been recognized. BTs for the KdV equation were obtained in Ref. [21]. Below, we write the BTs for the KdV equation: (14a)
q, + 4: = -2772 + (q - q’)“/2,
4, + 4:= 2(42, + 44: 3. DERIVATION
+ d’) - ((7- 4’)(4.,.r- 4,).
(14b)
OF THE INVERSE METHOD FROM BTs
We will show that BTs can be reduced to the fundamental equations of the inverse method. For this purpose an auxiliary function (or functions) is introduced. Then, the linearization for the auxiliary function (or functions) gives the equations of the inverse method [28]. Considering eqns (14a) and (14b), we introduce a function defined by
q'=q-2r,. Substitution
(15)
of eqn (15) into eqns (14a) and (14b) gives
r,-2rlr-r*-q=o
(16a)
and r,+2(4$+2qq
+&)I?+ -4?7?4 -2q4, -qy,-21+(4$q
+2q)r2=o.
(16b)
Using the relation (8) we obtain eqns (2) and (3). It is to be remarked that eqns (16a) and (16b) are a Riccati form of the inverse method, eqns (13a) and (13b). Thus, from eqn (15), the BTs for the KdV equation in the case of the class of the AKNS system take the form
qf=q -2r.,.
(17)
Now, choose a known solution of the KdV equation and substitute that solution into the corresponding matrices P and Q. Next, solve the system (2) for cp, and (p2. Then, by eqn (8) and the corresponding BTs give the new solution of the KdV equation. Two different choices (a simple function and a travelling wave function) for 4 are considered in the following sections.
1904
A.
4. THE KNOWN
H. KHATER
SOLUTION
CI nl.
q =q(x,r)
IS A SIMPLE
FUNCTION
In this case the system (2) (3) cannot be solved for the vector @ as a whole, but can bc solved in components cpl and (p2 separately. From eqns (2) and (3) after inserting the known solution q(x,t) of the KdV equation into the corresponding matrices P and Q, we will have the following system of PDEs for the unknowns cp, and cpz: (PI, = rlcpl + ~2,
=w -
Y(P2v
(1X)
17592,
i1w (20)
czar= AR + BP,, cp2/
= CR -
(21)
Av2.
These equations are compatible under the conditions of the assumed values of matrices P and Q connected with the considered KdV equation. Solve for q, from eqn (19). giving CPI= ; i(P2.r + Substituting
(22)
wP2).
this cp, into eqn (21), together with eqn (7), we get CP2,
- rp2, = ,ij (C, -
(23)
r,)92.
This is a linear first order PDE with cp2 as its unknown function; it can be solved by the method of characteristics. After cp2has been obtained from eqn (23), and substituting it into eqn (22) we obtain cpl. Thus we have obtained two general solutions cp, and qz, which contain an arbitrary function F. This arbitrary function can be determined by demanding that the two solutions ‘pl and (p2 satisfy either eqns (18) or (20), which will yield a second order linear ODE with the function F as its unknown. If we can solve for the function F, we will obtain the two particular solutions ‘p, and (p2. Finally, by applying eqn (8) and the BTs corresponding to the KdV equation, we obtain a new solution of the KdV equation. To be more specific. apply the last technique for the following example. 4.1.
Example
I
Let x+1 q = 3(2t + 1) ’
t # -l/2.
(24)
By direct calculation one can check that eqn (24) is a solution of the KdV equation Inserting eqn (24) into eqn (23), together with eqns (10) and (11) gives 4$ + Equation
(25) has the following
(12).
cpZ.I+ (P2r= j3(2t1+ J’F’.
system of ODES as its characteristic
de -zz dt
(~2
3(2t + 1)
equations:
(27)
Exact solutions for KdV equations
1905
Solving these two equations gives the general solution of the unknown (p2 in eqn (25), which reads (p* = (2t+ 1))“6F([); 5 = (x + 1)(2t + 1))“” - 3$(2t where F is an arbitrary function. Substituting general solution of cp,, which reads cp, = -(2t + 1)-“6F’(<)
(3%
+ l)*“,
eqns (24) and (28) into eqn (22) gives the + 77(2f+ l)““F([).
(29)
To determine the function F(t), we substitute from eqns (24), (28) and (29) into eqn (18), then F(t) must satisfy the following Airy equation [34]: (30) Therefore
we obtain the function F(t) as follows: F = c,Ai([)
+ c2Bi([),
where Ai(Q and Si(t) are two Airy functions, while cl and c2 are two arbitrary constants. After F(e) has been determined, eqns (28) and (29) lead to r = -(2t + 1>- ,,3 d(log F) d5-vj then substituting this I’ and eqn (24) into the BTs (17) yields the new solution 9’ of the KdV equation (12) corresponding to the known solution (24): x+1 “=3(2t+
1)
+ 2(2t + l))*‘”
d*(log F) dt2 ’
5. THE KNOWN SOLUTION IS A TRAVELLING
In this case we suppose that the components q = q(p);
then the components also functions of p:
r = r(p),
A, B and C of the matrix B = B(p)
A =A@);
Under these assumptions, solution. The quantity
the following
(32)
WAVE
q and r of the matrix P are functions of p:
where
p =x - kt,
Q determined and
(33)
by eqns (5), (6) and (7) are
C= C(p).
(34) result holds, which is crucial in the subsequent exact
/3 = (A + kq)2 + (B + kv)(C + kr)
(35)
is constant with respect to p (or x and t). dt dx W, -=-= --r C CC -rdcp2 Using eqns (33) and (34) and substituting into eqn (36) gives dt T=
dp
2dv,
(C + kr) = (C + kr);cp, ’
(37)
1906
A. H. KHATER
These equations yield the following
system of ODES: d(ln 6
= (C + w; 2(C + kr) J
&
-(C
(38)
+ kr) r .
dt=
Integrating
et al.
(39)
eqn (38) leads to p2 = k2(C + kr)“‘,
where k2 is an integration
constant. Integrating -t+k,
where k, is another integration
=
eqn (39) we get
r dp I
(41)
(C + kr) ’
constant. Denoting LT(p) =
and substituting
(40)
rdp I (C + kr)
(42)
eqn (42) into eqn (41), we have u(p)+t=k,.
(43)
From eqns (40) and (43) we obtain the general solution of eqn (23): cp2= (C + kr)“2F([),
(44)
5 = U(P) + t
(45) eqn (44) into eqn (42) gives the
where and F(t) is a differentiable general solution for cpI:
function
of 5. Substituting
qq = (C + kr)-“‘(F,’
+ (A + kq)F).
(46)
For the determination of the function F, eqns (44) and (46) are substituted and we find that F must satisfy the following second order ODE:
into eqn (lg),
FCC’- PF = 0, where /3 is a constant defined in eqn (35). According following three different solutions: F =c,{-t-cz, F = cl sinh ~(6 + cJ, F = c, sin w([ + c2),
(47)
to the sign of /3, eqn (43) will have the
when p =0,
(48)
when p > 0, w* = p,
(49)
when p
(50) where cl and c2 are integration constants. Substituting these solutions into eqns (46) and (44) respectively, we obtain the corresponding different solutions of the system (2), (3): (C + kr)-“2[(A + kv)(c,l+ c2) + cl] (C + kr)“*(c] 5 + c2) ’
1
c,(C + kr)-“2[(A c,(C + kr)-“2[(A
(51 a)
when ’ = OS
+ kq) sinh ~(5 + c2) + w cash ~(5 + cJ] when j.3 ~0, c,(C + kr)“‘sinh ~(5 + c2) I7 + kq) sin ~(5 + cz) + w cos ~(5 + c2)] 1, c,(C + kr)“‘sin w([ + c2)
when /3<0.
(51b) (SIC)
Exact solutions for KdV equations
1007
These results (eqns (51a), (51b) and (51~)) are valid for any NLEE contained in the AKNS system (2), (3), provided that they satisfy assumptions (33) and (34). Now, we apply the results obtained here and the known travelling wave solution of the KdV equation obtained previously to construct a new solution of the corresponding KdV equation by means of the BTs. The constant p defined by eqn (35) is zero and therefore the corresponding solution of the AKNS system (2), (3) is (51a-c). By substituting eqn (51a) into eqn (8) we get the common expression of I’ for this KdV equation:
c=c,/cz. This will be illustrated 5. I.
in the following example.
Example 2
Let 4 = 271set h2[17pl, where p = x - kt and k = 477’. By direct calculation we can check that eqn (53) is a solution Substituting eqn (53) into eqn (35), we obtain
of the KdV equation
(53) (1).
/5’= (A + kv)’ + (B + kq)(C + kr)
= [( -47$ - 2nq - 4, + 47172 - (47724 + 2qq., + k
+ 2$)(4-r72 + 29 - 471%
= [2712q2 + (qJ2 - 2qqr.3 - 497 = [ 16v’sec h’qp + 16r)‘sec h’vp - 16qh set h”qp - 32q’sec h4qp + 4877’sech’vp - 32q’sec h’r]p].
Substituting
eqn (53) into the relation
(45), we get
+k+,
C + kr
=
I
= L& Substituting
-dp +t
4q2 set h’(vp)
pTp + sinh (~vP) - lfiqJr)
in the relation (52), I‘=
cash’ vp 4712 = ~(-1
-4~’
1
set h*vp( -1 + tanh qp) + (~ + c)
+ tanh qp) -
I
417 cash’ qp (217~+ sinh 277~- 16n’(r + c))
and r, = q2 set h2qp -
8q2 cash vp sinh vp S$ cash’ vp( 1 + cash 2qp) (277~ + sinh 2qp - 16n3(t + c)) + (2r)p + sinh 2qp - 16$(t + c))’ ’
(54)
1008
A.
Then, substituting
from the relations
H. KHATER
(53) and (54) into eqn (17), we obtain
8q’sinh 271~ ’ = (2r]p + sinh 2vp - 16$(t
32q2 cosh4 qp + c)) - (211~ + sinh 2vp - 16v”(t + c))’
The solution (55) is displayed graphically
2
4
et al.
in Figs l(a) and (b).
6
x
6
x
X
Fig.
1. The
soliton
solutions
of the KdV
(55)
equation times
10
I?
1
(55) with (a) t) = 1 and c = -150: I = 0.0 and t = 1.0.
(b)
77 = 1, c = -150
and the
Exact solutions for KdV equations
I YOY
Acknowledgemenls-It is a pleasure to thank Professors D. Callebaut and W. Malfliet (University of Antwerp. Belgium) for critical comments on this work.
REFERENCES 1. Dodd, R. K., Eilbeck, J. C., Gibbon, J. D. and Morris. H. C., Solitons and Nonlinear Waves Equntion. Academic Press, New York, 1982. 2. Whitham, G. B., Linear and Nonlinear Waves. Wiley, New York, 1974. 3. Lamb, G. L., Elements of Soliton Theory. Wiley, New York, 1980. 4. Ablowitz, M. J. and Clarkson, P. A., So/irons, Nonlinear Evolution Equations and lnuer,se Scrrttering. Cambridge University Press, Cambridge, 1991. 5. Rogers, C. and Shadwick. W. E.. Biicklund Transformations and their Applicntions. Academic Press. New York. 1482. 6. Wiess, J., Tabor, M. and Garnevale, G., J. Math. Phys., 1983, 24, 522. 7. Khater, A. H., Astrophys. Space SC;.. 1989, 162, 151. 8. Khater. A. H.. El-Sabbagh, M. F. and Callebaut. D. K., Annul. Sot. Sci. de Bruxelles, 1987, TlOlIII, 99. 9. Khater, A. H., El-attary, M. A., El-Sabbagh, M. F. and Callebaut, D. K., Astrophys. Space SC;., 1988, 149, 217. IO. Khater, A. H.. El-Sabbagh. M. F. and Callebaut, D. K., Comput. Math. Applic., 1988. 17, 1379. Il. Olver. P. J., Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Vol. 107, Springer-Verlag, Berlin, 1986. 12. Hereman, W., Korpel, A. and Banerjee, P. P., Wuue Motion, 1985. 7, 283. 13. Hereman. W.. Banerjee. P.P., Korpel, A., Assanto, G., Van Immerzeel, A. and Mee-poel. A., J. Phys. A. 1986, 19,607. 14. Hereman. W., In Proceedings of Conference on Applied Mathematics, Cairo, Egypt. 1987. p. 291. IS. Hereman. W.. Banerjee, P. P. and Chatterjee, M. R., J. Phys. A, 1989, 22, 241. 16. Hereman, W. and Takaoka. M., J. Phys. A. 1990,23,4805. 17. Malfliet, W.. J. Phys. A. 1980, 13,2929. 18. Malfliet, W., J. Phys. A, 1991. 24, 5499. 19. Malfliet, W., Am. 1. Phys.. 1992, 60, 650. 20. Malfliet, W., J. fhys. A, 1993, 26, L.723. 21. Wahlquist, H. D. and Estabrook, F. B., Phys. Rev. Lett., 1973, 31, 1386. 22. Zakharov, V. E. and Shabat, A. B., Soviet Phys. JETP, 1972, 34, 1403. 23. Wadati, M., J. fhys. Sot. Jpn., 1974, 36, 1498. 24. Lamb, G. L.. J. Math. fhys., 1974, 15, 21.57. 25. Levi, 0.. Rognisco. D. and Sym. A., Lett. Nuovo Cim.. 1982, 33,401. 26. Porsezian, K., J. Phys. A, 1991, 24, 337. 27. Chan, W. L. and Zheng, Y. K., J. Austral. Maih. Sot. Ser. B, 1989. 30, 313. 28. Wadati. M., Sunuki, H. and Konno, K., Progr. Theor. Phys.. 1975,53,419. 29. Konno, K. and Wadati, M., Progr. Theoret. fhys., 1975. 53, 1652. 30. Khater. A. H.. Shamardan, A. B., Callebaut, D. K. and Ibrahim. R. S.. /MA J. Appl. Math.. 1997, 58, 51. 31. Khater. A. H. and Ibrahim. R. S., submitted to J. Marh. Annl. & Applic. 32. Eisenhart, L.. A Treatise on the Differential Geometry of Curues and Surfaces. Dover, New York, 1960. 33. Forsyth. A. R., Theory of Differential Equarions, Vol. 6. Dover, New York, 1959. 34. Airy, G. B.. Tides and waves. In Encyclopedia Metropolotana, Vol. 5. London, 1845. p. 241.
of Mathematics,
Faculty
0. H. EL-KALAAWY of Science,
Cairo
Liniversity.
Beni-Suef,
Egypt
and
M. A. HELAL Department
of Mathematics,
Faculty (Accepted
of Science, 3 April
Cairo
University.
Giza.
Egypt
1997)
Abstract-The
Korteweg-de Vries equation which includes nonlinear and dispersive terms quadratic in the wave amplitude is considered. The exact solutions can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system: that is. a linear eigenvaluc problem in the form of a system of first order partial differential equations. The method of characteristics is used and Backlund transformations (BTs) arc employed to generate two new solutions from the old one. 0 1997 Elscvier Science Ltd
1. INTRODUCTION
The problem of wave propagation may be considered as the one which stands at the frontier of nonlinear domains in both mathematical and physical sciences (e.g. fluid dynamics, plasma physics, solid-state physics, optics, etc.). Indeed, it was the first to be observed, analysed and termed as solitary wave or soliton [l, 21. During the last three decades or so an exciting and extremely active area of research has been devoted to constructing exact solutions for a wide class of nonlinear equations. This includes the most famous nonlinear equation, the Korteweg-de Vries (KdV) equation. Indeed, various branches of mathematics as well as physics have been developed, renewed or even begun to meet the needs of the nonlinear world. Certain important and novel subareas of research lead to the construction of exact solutions, such as the applications of the Jacobian elliptic function [3], the powerful inverse scattering transform (IST) [4], the Backlund transformations (BTs) [5], the Painleve analysis [&lo], the Lie group theoretic methods [ll], the direct algebraic method [12-161 and tangent hyperbolic method [17-201. In this paper, two classes of new exact analytical solution are generated by using the BT technique for the KdV equation qr + 6~7, + qvrr = 0. The outline
(1)
is as follows. First, we address the Ablowitz,
tcommunicated by Professor Wadati. fPresent address: Physics Department,
U.I.A.,
University 1901
of Antwerp.
Kaup, Newell, Segur (AKNS)
B-2610
Antwerp,
Belgium.
lYO2
A. H. KHATER ef ol
system and the BTs. Second, the derivation of the inverse method from the BTs is reported. Third, a class of exact solutions is generated from a known simple function solution. Fourth. another class of exact solutions is generated from a known travelling wave solution.
2. THE
AKNS
SYSTEM
AND
BACKLUND
TRANSFORMATIONS
Nonlinear partial differential equations integrable by the inverse scattering transform (IST) method form a wide class of soliton equations which possess many remarkable properties [4,21,22]. These soliton solutions can be obtained in different ways. The BT technique is one of the direct methods to generate a new solution of a nonhnear evolution equation (NLEE) from a known solution of that equation [5,23-261. Previously, Refs [27-311 had derived some BTs for the NLEEs of the AKNS class. These BTs explicitly express the new solutions in terms of the known solutions of the NLEEs and the corresponding wave functions which are solutions of the associated AKNS system. The AKNS system is a linear eigenvalue problem in the form of a system of first order partial differential equations. Therefore, the problem of obtaining new solutions by BTs is to obtain the wave function. The main aim of this section is not to derive BTs but rather to implement them in the construction of exact solutions for the KdV equation. It is known that many NLEEs can be derived from the following AKNS system eigenvalue problem, defined in the form a, = P@; P and Q are two 2 X 2 null-trace
a, = Q@,
where CD=
(2)
matrices: (3)
where n is a parameter, independent the integrability condition:
of x and t, while 9 and r are functions of x and t. From
P,-Q,+PQ-QP=O,
or in component
(4)
form: -A,+qC-rB=U,
(3
q, - B,v - 2qA + 2778 = 0,
(6)
r, - C, - 2qC + 2rA = 0.
(7)
A, B and C are functions of 7, q and r. A suitable choice of r, A, B and C in eqns (5)-(7) satisfy. For the KdV equation r = - 1.
Konno and Wadati [29] introduced
yields various NLEEs
which q must
the function &!?l ‘pz ’
and for any of the NLEEs
derived a BT with the following form: 4’ = 4 + W,rl),
where q is the old solution
and q’ is a new solution of the corresponding
NLEEs.
190.1
Exact solutions for KdV equations
To construct the KdV equation in the form
from the AKNS
system (2) and (5)-(7)
we take P and Q
(l(J) Q=[and, substituting
47f-227)q-q~ 4v2+ 2q
in eqns (5)-(7),
-4v2q -2w-Cl,, -2Y2 4g + 2774+ q, I
(11)
then eqn (7) gives
(12) 4, + WI, + 4rr.t = 0. In the discussion given later (P and Q) it will be found that the Riccati form of the inverse method is sometimes useful. Introducing the function I in eqn (8) eqns (2) and (3) are rewritten as
I-, =2T$+P+q, r,=2Ar+ecr*.
(13a) (13b)
The Backlund transformation originates from the transformation theory in differential geometry [32,33]. Its importance for the theory of nonlinear differential equations has been recognized. BTs for the KdV equation were obtained in Ref. [21]. Below, we write the BTs for the KdV equation: (14a)
q, + 4: = -2772 + (q - q’)“/2,
4, + 4:= 2(42, + 44: 3. DERIVATION
+ d’) - ((7- 4’)(4.,.r- 4,).
(14b)
OF THE INVERSE METHOD FROM BTs
We will show that BTs can be reduced to the fundamental equations of the inverse method. For this purpose an auxiliary function (or functions) is introduced. Then, the linearization for the auxiliary function (or functions) gives the equations of the inverse method [28]. Considering eqns (14a) and (14b), we introduce a function defined by
q'=q-2r,. Substitution
(15)
of eqn (15) into eqns (14a) and (14b) gives
r,-2rlr-r*-q=o
(16a)
and r,+2(4$+2qq
+&)I?+ -4?7?4 -2q4, -qy,-21+(4$q
+2q)r2=o.
(16b)
Using the relation (8) we obtain eqns (2) and (3). It is to be remarked that eqns (16a) and (16b) are a Riccati form of the inverse method, eqns (13a) and (13b). Thus, from eqn (15), the BTs for the KdV equation in the case of the class of the AKNS system take the form
qf=q -2r.,.
(17)
Now, choose a known solution of the KdV equation and substitute that solution into the corresponding matrices P and Q. Next, solve the system (2) for cp, and (p2. Then, by eqn (8) and the corresponding BTs give the new solution of the KdV equation. Two different choices (a simple function and a travelling wave function) for 4 are considered in the following sections.
1904
A.
4. THE KNOWN
H. KHATER
SOLUTION
CI nl.
q =q(x,r)
IS A SIMPLE
FUNCTION
In this case the system (2) (3) cannot be solved for the vector @ as a whole, but can bc solved in components cpl and (p2 separately. From eqns (2) and (3) after inserting the known solution q(x,t) of the KdV equation into the corresponding matrices P and Q, we will have the following system of PDEs for the unknowns cp, and cpz: (PI, = rlcpl + ~2,
=w -
Y(P2v
(1X)
17592,
i1w (20)
czar= AR + BP,, cp2/
= CR -
(21)
Av2.
These equations are compatible under the conditions of the assumed values of matrices P and Q connected with the considered KdV equation. Solve for q, from eqn (19). giving CPI= ; i(P2.r + Substituting
(22)
wP2).
this cp, into eqn (21), together with eqn (7), we get CP2,
- rp2, = ,ij (C, -
(23)
r,)92.
This is a linear first order PDE with cp2 as its unknown function; it can be solved by the method of characteristics. After cp2has been obtained from eqn (23), and substituting it into eqn (22) we obtain cpl. Thus we have obtained two general solutions cp, and qz, which contain an arbitrary function F. This arbitrary function can be determined by demanding that the two solutions ‘pl and (p2 satisfy either eqns (18) or (20), which will yield a second order linear ODE with the function F as its unknown. If we can solve for the function F, we will obtain the two particular solutions ‘p, and (p2. Finally, by applying eqn (8) and the BTs corresponding to the KdV equation, we obtain a new solution of the KdV equation. To be more specific. apply the last technique for the following example. 4.1.
Example
I
Let x+1 q = 3(2t + 1) ’
t # -l/2.
(24)
By direct calculation one can check that eqn (24) is a solution of the KdV equation Inserting eqn (24) into eqn (23), together with eqns (10) and (11) gives 4$ + Equation
(25) has the following
(12).
cpZ.I+ (P2r= j3(2t1+ J’F’.
system of ODES as its characteristic
de -zz dt
(~2
3(2t + 1)
equations:
(27)
Exact solutions for KdV equations
1905
Solving these two equations gives the general solution of the unknown (p2 in eqn (25), which reads (p* = (2t+ 1))“6F([); 5 = (x + 1)(2t + 1))“” - 3$(2t where F is an arbitrary function. Substituting general solution of cp,, which reads cp, = -(2t + 1)-“6F’(<)
(3%
+ l)*“,
eqns (24) and (28) into eqn (22) gives the + 77(2f+ l)““F([).
(29)
To determine the function F(t), we substitute from eqns (24), (28) and (29) into eqn (18), then F(t) must satisfy the following Airy equation [34]: (30) Therefore
we obtain the function F(t) as follows: F = c,Ai([)
+ c2Bi([),
where Ai(Q and Si(t) are two Airy functions, while cl and c2 are two arbitrary constants. After F(e) has been determined, eqns (28) and (29) lead to r = -(2t + 1>- ,,3 d(log F) d5-vj then substituting this I’ and eqn (24) into the BTs (17) yields the new solution 9’ of the KdV equation (12) corresponding to the known solution (24): x+1 “=3(2t+
1)
+ 2(2t + l))*‘”
d*(log F) dt2 ’
5. THE KNOWN SOLUTION IS A TRAVELLING
In this case we suppose that the components q = q(p);
then the components also functions of p:
r = r(p),
A, B and C of the matrix B = B(p)
A =A@);
Under these assumptions, solution. The quantity
the following
(32)
WAVE
q and r of the matrix P are functions of p:
where
p =x - kt,
Q determined and
(33)
by eqns (5), (6) and (7) are
C= C(p).
(34) result holds, which is crucial in the subsequent exact
/3 = (A + kq)2 + (B + kv)(C + kr)
(35)
is constant with respect to p (or x and t). dt dx W, -=-= --r C CC -rdcp2 Using eqns (33) and (34) and substituting into eqn (36) gives dt T=
dp
2dv,
(C + kr) = (C + kr);cp, ’
(37)
1906
A. H. KHATER
These equations yield the following
system of ODES: d(ln 6
= (C + w; 2(C + kr) J
&
-(C
(38)
+ kr) r .
dt=
Integrating
et al.
(39)
eqn (38) leads to p2 = k2(C + kr)“‘,
where k2 is an integration
constant. Integrating -t+k,
where k, is another integration
=
eqn (39) we get
r dp I
(41)
(C + kr) ’
constant. Denoting LT(p) =
and substituting
(40)
rdp I (C + kr)
(42)
eqn (42) into eqn (41), we have u(p)+t=k,.
(43)
From eqns (40) and (43) we obtain the general solution of eqn (23): cp2= (C + kr)“2F([),
(44)
5 = U(P) + t
(45) eqn (44) into eqn (42) gives the
where and F(t) is a differentiable general solution for cpI:
function
of 5. Substituting
qq = (C + kr)-“‘(F,’
+ (A + kq)F).
(46)
For the determination of the function F, eqns (44) and (46) are substituted and we find that F must satisfy the following second order ODE:
into eqn (lg),
FCC’- PF = 0, where /3 is a constant defined in eqn (35). According following three different solutions: F =c,{-t-cz, F = cl sinh ~(6 + cJ, F = c, sin w([ + c2),
(47)
to the sign of /3, eqn (43) will have the
when p =0,
(48)
when p > 0, w* = p,
(49)
when p
(50) where cl and c2 are integration constants. Substituting these solutions into eqns (46) and (44) respectively, we obtain the corresponding different solutions of the system (2), (3): (C + kr)-“2[(A + kv)(c,l+ c2) + cl] (C + kr)“*(c] 5 + c2) ’
1
c,(C + kr)-“2[(A c,(C + kr)-“2[(A
(51 a)
when ’ = OS
+ kq) sinh ~(5 + c2) + w cash ~(5 + cJ] when j.3 ~0, c,(C + kr)“‘sinh ~(5 + c2) I7 + kq) sin ~(5 + cz) + w cos ~(5 + c2)] 1, c,(C + kr)“‘sin w([ + c2)
when /3<0.
(51b) (SIC)
Exact solutions for KdV equations
1007
These results (eqns (51a), (51b) and (51~)) are valid for any NLEE contained in the AKNS system (2), (3), provided that they satisfy assumptions (33) and (34). Now, we apply the results obtained here and the known travelling wave solution of the KdV equation obtained previously to construct a new solution of the corresponding KdV equation by means of the BTs. The constant p defined by eqn (35) is zero and therefore the corresponding solution of the AKNS system (2), (3) is (51a-c). By substituting eqn (51a) into eqn (8) we get the common expression of I’ for this KdV equation:
c=c,/cz. This will be illustrated 5. I.
in the following example.
Example 2
Let 4 = 271set h2[17pl, where p = x - kt and k = 477’. By direct calculation we can check that eqn (53) is a solution Substituting eqn (53) into eqn (35), we obtain
of the KdV equation
(53) (1).
/5’= (A + kv)’ + (B + kq)(C + kr)
= [( -47$ - 2nq - 4, + 47172 - (47724 + 2qq., + k
+ 2$)(4-r72 + 29 - 471%
= [2712q2 + (qJ2 - 2qqr.3 - 497 = [ 16v’sec h’qp + 16r)‘sec h’vp - 16qh set h”qp - 32q’sec h4qp + 4877’sech’vp - 32q’sec h’r]p].
Substituting
eqn (53) into the relation
(45), we get
+k+,
C + kr
=
I
= L& Substituting
-dp +t
4q2 set h’(vp)
pTp + sinh (~vP) - lfiqJr)
in the relation (52), I‘=
cash’ vp 4712 = ~(-1
-4~’
1
set h*vp( -1 + tanh qp) + (~ + c)
+ tanh qp) -
I
417 cash’ qp (217~+ sinh 277~- 16n’(r + c))
and r, = q2 set h2qp -
8q2 cash vp sinh vp S$ cash’ vp( 1 + cash 2qp) (277~ + sinh 2qp - 16n3(t + c)) + (2r)p + sinh 2qp - 16$(t + c))’ ’
(54)
1008
A.
Then, substituting
from the relations
H. KHATER
(53) and (54) into eqn (17), we obtain
8q’sinh 271~ ’ = (2r]p + sinh 2vp - 16$(t
32q2 cosh4 qp + c)) - (211~ + sinh 2vp - 16v”(t + c))’
The solution (55) is displayed graphically
2
4
et al.
in Figs l(a) and (b).
6
x
6
x
X
Fig.
1. The
soliton
solutions
of the KdV
(55)
equation times
10
I?
1
(55) with (a) t) = 1 and c = -150: I = 0.0 and t = 1.0.
(b)
77 = 1, c = -150
and the
Exact solutions for KdV equations
I YOY
Acknowledgemenls-It is a pleasure to thank Professors D. Callebaut and W. Malfliet (University of Antwerp. Belgium) for critical comments on this work.
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