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Modified Korteweg-de Vries hierarchy with hodograph transformation: Camassa–Holm and Harry–Dym hierarchies
Mathematics and Computers in Simulation 55 (2001) 483–491
Modified Korteweg-de Vries hierarchy with hodograph transformation: Camassa–Holm and Harry–Dym hierarchies R.A. Kraenkel a,∗ , A.I. Zenchuk b a
b
Instituto de F´ısica Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sao Paulo, Brazil Department of Mathematics, The University of Arizona, 617 N. Santa Rita, P.O. Box 210087, Tucson, AZ 85721, USA Received 1 October 2000; accepted 31 October 2000
Abstract In this paper, we present relations between Camassa–Holm (CH), Harry–Dym (HD) and modified Korteweg-de Vries (mKdV) hierarchies, which are given by the hodograph type transformation. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Hierarchies; Harry–Dym hierarchies; Camassa–Holm equation; Integrable systems
1. Introduction A very productive method for the investigation of the new equations of mathematical physics is using their relation with certain other equations, whose properties are known. Very widespread type of relation is the hodograph type transformation. For instance, the linear wave equation ut = ux , with solution u = f (t − x), leads, under the transformation t → t + ϕ(x, t) with ϕ = tv, to the shock-wave equation vt = (1 + v)vx . Other examples are the C-integrable systems [1] which represent a large class of such equations. In the above examples the new nonlinear partial differential equations (PDE) are related with the known linear systems. But in the set of cases we are able to transform (by using, say, hodograph transformation) the new system of nonlinear PDE to the integrable system of nonlinear PDE rather than to the linear one. ∗ Corresponding author. E-mail addresses: [email protected] (R.A. Kraenkel), [email protected] (A.I. Zenchuk).
0378-4754/01/$20.00 © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 0 ) 0 0 3 0 8 - 6
484
R.A. Kraenkel, A.I. Zenchuk / Mathematics and Computers in Simulation 55 (2001) 483–491
In this case the equation under transformation can be considered as an integrable one, since it possesses the set of properties which are typical for integrable systems. ¯ The expansion of this procedure to the nonlinear PDE solvable through the ∂-problem has been done in [2]. The Camassa–Holm equation (CH) is one of the remarkable equations which can be solved by this method [3,4]. In [4] we give the hodograph transformation between CH mt − κux − 2mux − umx = 0,
m = uxx − u,
κ = const,
(1)
and Sinh–Gordon equation with nonlocal deformation [5] 1 χ 2 TX
− c1 eχ − c2 e−χ + c3 (∂X−1 (eχ )∂X−1 (e−χ ))X = 0,
ci = const, i = 1, 2, 3,
(2)
The important fact is that the spectral problem (6) for Eq. (2) looks exactly like the spectral problem for the mKdV (not for KdV). In other words, the Sinh–Gordon equation with such a deformation is the commuting flow of the mKdV. So the natural way to analyze the integrability properties of CH-hierarchy is to relate these properties with ones for mKdV-hierarchy. This relation implicitly has been described in [4] (although we did not mention there about mKdV hierarchy). Here, we represent it in a little different form to emphasize the relations between CH and mKdV hierarchies. By the way we reproduce the relation between two above hierarchies and Harry–Dym (HD) hierarchy [6]. The CH (1) appeared first in a physical context as describing the shallow-water approximation in inviscid hydrodynamics [7]. The variable u(x, t) represents the fluid velocity in the horizontal direction x, and κ is a constant. Although first derived by Hamiltonian approximation methods it can also be obtained by using standard asymptotic methods [8]. Much of the interest on this equation comes from two remarkable facts: (i) it is a completely integrable equation [7,9], consequently allowing for its study the use of many peculiar properties of these systems [10,11]; (ii) it possesses peaked solitary-wave solutions (termed peakons) in the limit κ → 0. Peakons are solutions presenting a finite discontinuity in its first derivative, as the solution found in [7] u(x, t) = c exp(−|x − ct|),
(3)
where c is an arbitrary constant. Further studies led to new solutions like the billiard solutions [12] and the cuspon solutions [3,13,14] (solutions with the first derivative going to infinity at a given point [15]). Also the term with can be set to zero in Eq. (1) through the map u → u → κ, x → x + κt. This equation with κ 6= 0 has essentially different classes of solutions which vanish at the infinity then with κ = 0: peakons if κ = 0 and cuspons otherwise. In [4] we have found that the linear overdetermined system for CH equation (Eq. (1)) with κ = 4 [7–9] Ψxx − Ψx +
m−2 Ψ = 0, 2Ω
m = uxx − u,
Ψt − (u − Ω)Ψx + 21 (ux + u)Ψ = 0,
(4) (5)
can be obtained from the following linear system 1 Ψ = 0. Ω ΨT + V ΩΨX + W Ψ = 0, ΨXX − U2 ΨX −
(6) (7)
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485
through the hodograph transformation X = Φ(x, t),
T = t,
(8)
with u given by u=
Φt , Φx
(9)
and Φ satisfies the equation U2 Φx +
Φxx = 1. Φx
(10)
The compatibility condition of the system (6) and (7) gives us the Sinh–Gordon equation with nonlocal deformation [4,5] (2) with U2 = χX . Since the CH- and mKdV-hierarchies are generated by the spectral problems (4) and (6), respectively, which are related by the hodograph transformation (8) with function Φ satisfying (10), these hierarchies are also related through the same hodograph transformation This is the basic reason to consider the hierarchy, related with the mKdV-hierarchy through the transformation (8) with arbitrary function Φ of new independent variables (Section 2). The resultant hierarchy will also depend on the arbitrary function Φ. One can impose an additional equation which would fix this function. For example, if Φ satisfies Eq. (10), one has the CH-hierarchy. Another choice of the additional equation leads to the HD-hierarchy (see Examples in the end of the Section 2). Conclusions are given in Section 3.
2. The mKdV-hierarchy with hodograph transformation The mKdV 2 ∂T3 U10 + 41 ∂XXX U10 − 38 U10 ∂X U10 = 0,
(11)
is known to be solved through the spectral problem [16,17] LΨ ≡ ΨaXX − U10 ΨaX −
1 Ψa = 0, Ω
(12)
where Ω represents a spectral parameter. The time evolution is introduced by another linear equation A3 Ψ ≡ ΨT3 + ΨXXX + P32 ΨXX + P31 ΨX = 0,
(13)
so that the compatibility of these two equations leads to the mKdV (11). The classical (“positive”) hierarchy describes the evolution with respect to the set of parameters Tk , k > 3 and is given by the general system of linear equations [16,17] Ak Ψ ≡ ΨTk +
∂Xk Ψ
+
k−1 X n=1
Pkn ∂Xn Ψ = 0.
(14)
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So that the nonlinear equations of this hierarchy are represented by the commutator [16,17] [L, Ak ] = 0.
(15)
Along with this “positive” hierarchy the mKdV possesses so called “negative” hierarchy, which is described by the linear system A−1 Ψ ≡ ΨT−1 + W11 ΩΨX + W10 ΨX + W00 Ψ = 0. A−k Ψ ≡ ∂T−k Ψ +
k k−1 X X (−k) n (−k) n W1n Ω ∂X Ψ + W0n Ω Ψ = 0, n=0
(16) (17)
n=0
and the compatibility of the Eqs.(6), (16) and (17) requires more complicated relation [L, A−k ] = B k L,
(18)
where Bk is differential operator. For our convenience we eliminate the derivatives ∂Xk Ψ, k ≥ 2 from the Eqs. (13) and (14) by using the spectral problem (6). So that they take the form 1 V01 ΨX + Ψ = 0. (19) A3 Ψ ≡ ΨT3 + V10 + Ω Ω Ak Ψ ≡ ∂Tk Ψ +
k−2 k X V n=0
1n ∂X W Ψ Ωn
+
k−2 k X V
0n
Ωn n=0
Ψ = 0.
(20)
Here, potentials Vij , Vijk are related with potentials Pij , Pijk . For instance 2 + P32 U10 + P31 , V10 = ∂X U10 + U10
V01 = U10 + P32 .
The hierarchies (15) and (18) (mainly the “positive” hierarchy (15)) are well known and the spectral problem (6) is well investigated [16,17]. So, it would be remarkable to use the properties of the above hierarchies to get the properties of other integrable hierarchies, which are related with these ones through a change of variables with arbitrary real function X = Φ(x, tk ),
Tk = tk .
(21)
We call this transformation hodograph-type transformation to emphasize the role of the arbitrary function of the independent variable as the new function of the nonlinear system. Under these transformation the above system of linear Eqs. (6), (13), (14), (16) and (17) will be rewritten in the form ˜ ˜ ≡ Ψxx − U˜ 10 Ψx − U01 Ψ = 0, LΨ Ω ! ˜11 V V˜01 Ψx + Ψ = 0. A˜ 3 Ψ ≡ Ψt3 + V˜10 + Ω Ω
(22) (23)
R.A. Kraenkel, A.I. Zenchuk / Mathematics and Computers in Simulation 55 (2001) 483–491
A˜ k Ψ ≡ ∂tk Ψ +
k−2 ˜ k X V n=0
1n Ψx Ωn
+
k−2 ˜ k X V
0n
Ωn n=0
Ψ = 0.
A˜ −1 Ψ ≡ Ψt−1 + W˜ 11 ΩΨx + W˜ 10 Ψx + W˜ 00 Ψ = 0, A˜ −k Ψ ≡ Ψtk +
k k X X k k Ψx + Ψ = 0. W˜ 1n W˜ 0n n=0
487
(24) (25) (26)
n=0
We emphasize that all potentials of the last system (22)–(26) are related with the potentials of the system (6, 13, 14, 16, 17) and the function Φ which is an arbitrary function of independent variables. For example U˜ 01 = Φx2 ,
Φxx U˜ 10 = U10 Φx + . Φx
1 , V˜11 = Φx
V˜01 = V01 ,
W11 , W˜ 11 = Φx
(27)
V10 Φt3 V˜10 = − , Φx Φx
W10 Φt−1 W˜ 10 = − , Φx Φx
W˜ 00 = W00 .
(28) (29)
The compatibility conditions of the system (22)–(26) gets a different form: instead of (15) and (18) one has now ˜ A˜ k ] = B˜ k L, ˜ [L,
(30)
˜ ˜ A˜ −k ] = B˜ −k L, [L,
(31)
with Bk , B−k are the differential operators. The system (30) and (31) represents the hierarchy which is related with mKdV hierarchy (15) and (18) through the transformation (21) with arbitrary function Φ. This function leads to an additional freedom in the system of nonlinear PDE (30) and (31). So one can give the extra equation to this system which would fix the function Φ. This additional equation can be considered as the reduction on the general system (30) and (31). At least two particular reductions have physical interest ˜ A˜ −1 ] = 1. U˜ 2 = 1. In this case the system (30) and (31) results to the CH-hierarchy with the equation [L, ˜ B−1 L is the CH itself. ˜ A˜ 3 ] = B3 L˜ give the HD equation. 2. U˜ 2 = 0. In this case one has the HD hierarchy with equation [L, Both these cases will be considered in the end of the next section. 2.1. Two simplest terms from the hierarchy (30) and (31) The CH and HD hierarchies are considered in the set of papers [6–9,12]. Here, we emphasize their relation with mKdV hierarchy. As has been pointed in the previous section, this relation is provided by the change of variables (21). Here, we consider closely the linear Eqs. (12), (13) and (16) since namely these equations are related with CH and HD equations.
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R.A. Kraenkel, A.I. Zenchuk / Mathematics and Computers in Simulation 55 (2001) 483–491
The compatibility conditions of the Eqs. (12) and (13) give us the mKdV (11), while the compatibility of the system (12) and (16) give us the negative flow of the mKdV hierarchy, which can be represented by the system ∂T−1 U10 + 2∂X W00 = 0. 2∂X W11 − U10 ∂X W00 + ∂XX W00 = 0. ∂X (W11 U10 ) + ∂XX W11 = 0. This last system can be rewritten in the form of Sinh–Gordon equation with nonlocal deformation (2) [4]: U2 = U10 ,
T = T−1 .
We have seen in the previous section that under the transformation (21) the linear system (12), (16) and (19) becomes of the form (22), (23) and (25) with the compatibility condition for the Eqs. (22) and (23) looks like ∂t3 U˜ 10 + ∂x (U˜ 10 V˜10 ) + ∂XX V˜10 = 0,
(32)
∂x (V˜11 U10 ) + 2∂x V˜01 + ∂xx V˜11 = 0,
(33)
∂t3 U˜ 01 + V˜10 ∂x U˜ 01 − ∂x V˜01 U˜ 10 + 2U˜ 01 ∂x V˜10 + ∂xx V˜01 = 0,
(34)
V˜11 ∂x U˜ 01 + 2∂x V˜11 U˜ 01 = 0,
(35)
while the compatibility condition for the system (22) and (25) is given below ∂t−1 U˜ 10 + ∂x (U˜ 10 W˜ 10 ) + 2∂x W˜ 00 + ∂xx W˜ 10 = 0,
(36)
∂x (W˜ 11 U˜ 10 ) + ∂xx W˜ 11 = 0,
(37)
2U˜ 01 ∂x W˜ 11 − U˜ 10 ∂x W˜ 00 + W˜ 11 ∂x U˜ 01 + ∂xx W˜ 00 = 0,
(38)
∂t−1 U˜ 01 + 2U˜ 01 ∂x W˜ 10 + W˜ 10 ∂x U˜ 01 = 0,
(39)
Thus, one has two systems of equations. The remarkable fact is that these systems depend implicitly on the arbitrary function Φ in accordance with (27)–(29). Thus, one can add one more arbitrary equation to these systems (the same for both of them) which would fix the function Φ. This equation can be considered as the reduction of the system (32)–(39). Consider several useful examples. Example 1 (CH hierarchy). Let us consider the reduction of the form U˜ 10 = c1 = const 6= 0,
(40)
as an equation to define Φ. Then, Eqs. (36) and (37) give us W˜ 11 = c2 = const,
W˜ 00 = − 21 (c1 W˜ 10 + ∂x W˜ 10 ) + c3 , c3 = const,
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489
(generally speaking, c2,3 are arbitrary functions of parameters tk ; here we consider them to be constants) So finally we get the coupled Eqs. (38) and (39) in the form 1 2 ˜ (c W10 − ∂xx W˜ 10 ) + c4 . U˜ 01 = − 2c2 1
(41)
∂t−1 U˜ 01 + W˜ 10 ∂x U˜ 01 + 2U˜ 01 ∂x W˜ 10 = 0,
(42)
which is the CH (1) with u = W˜ 10 , κ = −2c2 c4 , t = −t−1 if c1 = 1. The same reduction (40) in the Eqs. (32) and (33) reads V10 = b2 ,
V˜01 = − 21 (c1 V˜11 + ∂x V˜11 ) + b3 , b2,3 = const,
and finally we have Eq. (34) in the form (∂t3 + b2 ∂x )U˜ 01 + 21 (c12 ∂x − ∂xxx )V˜11 = 0, (43) q with V˜11 = b4 / U˜ 01 is defined from (35) (b4 = const). This equation represents the commuting flow of CH equation, if c1 = 1. A remarkable fact is that the reduction (40) establishes simple mutual relation between potentials of the CH and mKdV spectral problems. To prove this let us turn to the relation between this potentials (27), which reads under the above reduction ∂x (χ + ln Φx − c1 x) = 0, U˜ 10 (Φ, t) = ∂X (χ )
X=Φ(x,tk )
Φx = C(tk ) exp(c1 x − χ),
or ,
Φx2 = U˜ 01 (x, tk ),
(44) (45)
(C(tk ) is an arbitrary function). For U˜ 01 ≥ 0, this system give us the way to reconstruct the potential of the spectral problem for mKdV through the known potential of the spectral problem for CH and vice versa. In fact, by knowing the potential U01 we can find the function Φ explicitly by integrating Eq. (45) (numerically in general case). Due to the integrations we get the family of the functions Φ(x, tk , c(tk )) ˜ which is parametrized by the arbitrary function c(tk ): Φ(x, tk ) = φ((x, tk ) + c(tk ). Then, Eq. (44) give us the function χ as a function of (x, tk ): χ = f (x, tk , c(tk ), C(tk )), which is parametrized by the function C(tk ). The functions C(tk ) and c(tk ) can be fixed by the additional conditions imposed on the both potential U˜ 01 (x, tk ) and potential U01 (X, Tk ) (initial or boundary conditions). Now one can reverse the Eq. (21) to express x in terms of X, Tk = tk : x = Φ −1 (X, Tk , c(Tk )). Since Φ(x, tk ) is monotonic function of x, it can be done (at least numerically) for all values tk . So one gets the function χ(X, Tk ) related with the given potential U˜ 10 (x, tk ): χ = f (Φ −1 (X, Tk , c(Tk )), Tk , C(Tk )), and the potential U01 = χX . Conversely, if one has the potential U01 of the spectral problem for mKdV, one can solve the first-order ordinary differential Eq. (44) with separated variables to find function Φ, that can be done for any potential U01 (numerically in general case). The solution is parametrized by arbitrary functions C(Tk = tk ) and c(Tk = tk ). After this one should use Eq. (45) to find the potential U˜ 01 , which will be positive, since Φ is a real function.
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Note that complete CH-hierarchy corresponds to the hierarchy (30) and (31) under the reduction (40). The overdetermined linear system (22), (23) and (25) for this case takes the simpler form ! ˜ 01 ˜11 U V V˜01 ˜ ≡ Ψxx − c1 Ψx − LΨ Ψ = 0, A˜ 3 Ψ ≡ ΨT3 + b2 + ΨX + Ψ = 0, Ω Ω Ω A˜ −1 Ψ ≡ Ψτ−1 + c2 Ω∂X Ψ + W˜ 10 ∂X Ψ + W˜ 00 Ψ = 0
(46)
Example 2 (HD hierarchy). In the particular case c1 = 0, Eq. (43) gives the HD equation [6] −1/2
(∂t3 + b2 ∂x )U˜ 01 − 21 ∂xxx (U˜ 01
) = 0,
which is of physical interest as well. In this case Eqs. (41) and (42) represent the commuting flow of HD-hierarchy. All formulas of the previous example are true here with substitution c1 = 0. Example 3. We would like to give one more example of an integrable system, which results from the system (32)–(39). If the function Φ satisfies the equation 2 U˜ 01 = U˜ 10 ,
(47)
then the system (32)–(35) leads to (with zero integrability constants) ∂xx (V˜11 − 4V˜10 ) − V˜11 ∂xxx V˜11 = 0,
2 ∂t3 V˜11 − V˜11 ∂x V˜10 + V˜10 ∂x V˜11 − V˜11 ∂xx V˜10 = 0,
and the system (36)–(39) reads ∂t−1 U˜ 10 + ∂x (W˜ 10 U˜ 10 ) = 0. ∂x (W˜ 11 ∂xx W˜ 10 − 2(∂x W˜ 11 )2 = 0. ∂x W˜ 11 U˜ 10 = − . W˜ 11 The linear system for this case keeps the form (22), (23) and (25) with substitution (47). In this case there is also the simple relation between the potentials of two spectral problems: for mKdV and for the above nonlinear system. Since the proof of this statement is straightforward, we don’t represent it here. 3. Conclusions We represent the CH hierarchy as one of the remarkable examples where the change of variables with arbitrary function plays the leading role in studying the integrability properties of new nonlinear PDE. Being related with the known integrable hierarchy through the simple hodograph transformation, it can be investigated indirectly by using the knowledge about integrability of mKdV [3,4,18,19]. We would like to emphasize one more feature of this approach. The system (32)–(39) is complete system of nonlinear PDE of eight equations with eight functions. Although it allows an additional equation for
R.A. Kraenkel, A.I. Zenchuk / Mathematics and Computers in Simulation 55 (2001) 483–491
491
the function Φ, this system itself can be treated as nonlinear (2 + 1)-dimensional integrable system of PDE. This is “weak” multidimensions, since it comes from (1 + 1)-dimensions due to the change of variables (21) with arbitrary function Φ. One of the authors (A. Zenchuk) thanks FAPESP (Brazil) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
V.E. Zakharov, What is integrability, Springer, Berlin, 1990, p. 33. A. Degasperis, S. Manakov, A. Zenchuk, Phys. Lett. A 249 (1998) 307. A. Zenchuk, JETP Lett. 68 (1998) 715. R.A. Kraenkel, A.I. Zenchuk, J. Phys. A: Math. Gen. 32 (1999) 4733. A.I. Zenchuk, Theor. Math. Phys. 110 (1997) 183. F. Calogero, A. Degasperis, Spectral Transform and Solitons I, North-Holland, Amsterdam, 1982. R. Camassa, D.D. Holm, Phys. Rev. Lett. 71 (1993) 1661. A.S. Fokas, Physica D 87 (1995) 145. B. Fuchssteiner, A.S. Fokas, Physica D 4 (1981) 47. B. Fuchssteiner, Physica D 95 (1996) 229. H.H. Dai, M. Pavlov, J. Phys. Soc. Jpn. 67 (1998) 3655. M.S. Alber, R. Camassa, D.D. Holm, J.E. Marsden, Lett. Math. Phys. 32 (1994) 137. Y.A. Li, P.J. Olver, Discrete Cont. Dyn. Syst. 3 (1997) 419. Y.A. Li, P.J. Olver, Discrete Cont. Dyn. Syst. 4 (1998) 159. M. Wadati, Y.H. Ichikowa, T. Shimizu, Prog. Theor. Phys. 64 (1980) 1959. M.J. Ablowitz, H. Segur, Solitons and Inverse Scattering Transform, SIAM, Philadelphia, 1981. V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.P. Pitaevsky, Theory of Solitons. The Inverse Problem Method, Plenum Press, New York, 1984. [18] R.A. Kraenkel, A.I. Zenchuk, Two-dimensional integrable generalizations of the Camassa–Holm equation: ((F(2+1 (/F()-dimensional hierarch, Phys. Lett. A 260 (1999) 218. [19] M.C. Ferreira, R.A. Kraenkel, A.J. Zenchuk, Soliton–cuspon interaction for the Camassa–Holm equation, J. Phys. A: Math. Gen. 32 (1999) 8665.
Modified Korteweg-de Vries hierarchy with hodograph transformation: Camassa–Holm and Harry–Dym hierarchies R.A. Kraenkel a,∗ , A.I. Zenchuk b a
b
Instituto de F´ısica Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sao Paulo, Brazil Department of Mathematics, The University of Arizona, 617 N. Santa Rita, P.O. Box 210087, Tucson, AZ 85721, USA Received 1 October 2000; accepted 31 October 2000
Abstract In this paper, we present relations between Camassa–Holm (CH), Harry–Dym (HD) and modified Korteweg-de Vries (mKdV) hierarchies, which are given by the hodograph type transformation. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Hierarchies; Harry–Dym hierarchies; Camassa–Holm equation; Integrable systems
1. Introduction A very productive method for the investigation of the new equations of mathematical physics is using their relation with certain other equations, whose properties are known. Very widespread type of relation is the hodograph type transformation. For instance, the linear wave equation ut = ux , with solution u = f (t − x), leads, under the transformation t → t + ϕ(x, t) with ϕ = tv, to the shock-wave equation vt = (1 + v)vx . Other examples are the C-integrable systems [1] which represent a large class of such equations. In the above examples the new nonlinear partial differential equations (PDE) are related with the known linear systems. But in the set of cases we are able to transform (by using, say, hodograph transformation) the new system of nonlinear PDE to the integrable system of nonlinear PDE rather than to the linear one. ∗ Corresponding author. E-mail addresses: [email protected] (R.A. Kraenkel), [email protected] (A.I. Zenchuk).
0378-4754/01/$20.00 © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 0 ) 0 0 3 0 8 - 6
484
R.A. Kraenkel, A.I. Zenchuk / Mathematics and Computers in Simulation 55 (2001) 483–491
In this case the equation under transformation can be considered as an integrable one, since it possesses the set of properties which are typical for integrable systems. ¯ The expansion of this procedure to the nonlinear PDE solvable through the ∂-problem has been done in [2]. The Camassa–Holm equation (CH) is one of the remarkable equations which can be solved by this method [3,4]. In [4] we give the hodograph transformation between CH mt − κux − 2mux − umx = 0,
m = uxx − u,
κ = const,
(1)
and Sinh–Gordon equation with nonlocal deformation [5] 1 χ 2 TX
− c1 eχ − c2 e−χ + c3 (∂X−1 (eχ )∂X−1 (e−χ ))X = 0,
ci = const, i = 1, 2, 3,
(2)
The important fact is that the spectral problem (6) for Eq. (2) looks exactly like the spectral problem for the mKdV (not for KdV). In other words, the Sinh–Gordon equation with such a deformation is the commuting flow of the mKdV. So the natural way to analyze the integrability properties of CH-hierarchy is to relate these properties with ones for mKdV-hierarchy. This relation implicitly has been described in [4] (although we did not mention there about mKdV hierarchy). Here, we represent it in a little different form to emphasize the relations between CH and mKdV hierarchies. By the way we reproduce the relation between two above hierarchies and Harry–Dym (HD) hierarchy [6]. The CH (1) appeared first in a physical context as describing the shallow-water approximation in inviscid hydrodynamics [7]. The variable u(x, t) represents the fluid velocity in the horizontal direction x, and κ is a constant. Although first derived by Hamiltonian approximation methods it can also be obtained by using standard asymptotic methods [8]. Much of the interest on this equation comes from two remarkable facts: (i) it is a completely integrable equation [7,9], consequently allowing for its study the use of many peculiar properties of these systems [10,11]; (ii) it possesses peaked solitary-wave solutions (termed peakons) in the limit κ → 0. Peakons are solutions presenting a finite discontinuity in its first derivative, as the solution found in [7] u(x, t) = c exp(−|x − ct|),
(3)
where c is an arbitrary constant. Further studies led to new solutions like the billiard solutions [12] and the cuspon solutions [3,13,14] (solutions with the first derivative going to infinity at a given point [15]). Also the term with can be set to zero in Eq. (1) through the map u → u → κ, x → x + κt. This equation with κ 6= 0 has essentially different classes of solutions which vanish at the infinity then with κ = 0: peakons if κ = 0 and cuspons otherwise. In [4] we have found that the linear overdetermined system for CH equation (Eq. (1)) with κ = 4 [7–9] Ψxx − Ψx +
m−2 Ψ = 0, 2Ω
m = uxx − u,
Ψt − (u − Ω)Ψx + 21 (ux + u)Ψ = 0,
(4) (5)
can be obtained from the following linear system 1 Ψ = 0. Ω ΨT + V ΩΨX + W Ψ = 0, ΨXX − U2 ΨX −
(6) (7)
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through the hodograph transformation X = Φ(x, t),
T = t,
(8)
with u given by u=
Φt , Φx
(9)
and Φ satisfies the equation U2 Φx +
Φxx = 1. Φx
(10)
The compatibility condition of the system (6) and (7) gives us the Sinh–Gordon equation with nonlocal deformation [4,5] (2) with U2 = χX . Since the CH- and mKdV-hierarchies are generated by the spectral problems (4) and (6), respectively, which are related by the hodograph transformation (8) with function Φ satisfying (10), these hierarchies are also related through the same hodograph transformation This is the basic reason to consider the hierarchy, related with the mKdV-hierarchy through the transformation (8) with arbitrary function Φ of new independent variables (Section 2). The resultant hierarchy will also depend on the arbitrary function Φ. One can impose an additional equation which would fix this function. For example, if Φ satisfies Eq. (10), one has the CH-hierarchy. Another choice of the additional equation leads to the HD-hierarchy (see Examples in the end of the Section 2). Conclusions are given in Section 3.
2. The mKdV-hierarchy with hodograph transformation The mKdV 2 ∂T3 U10 + 41 ∂XXX U10 − 38 U10 ∂X U10 = 0,
(11)
is known to be solved through the spectral problem [16,17] LΨ ≡ ΨaXX − U10 ΨaX −
1 Ψa = 0, Ω
(12)
where Ω represents a spectral parameter. The time evolution is introduced by another linear equation A3 Ψ ≡ ΨT3 + ΨXXX + P32 ΨXX + P31 ΨX = 0,
(13)
so that the compatibility of these two equations leads to the mKdV (11). The classical (“positive”) hierarchy describes the evolution with respect to the set of parameters Tk , k > 3 and is given by the general system of linear equations [16,17] Ak Ψ ≡ ΨTk +
∂Xk Ψ
+
k−1 X n=1
Pkn ∂Xn Ψ = 0.
(14)
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So that the nonlinear equations of this hierarchy are represented by the commutator [16,17] [L, Ak ] = 0.
(15)
Along with this “positive” hierarchy the mKdV possesses so called “negative” hierarchy, which is described by the linear system A−1 Ψ ≡ ΨT−1 + W11 ΩΨX + W10 ΨX + W00 Ψ = 0. A−k Ψ ≡ ∂T−k Ψ +
k k−1 X X (−k) n (−k) n W1n Ω ∂X Ψ + W0n Ω Ψ = 0, n=0
(16) (17)
n=0
and the compatibility of the Eqs.(6), (16) and (17) requires more complicated relation [L, A−k ] = B k L,
(18)
where Bk is differential operator. For our convenience we eliminate the derivatives ∂Xk Ψ, k ≥ 2 from the Eqs. (13) and (14) by using the spectral problem (6). So that they take the form 1 V01 ΨX + Ψ = 0. (19) A3 Ψ ≡ ΨT3 + V10 + Ω Ω Ak Ψ ≡ ∂Tk Ψ +
k−2 k X V n=0
1n ∂X W Ψ Ωn
+
k−2 k X V
0n
Ωn n=0
Ψ = 0.
(20)
Here, potentials Vij , Vijk are related with potentials Pij , Pijk . For instance 2 + P32 U10 + P31 , V10 = ∂X U10 + U10
V01 = U10 + P32 .
The hierarchies (15) and (18) (mainly the “positive” hierarchy (15)) are well known and the spectral problem (6) is well investigated [16,17]. So, it would be remarkable to use the properties of the above hierarchies to get the properties of other integrable hierarchies, which are related with these ones through a change of variables with arbitrary real function X = Φ(x, tk ),
Tk = tk .
(21)
We call this transformation hodograph-type transformation to emphasize the role of the arbitrary function of the independent variable as the new function of the nonlinear system. Under these transformation the above system of linear Eqs. (6), (13), (14), (16) and (17) will be rewritten in the form ˜ ˜ ≡ Ψxx − U˜ 10 Ψx − U01 Ψ = 0, LΨ Ω ! ˜11 V V˜01 Ψx + Ψ = 0. A˜ 3 Ψ ≡ Ψt3 + V˜10 + Ω Ω
(22) (23)
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A˜ k Ψ ≡ ∂tk Ψ +
k−2 ˜ k X V n=0
1n Ψx Ωn
+
k−2 ˜ k X V
0n
Ωn n=0
Ψ = 0.
A˜ −1 Ψ ≡ Ψt−1 + W˜ 11 ΩΨx + W˜ 10 Ψx + W˜ 00 Ψ = 0, A˜ −k Ψ ≡ Ψtk +
k k X X k k Ψx + Ψ = 0. W˜ 1n W˜ 0n n=0
487
(24) (25) (26)
n=0
We emphasize that all potentials of the last system (22)–(26) are related with the potentials of the system (6, 13, 14, 16, 17) and the function Φ which is an arbitrary function of independent variables. For example U˜ 01 = Φx2 ,
Φxx U˜ 10 = U10 Φx + . Φx
1 , V˜11 = Φx
V˜01 = V01 ,
W11 , W˜ 11 = Φx
(27)
V10 Φt3 V˜10 = − , Φx Φx
W10 Φt−1 W˜ 10 = − , Φx Φx
W˜ 00 = W00 .
(28) (29)
The compatibility conditions of the system (22)–(26) gets a different form: instead of (15) and (18) one has now ˜ A˜ k ] = B˜ k L, ˜ [L,
(30)
˜ ˜ A˜ −k ] = B˜ −k L, [L,
(31)
with Bk , B−k are the differential operators. The system (30) and (31) represents the hierarchy which is related with mKdV hierarchy (15) and (18) through the transformation (21) with arbitrary function Φ. This function leads to an additional freedom in the system of nonlinear PDE (30) and (31). So one can give the extra equation to this system which would fix the function Φ. This additional equation can be considered as the reduction on the general system (30) and (31). At least two particular reductions have physical interest ˜ A˜ −1 ] = 1. U˜ 2 = 1. In this case the system (30) and (31) results to the CH-hierarchy with the equation [L, ˜ B−1 L is the CH itself. ˜ A˜ 3 ] = B3 L˜ give the HD equation. 2. U˜ 2 = 0. In this case one has the HD hierarchy with equation [L, Both these cases will be considered in the end of the next section. 2.1. Two simplest terms from the hierarchy (30) and (31) The CH and HD hierarchies are considered in the set of papers [6–9,12]. Here, we emphasize their relation with mKdV hierarchy. As has been pointed in the previous section, this relation is provided by the change of variables (21). Here, we consider closely the linear Eqs. (12), (13) and (16) since namely these equations are related with CH and HD equations.
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The compatibility conditions of the Eqs. (12) and (13) give us the mKdV (11), while the compatibility of the system (12) and (16) give us the negative flow of the mKdV hierarchy, which can be represented by the system ∂T−1 U10 + 2∂X W00 = 0. 2∂X W11 − U10 ∂X W00 + ∂XX W00 = 0. ∂X (W11 U10 ) + ∂XX W11 = 0. This last system can be rewritten in the form of Sinh–Gordon equation with nonlocal deformation (2) [4]: U2 = U10 ,
T = T−1 .
We have seen in the previous section that under the transformation (21) the linear system (12), (16) and (19) becomes of the form (22), (23) and (25) with the compatibility condition for the Eqs. (22) and (23) looks like ∂t3 U˜ 10 + ∂x (U˜ 10 V˜10 ) + ∂XX V˜10 = 0,
(32)
∂x (V˜11 U10 ) + 2∂x V˜01 + ∂xx V˜11 = 0,
(33)
∂t3 U˜ 01 + V˜10 ∂x U˜ 01 − ∂x V˜01 U˜ 10 + 2U˜ 01 ∂x V˜10 + ∂xx V˜01 = 0,
(34)
V˜11 ∂x U˜ 01 + 2∂x V˜11 U˜ 01 = 0,
(35)
while the compatibility condition for the system (22) and (25) is given below ∂t−1 U˜ 10 + ∂x (U˜ 10 W˜ 10 ) + 2∂x W˜ 00 + ∂xx W˜ 10 = 0,
(36)
∂x (W˜ 11 U˜ 10 ) + ∂xx W˜ 11 = 0,
(37)
2U˜ 01 ∂x W˜ 11 − U˜ 10 ∂x W˜ 00 + W˜ 11 ∂x U˜ 01 + ∂xx W˜ 00 = 0,
(38)
∂t−1 U˜ 01 + 2U˜ 01 ∂x W˜ 10 + W˜ 10 ∂x U˜ 01 = 0,
(39)
Thus, one has two systems of equations. The remarkable fact is that these systems depend implicitly on the arbitrary function Φ in accordance with (27)–(29). Thus, one can add one more arbitrary equation to these systems (the same for both of them) which would fix the function Φ. This equation can be considered as the reduction of the system (32)–(39). Consider several useful examples. Example 1 (CH hierarchy). Let us consider the reduction of the form U˜ 10 = c1 = const 6= 0,
(40)
as an equation to define Φ. Then, Eqs. (36) and (37) give us W˜ 11 = c2 = const,
W˜ 00 = − 21 (c1 W˜ 10 + ∂x W˜ 10 ) + c3 , c3 = const,
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(generally speaking, c2,3 are arbitrary functions of parameters tk ; here we consider them to be constants) So finally we get the coupled Eqs. (38) and (39) in the form 1 2 ˜ (c W10 − ∂xx W˜ 10 ) + c4 . U˜ 01 = − 2c2 1
(41)
∂t−1 U˜ 01 + W˜ 10 ∂x U˜ 01 + 2U˜ 01 ∂x W˜ 10 = 0,
(42)
which is the CH (1) with u = W˜ 10 , κ = −2c2 c4 , t = −t−1 if c1 = 1. The same reduction (40) in the Eqs. (32) and (33) reads V10 = b2 ,
V˜01 = − 21 (c1 V˜11 + ∂x V˜11 ) + b3 , b2,3 = const,
and finally we have Eq. (34) in the form (∂t3 + b2 ∂x )U˜ 01 + 21 (c12 ∂x − ∂xxx )V˜11 = 0, (43) q with V˜11 = b4 / U˜ 01 is defined from (35) (b4 = const). This equation represents the commuting flow of CH equation, if c1 = 1. A remarkable fact is that the reduction (40) establishes simple mutual relation between potentials of the CH and mKdV spectral problems. To prove this let us turn to the relation between this potentials (27), which reads under the above reduction ∂x (χ + ln Φx − c1 x) = 0, U˜ 10 (Φ, t) = ∂X (χ )
X=Φ(x,tk )
Φx = C(tk ) exp(c1 x − χ),
or ,
Φx2 = U˜ 01 (x, tk ),
(44) (45)
(C(tk ) is an arbitrary function). For U˜ 01 ≥ 0, this system give us the way to reconstruct the potential of the spectral problem for mKdV through the known potential of the spectral problem for CH and vice versa. In fact, by knowing the potential U01 we can find the function Φ explicitly by integrating Eq. (45) (numerically in general case). Due to the integrations we get the family of the functions Φ(x, tk , c(tk )) ˜ which is parametrized by the arbitrary function c(tk ): Φ(x, tk ) = φ((x, tk ) + c(tk ). Then, Eq. (44) give us the function χ as a function of (x, tk ): χ = f (x, tk , c(tk ), C(tk )), which is parametrized by the function C(tk ). The functions C(tk ) and c(tk ) can be fixed by the additional conditions imposed on the both potential U˜ 01 (x, tk ) and potential U01 (X, Tk ) (initial or boundary conditions). Now one can reverse the Eq. (21) to express x in terms of X, Tk = tk : x = Φ −1 (X, Tk , c(Tk )). Since Φ(x, tk ) is monotonic function of x, it can be done (at least numerically) for all values tk . So one gets the function χ(X, Tk ) related with the given potential U˜ 10 (x, tk ): χ = f (Φ −1 (X, Tk , c(Tk )), Tk , C(Tk )), and the potential U01 = χX . Conversely, if one has the potential U01 of the spectral problem for mKdV, one can solve the first-order ordinary differential Eq. (44) with separated variables to find function Φ, that can be done for any potential U01 (numerically in general case). The solution is parametrized by arbitrary functions C(Tk = tk ) and c(Tk = tk ). After this one should use Eq. (45) to find the potential U˜ 01 , which will be positive, since Φ is a real function.
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Note that complete CH-hierarchy corresponds to the hierarchy (30) and (31) under the reduction (40). The overdetermined linear system (22), (23) and (25) for this case takes the simpler form ! ˜ 01 ˜11 U V V˜01 ˜ ≡ Ψxx − c1 Ψx − LΨ Ψ = 0, A˜ 3 Ψ ≡ ΨT3 + b2 + ΨX + Ψ = 0, Ω Ω Ω A˜ −1 Ψ ≡ Ψτ−1 + c2 Ω∂X Ψ + W˜ 10 ∂X Ψ + W˜ 00 Ψ = 0
(46)
Example 2 (HD hierarchy). In the particular case c1 = 0, Eq. (43) gives the HD equation [6] −1/2
(∂t3 + b2 ∂x )U˜ 01 − 21 ∂xxx (U˜ 01
) = 0,
which is of physical interest as well. In this case Eqs. (41) and (42) represent the commuting flow of HD-hierarchy. All formulas of the previous example are true here with substitution c1 = 0. Example 3. We would like to give one more example of an integrable system, which results from the system (32)–(39). If the function Φ satisfies the equation 2 U˜ 01 = U˜ 10 ,
(47)
then the system (32)–(35) leads to (with zero integrability constants) ∂xx (V˜11 − 4V˜10 ) − V˜11 ∂xxx V˜11 = 0,
2 ∂t3 V˜11 − V˜11 ∂x V˜10 + V˜10 ∂x V˜11 − V˜11 ∂xx V˜10 = 0,
and the system (36)–(39) reads ∂t−1 U˜ 10 + ∂x (W˜ 10 U˜ 10 ) = 0. ∂x (W˜ 11 ∂xx W˜ 10 − 2(∂x W˜ 11 )2 = 0. ∂x W˜ 11 U˜ 10 = − . W˜ 11 The linear system for this case keeps the form (22), (23) and (25) with substitution (47). In this case there is also the simple relation between the potentials of two spectral problems: for mKdV and for the above nonlinear system. Since the proof of this statement is straightforward, we don’t represent it here. 3. Conclusions We represent the CH hierarchy as one of the remarkable examples where the change of variables with arbitrary function plays the leading role in studying the integrability properties of new nonlinear PDE. Being related with the known integrable hierarchy through the simple hodograph transformation, it can be investigated indirectly by using the knowledge about integrability of mKdV [3,4,18,19]. We would like to emphasize one more feature of this approach. The system (32)–(39) is complete system of nonlinear PDE of eight equations with eight functions. Although it allows an additional equation for
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the function Φ, this system itself can be treated as nonlinear (2 + 1)-dimensional integrable system of PDE. This is “weak” multidimensions, since it comes from (1 + 1)-dimensions due to the change of variables (21) with arbitrary function Φ. One of the authors (A. Zenchuk) thanks FAPESP (Brazil) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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