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Positon and positon-like solutions of the Korteweg-de Vries and Sine-Gordon equations
Chnm,
Solrtom
Pergamon
& Fractds
Vol. 5, No. 12. pp. 2229 2233, ,995 Elsevier Science Ltd Printed m Great Bntam 0960-0779/95$930 + 0.00
O%O-0779(94)EOO97-9
Positon and Positon-like Solutions of the Korteweg-de Sine-Gordon Equations M. JAWORSKI
Vries and
and J. ZAGRODZIfiSKI
Institute of Physics, Polish Academy of Sciences, Al. Lotnikow
32/46, 02-668 Warszawa, Poland
(Received 2 July 1993)
Abstract-It is shown that the positon solution, reported recently for the Korteweg-de Vries and other completely integrable equations, can be regarded as a limiting case of the well-known 2-soliton formula. The existence of integrals of motion related to singular positon solutions is also discussed.
In a series of recent papers [l, 21, Matveev introduced the concept of a positon as a new solution of the Korteweg-de Vries (KdV) equation. Subsequently, positon-like solutions were derived also for other completely integrable systems, e.g. Sine-Gordon (SG) and modified KdV equation [3,4]. For the KdV equation ut - 6uu, + u,
= 0
(1)
the canonical form of the one-positon solution is given by [l] u = -2aZln(sin26
- 2xy) =
322 sin i?(sin 6 - xy cos 8) ’ (sin26 - 2~y)~
(2)
where 6 = x(x + Xl + 43Zt), y = x + y, + 12&, and xi, yl denote arbitrary constants. Positons exhibit many interesting properties which differ essentially from those of solitons: (i) positons are weakly localized, in contrast to exponentially decaying soliton solutions, (ii) positon so1ut’ion is super-reflectionless, i.e. the reflection and transmission coefficients of the associated linear spectral problem are given by b = 0, a = +l, respectively, (iii) the eigenvalue of the spectral problem is positive (embedded in the continuous spectrum), (iv) positons are completely transparent to other interacting objects. In particular, two positons remain unchanged after mutual collision. On the other hand, during the soliton-positon collision the soliton remains unchanged, while both the carrier-wave of the positon and its envelope experience finite phase-shifts. The results of references [l-4] have been derived using the method of Darboux transformation and the positons presented as new fundamental solutions of the KdV and other completely integrable equations. 2229
2230
M. JAWORSKI
and J. ZAGRODZIfiSKI
The aim of this paper is twofold. First, we show that the positons belong, in fact, to the ‘Hirota family’, and can be derived from the standard 2-soliton solutions by a proper limiting procedure. Second, we discuss the existence of some integrals of motion related to singular positon-like solutions. The 2-soliton solution of the KdV equation can be written as [5,6] 24= -2azlnr.,
(3)
where
=&+&
r = 1 + eE1+ e52+ E = -2k,x
’
+ 8k3,t + E’,“‘,
m = 1,2,
and k,, E$’ are free parameters. Substituting k, = a + ib, k2 = a - ib into equation (3) we obtain the breather-like solution [7] which can be roughly described as an oscillating wave-packet modulated by a singular soliton-like envelope. Moreover, choosing
and denoting y = x + 4(3b= - a’)t, 6 = b[x + 4(b2 - 3a=)t]
we obtain immediately
1
2a - 2”Ysin2fi = -- 2a e-2ay r 2z 1 - ep40Y- -e sin26 - -&-(1 - e-4ny) , b
b 1
(4)
where the multiplicative factor -2a/b as a constant does not affect the solution (3) and can be neglected. Taking the limit a + 0 and replacing b by x to make the notation consistent with Ref. [l] we find r = sin26 - 2xy,
(5)
where y = x + 12yt, 6 = x(x + 429. Substituting (5) into Matveev’s result [ 11. The eigenvalues of hence in the limit a + continuous spectrum. The same limiting Indeed, denoting
(3) we obtain finally the positon solution (2) in agreement with the associated linear problem are given by A, = - ki (m = 1, 2), 0 we obtain Am= b2 > 0, i.e. a positive bound state embedded in the procedure can be applied to the case k, = a + ib, k2 = -a + ib. /3 = x - 4(3a= - b=)t, 5; = a[x - 4(a2 - 3b=)t]
and taking g(O) m = In (b/a) T in/2
2231
Positon solutions of KdV and SG equations
we find t=l-ee-
i4bB+ i(2b/u)e-‘*bS sinh25 = i(26/u)[e-‘2bfi’sinh2~ + (a/i2b)(l
- e-abp)]. (6)
Taking the limit b + 0 and replacing a by
we obtain
K
r = sinh25; +
~KP,
(7)
where c =
K(X
-
4K2t),
p = x - 12l?t, and the proportionality constant i2b/u has been neglected, as before. The expression (7), when substituted into (3), yields a negaton solution reported also in [2]. In contrast to the positon case, the negaton solution decays exponentially for x + km, and the eigenvalue of the associated spectral problem is negative: A, = - ki = -a* < 0, as b + 0. It is clear from (3), (5) and (7) that both positon and negaton solutions are singular. As a result, the ‘mass’ defined as the principal value [2] m m=P u(x, t) dx (8) I -cc is infinite in both cases because of the second order pole in the solution (3). In order to avoid this difficulty, we suggest here another definition by assuming the space-variable to be x + i& and next taking the limit E+ 0 [7]: m
E = lim u(x + i.5, t)dx. E--t0 I -m In contrast to the previous definition, now the mass is finite. Moreover, we find E = 0 for the positon and 5i= -8~ for the negaton, and the difference between the two cases is clearly visible. The zero mass of the positon is consistent with the super-reflectionless property as well as with the fact that other objects interacting with positons do not undergo any phase-shifts. On the other hand, the mass of the negaton is negative and two times larger than that of a single soliton having the same spectral parameter K. Thus, the negaton can be considered as a degenerate bound state of two singular solitons in the limit when two points of the discrete spectrum coincide. A similar approach can be applied to other constants of motion as well as to eigenfunctions of the associated linear problem. In this latter case, shifting the x-axis into the complex domain makes the eigenfunction continuous and removes any ambiguity related to its singular behaviour for x real. Moreover, the normalization integral becomes finite when taken as /@ dx rather than /]r/~l’dx [7]. As the next example, let us consider briefly a positon-like solution of the SG equation in the light-cone coordinates: As the next example, let us consider briefly a positon-like solution of the SG equation in the light-cone coordinates: a,, = 4sinQ.
(10)
It is convenient to write the 2-soliton solution of equation (10) in the form [8,9] @ = 2i In r-/r+,
(11)
2232
M. JAWORSKI and J. ZAGRODZINSKI
where
and A,,,, g’,“’ are free parameters. For Amimaginary (A, = ipm, ,LQ> &) we assume E(O) = In- p1+l4 m
.lr +. l--,
2
h-P2
hence denoting xm = P?nZ- thm we obtain t+
= 1=
ew1+x2)
2iei(Xl+X2)
-
Yl + + -@
!J2
Pl
-
P2
+
x2)
-
sin
(xl
2i,r1
_
e2ixz >
+
1
Pl --
+
I4
Pl
-
P2
sin
(xl
-
1
x2) .
(12)
In the limit when h + pl, since also x2 + xl, we find t-+ = -2ie2ix1(sin231, + 2r,),
(13)
where r1
=
PlZ
+
t/k
Both xl and rl are real, thus substituting (13) into (11) we obtain a purely imaginary positon-like solution of the SG equation (10) in full agreement with reference [3]: sin2g - 2rr (14) sin231, 4%’ Summarizing, in this paper we show that the positon-like (and related) solutions of the KdV and SG equations can be derived in the framework of Hirota formalism by taking a proper limit of the standard 2-soliton formula. Moreover, owing to a new definition we obtain relevant integrals of motion which remain finite for singular solutions. It is clear that the limiting procedures discussed here are not restricted to the above examples, but can be easily extended to other completely integrable systems. @ = -2iln
Acknowledgements-The authors wish to thank Prof. R. Hirota for helpful and stimulating comments. This work was supported by grant No. PB 2-0480-91-01.
REFERENCES 1. V. B. Matveev, Generalized Wronskian formulas for the solution of the Korteweg-de Vries equation: first applications, Whys.Letr. A166, 205 (1992). 2. V. B. Matveev, Theory of positons, MPI Metallforschung, Stuttgart, Preprint (1992). 3. R. Beutler, Positon solutions of the Enneper (sine-Gordon) equation, MPI Metallforschung, Stuttgart, Preprint (1992). 4. A. Stahlhofen, Positons of the modified KdV equation, Annalen der Physik 1, 554 (1992). 5. R. Hirota, “Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett. 27, 1192 (1971).
Positon solutions of KdV and SG equations
2233
6. C. S. Gardner, .I. M. Greene, M. D. Kruskal and R. M. Miura, The Korteweg-de Vries equation and generalizations, Comm. Pure Appl. Math. 27, 97 (1974). 7. M. Jaworski, A note on singular solutions of the Korteweg-de Vries equation, Phys. Left. AlOO, 321 (1984). 8. M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, (SIAM, Philadelphia, PA (1981)). 9. J. Pelka and J. Zagrodzinski, Phenomenological electrodynamics of the Josephson junction, Physicu BlJ4, 125 (1989).
Solrtom
Pergamon
& Fractds
Vol. 5, No. 12. pp. 2229 2233, ,995 Elsevier Science Ltd Printed m Great Bntam 0960-0779/95$930 + 0.00
O%O-0779(94)EOO97-9
Positon and Positon-like Solutions of the Korteweg-de Sine-Gordon Equations M. JAWORSKI
Vries and
and J. ZAGRODZIfiSKI
Institute of Physics, Polish Academy of Sciences, Al. Lotnikow
32/46, 02-668 Warszawa, Poland
(Received 2 July 1993)
Abstract-It is shown that the positon solution, reported recently for the Korteweg-de Vries and other completely integrable equations, can be regarded as a limiting case of the well-known 2-soliton formula. The existence of integrals of motion related to singular positon solutions is also discussed.
In a series of recent papers [l, 21, Matveev introduced the concept of a positon as a new solution of the Korteweg-de Vries (KdV) equation. Subsequently, positon-like solutions were derived also for other completely integrable systems, e.g. Sine-Gordon (SG) and modified KdV equation [3,4]. For the KdV equation ut - 6uu, + u,
= 0
(1)
the canonical form of the one-positon solution is given by [l] u = -2aZln(sin26
- 2xy) =
322 sin i?(sin 6 - xy cos 8) ’ (sin26 - 2~y)~
(2)
where 6 = x(x + Xl + 43Zt), y = x + y, + 12&, and xi, yl denote arbitrary constants. Positons exhibit many interesting properties which differ essentially from those of solitons: (i) positons are weakly localized, in contrast to exponentially decaying soliton solutions, (ii) positon so1ut’ion is super-reflectionless, i.e. the reflection and transmission coefficients of the associated linear spectral problem are given by b = 0, a = +l, respectively, (iii) the eigenvalue of the spectral problem is positive (embedded in the continuous spectrum), (iv) positons are completely transparent to other interacting objects. In particular, two positons remain unchanged after mutual collision. On the other hand, during the soliton-positon collision the soliton remains unchanged, while both the carrier-wave of the positon and its envelope experience finite phase-shifts. The results of references [l-4] have been derived using the method of Darboux transformation and the positons presented as new fundamental solutions of the KdV and other completely integrable equations. 2229
2230
M. JAWORSKI
and J. ZAGRODZIfiSKI
The aim of this paper is twofold. First, we show that the positons belong, in fact, to the ‘Hirota family’, and can be derived from the standard 2-soliton solutions by a proper limiting procedure. Second, we discuss the existence of some integrals of motion related to singular positon-like solutions. The 2-soliton solution of the KdV equation can be written as [5,6] 24= -2azlnr.,
(3)
where
=&+&
r = 1 + eE1+ e52+ E = -2k,x
’
+ 8k3,t + E’,“‘,
m = 1,2,
and k,, E$’ are free parameters. Substituting k, = a + ib, k2 = a - ib into equation (3) we obtain the breather-like solution [7] which can be roughly described as an oscillating wave-packet modulated by a singular soliton-like envelope. Moreover, choosing
and denoting y = x + 4(3b= - a’)t, 6 = b[x + 4(b2 - 3a=)t]
we obtain immediately
1
2a - 2”Ysin2fi = -- 2a e-2ay r 2z 1 - ep40Y- -e sin26 - -&-(1 - e-4ny) , b
b 1
(4)
where the multiplicative factor -2a/b as a constant does not affect the solution (3) and can be neglected. Taking the limit a + 0 and replacing b by x to make the notation consistent with Ref. [l] we find r = sin26 - 2xy,
(5)
where y = x + 12yt, 6 = x(x + 429. Substituting (5) into Matveev’s result [ 11. The eigenvalues of hence in the limit a + continuous spectrum. The same limiting Indeed, denoting
(3) we obtain finally the positon solution (2) in agreement with the associated linear problem are given by A, = - ki (m = 1, 2), 0 we obtain Am= b2 > 0, i.e. a positive bound state embedded in the procedure can be applied to the case k, = a + ib, k2 = -a + ib. /3 = x - 4(3a= - b=)t, 5; = a[x - 4(a2 - 3b=)t]
and taking g(O) m = In (b/a) T in/2
2231
Positon solutions of KdV and SG equations
we find t=l-ee-
i4bB+ i(2b/u)e-‘*bS sinh25 = i(26/u)[e-‘2bfi’sinh2~ + (a/i2b)(l
- e-abp)]. (6)
Taking the limit b + 0 and replacing a by
we obtain
K
r = sinh25; +
~KP,
(7)
where c =
K(X
-
4K2t),
p = x - 12l?t, and the proportionality constant i2b/u has been neglected, as before. The expression (7), when substituted into (3), yields a negaton solution reported also in [2]. In contrast to the positon case, the negaton solution decays exponentially for x + km, and the eigenvalue of the associated spectral problem is negative: A, = - ki = -a* < 0, as b + 0. It is clear from (3), (5) and (7) that both positon and negaton solutions are singular. As a result, the ‘mass’ defined as the principal value [2] m m=P u(x, t) dx (8) I -cc is infinite in both cases because of the second order pole in the solution (3). In order to avoid this difficulty, we suggest here another definition by assuming the space-variable to be x + i& and next taking the limit E+ 0 [7]: m
E = lim u(x + i.5, t)dx. E--t0 I -m In contrast to the previous definition, now the mass is finite. Moreover, we find E = 0 for the positon and 5i= -8~ for the negaton, and the difference between the two cases is clearly visible. The zero mass of the positon is consistent with the super-reflectionless property as well as with the fact that other objects interacting with positons do not undergo any phase-shifts. On the other hand, the mass of the negaton is negative and two times larger than that of a single soliton having the same spectral parameter K. Thus, the negaton can be considered as a degenerate bound state of two singular solitons in the limit when two points of the discrete spectrum coincide. A similar approach can be applied to other constants of motion as well as to eigenfunctions of the associated linear problem. In this latter case, shifting the x-axis into the complex domain makes the eigenfunction continuous and removes any ambiguity related to its singular behaviour for x real. Moreover, the normalization integral becomes finite when taken as /@ dx rather than /]r/~l’dx [7]. As the next example, let us consider briefly a positon-like solution of the SG equation in the light-cone coordinates: As the next example, let us consider briefly a positon-like solution of the SG equation in the light-cone coordinates: a,, = 4sinQ.
(10)
It is convenient to write the 2-soliton solution of equation (10) in the form [8,9] @ = 2i In r-/r+,
(11)
2232
M. JAWORSKI and J. ZAGRODZINSKI
where
and A,,,, g’,“’ are free parameters. For Amimaginary (A, = ipm, ,LQ> &) we assume E(O) = In- p1+l4 m
.lr +. l--,
2
h-P2
hence denoting xm = P?nZ- thm we obtain t+
= 1=
ew1+x2)
2iei(Xl+X2)
-
Yl + + -@
!J2
Pl
-
P2
+
x2)
-
sin
(xl
2i,r1
_
e2ixz >
+
1
Pl --
+
I4
Pl
-
P2
sin
(xl
-
1
x2) .
(12)
In the limit when h + pl, since also x2 + xl, we find t-+ = -2ie2ix1(sin231, + 2r,),
(13)
where r1
=
PlZ
+
t/k
Both xl and rl are real, thus substituting (13) into (11) we obtain a purely imaginary positon-like solution of the SG equation (10) in full agreement with reference [3]: sin2g - 2rr (14) sin231, 4%’ Summarizing, in this paper we show that the positon-like (and related) solutions of the KdV and SG equations can be derived in the framework of Hirota formalism by taking a proper limit of the standard 2-soliton formula. Moreover, owing to a new definition we obtain relevant integrals of motion which remain finite for singular solutions. It is clear that the limiting procedures discussed here are not restricted to the above examples, but can be easily extended to other completely integrable systems. @ = -2iln
Acknowledgements-The authors wish to thank Prof. R. Hirota for helpful and stimulating comments. This work was supported by grant No. PB 2-0480-91-01.
REFERENCES 1. V. B. Matveev, Generalized Wronskian formulas for the solution of the Korteweg-de Vries equation: first applications, Whys.Letr. A166, 205 (1992). 2. V. B. Matveev, Theory of positons, MPI Metallforschung, Stuttgart, Preprint (1992). 3. R. Beutler, Positon solutions of the Enneper (sine-Gordon) equation, MPI Metallforschung, Stuttgart, Preprint (1992). 4. A. Stahlhofen, Positons of the modified KdV equation, Annalen der Physik 1, 554 (1992). 5. R. Hirota, “Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett. 27, 1192 (1971).
Positon solutions of KdV and SG equations
2233
6. C. S. Gardner, .I. M. Greene, M. D. Kruskal and R. M. Miura, The Korteweg-de Vries equation and generalizations, Comm. Pure Appl. Math. 27, 97 (1974). 7. M. Jaworski, A note on singular solutions of the Korteweg-de Vries equation, Phys. Left. AlOO, 321 (1984). 8. M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, (SIAM, Philadelphia, PA (1981)). 9. J. Pelka and J. Zagrodzinski, Phenomenological electrodynamics of the Josephson junction, Physicu BlJ4, 125 (1989).