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A multidimensional superposition principle: classical solitons I
1 January 2001
Physics Letters A 278 (2001) 198–208 www.elsevier.nl/locate/pla
A multidimensional superposition principle: classical solitons I Alexander A. Alexeyev Laboratory of Computer Physics and Mathematical Simulation, Research Division, Room 247, Faculty of Physics–Mathematics and Natural Sciences, Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya str., Moscow 117198, Russia Received 5 August 1999; received in revised form 22 May 2000; accepted 20 November 2000 Communicated by A.R. Bishop
Abstract A concept that easily explains both classical solitonic and more complex wave interactions is proposed for differential equations. Nonlinear PDEs associated with the Riccati equations via ‘truncated expansions’ are considered, and the existence of the KdV-type soliton/kink is shown. Other interactions, including inelastic ones, are indicated. 2001 Elsevier Science B.V. All rights reserved. PACS: 02.30; 03.40.K Keywords: Soliton; Nonlinear interaction; Superposition; Nonlinear PDE
1. Introduction Owing to the extensive studies of integrable nonlinear models performed during the previous decades, especially for such famous equations as the KdV, MKdV and SG equations, nowadays there exists the deep understanding of soliton behaviour in such systems. In many respects we owe this to the inverse scattering transform [1]. Although for an arbitrary potential the related equations and algebra are too difficult to handle, so ultimately asymptotic analysis is applied [2], in reality this is not essential. The theory gives a comprehensive description of the main features, and computer simulation can be used for refinements. What is more important, the approach itself is rather restrictive, and finding a suitable spectral problem and generalization to (n + 1) cases are also open questions at the present time. Because of this, a number of direct
E-mail address: [email protected] (A.A. Alexeyev).
techniques have been invented (see, e.g., [3–6]). They may be effective for the construction of solutions even when the ITS is not applicable. But they do not explain the nature of solitonic interactions, while in the IST solitons are embedded and emerge from the discrete spectrum. Thus the present Letter is an attempt to propose a direct concept of solitons in some sense alternative to the IST in which the notion of solitons would be incorporated by definition and be, perhaps, more tractable. In addition, a different view to solitonic interactions could indicate novel aspects of nonlinear dynamics as a whole.
2. Theoretical background Let us suppose that there is some nonlinear PDE, assumed to be in (1 + 1) dimensions, and for the simplicity not depending explicitly on the independent
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 7 5 - 1
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
variables, ∂ ∂ , ; u(x, t) = 0, E ∂x ∂t
(1)
and construct its (2 + 2)-dimensional adjoint equation by making the formal replacement ∂ ∂ ∂ ∂ + , + ; u(x1 , x2 , t1 , t2 ) = 0. (2) E ∂x1 ∂x2 ∂t1 ∂t2 Keeping in mind that u(x, t) = u(x1 , x2 , t1 , t2 )|x1 =x2 =x, t1 =t2 =t
(3)
we could investigate the latter equation instead of the original one, and because of non-uniqueness of map (3) any subset of the related solutions of (2) with a suitable representation in terms of x1 , x2 , t1 , t2 could be used. In particular, we can restrict ourselves to solutions of (2) associated with additional compatible differential constraints (see, e.g., [7]) such that the equations only with the derivatives with respect to x1 , t1 , ∂ ∂ , ; u(x1, x2 , t1 , t2 ) = 0, i = 1, n1 , G1i ∂x1 ∂t1 (4) and/or x2 , t2 , ∂ ∂ , ; u(x1, x2 , t1 , t2 ) = 0, j = 1, n2 , G2j ∂x2 ∂t2 (5) can be isolated from them, possibly along with other relations ∂ ∂ ∂ ∂ , , , ; u(x1, x2 , t1 , t2 ) = 0, G3l ∂x1 ∂x2 ∂t1 ∂t2 l = 1, n3 ,
n1 , n2 , n3 ∈ N.
(6)
We lose nothing here from the viewpoint of u(x, t) in (3). However, in view of (4)–(6) the variables x1 , t1 and x2 , t2 appear to be separated, the full solution (process) u(x1 , x2 , t1 , t2 ) and, respectively, u(x, t) can be presented as superposition of the two separated solutions (processes) proceeding in the different (x1 , t1 )and (x2 , t2 )-spaces and could be obtained by consecutive solving with respect to x1 , t1 and x2 , t2 . Such a paradigm will be called a multidimensional superposition principle in contrast to a conventional algebraic superposition for ODEs [8]. In so doing, an observer in our (1 + 1) real world sees only a projection of events occurring in that (2 + 2)-dimensional
199
world; only the section x1 = x2 , t1 = t2 is accessible, and these processes are not distinguishable for him. Although the consecutive integration mentioned can really simplify the construction of solutions in a number of cases, the properties of such multidimensional systems themselves and their reflection in the features of the projected solution u(x, t) can appear to be more essential from the viewpoint of understanding and describing nonlinear phenomena and their applications. For instance, if u corresponds to an elastic interaction of two localized waves, then their mutual modulations and deformations are determined by the associated parts of (4)–(6), and the asymptotic states before and after the interaction, u(x1 , x2 , t1 , t2 ) → ubefore, after (x1 , t1 ), u(x1 , x2 , t1 , t2 ) → ubefore, after (x2 , t2 ), are described by the related degeneracies of (4)– (6). From these overdetermined systems one can judge what classes of waves could take part in such interactions and what changes of them are possible in principle. As for the former, several nontrivial situations are possible. Firstly, one degenerate system describes a special solution, and another is reduced to the only equation of the initial type (1) (a conventional soliton case). Secondly, both such systems describe some special solutions. As a result, ‘solitons’ will interact with one perturbations elastically and inelastically with others. Thirdly, both systems are reduced to equations of the form (1). Linear equations are trivially shown to be among such models. As for changes in the waves, an interaction switches a wave from one state to another, in the simplest cases differing by the values of its parameters. Obviously, classical soliton interactions corresponds to changing only a trivial parameter, namely the phase. While in [9] the case with the wavenumbers and velocities changes is adduced. Note in conclusion that all the aforesaid can be extended to Eqs. (1), (2) and (4)–(6) with explicit dependence on the independent variables. In these cases, however, it is necessary in addition to take into account known functions (coefficients) and their properties in (4)–(6) and (1), (2) (for (4)–(6) to reveal the degenerations, and for (1), (2) to chose corresponding conditions directly in the form f (x1 , x2 , t1 ,
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t2 )|x1 =x2 =x, t1 =t2 =t = f (x, t) or when possible through the introduction of auxiliary differential equations that are to be satisfied).
3. The multidimensional superposition and singular manifold technique Both finding sets of differential constraints possessed by an equation itself [7,10], and the investigation of overdetermined systems [11,12] in particular are still open questions for contemporary mathematics as a whole, and frequently possible in full volume only for simple enough cases. While within the framework of the invariant version of the so-called singular manifold approach [13] a number of nonlinear PDEs of various kinds can be reduced by use of the series u(x, t) =
0 X
wi (S, C, Sx , Cx , St , Ct , . . .)V i (x, t),
i=m
m ∈ N,
(7)
to the system of the ODEs Vx = −V 2 − S/2, 1 Vt = CV 2 − Cx V + (CS + Cxx ), 2 S = S(x, t), C = C(x, t),
(8) (9)
with the formal compatibility condition St + Cxxx + 2SCx + CSx = 0 and some extra restriction to the functions S and C, ∂ ∂ , ; S, C; x, t = 0 M (10) ∂x ∂t (a ‘singular manifold equation’), which and determines the type of an equation under consideration. Further study of those system may arrive at (auto)Bäcklund transformations, Lax pairs, etc. for the initial equations [13,14]. Obviously, various NPDEs may be associated with one and the same SME via different expansions (7). (The simplest one of them is an equation in V itself. When an SME is of the form C = M(S, Sx , St , . . . ; x, t), it directly follows from (9). In other cases, set (8)–(10) can in principle [11,12] be transformed to isolate an equation only in V .) By this means the properties of all these equations appear to closely connected, and the explicit maps (see the famous Miura one) can be derived using (7)–(10). As
was shown in [15], all of this technique itself is of an algebraic nature, and more complicated structures than (7)–(9) should frequently be used for a successful outcome. Here our attention will be concentrated on the above case, and the function V with (8), (9) will be considered from the viewpoint proposed in the previous section. The adjoint equations (2) for (8), (9) are as follows: Vx1 + Vx2 = −V 2 − S/2,
(11)
Vt1 + Vt2 = CV − (Cx1 + Cx2 )V 1 + (CS + Cx1 x1 + 2Cx1 x2 + Cx2 x2 ), 2 2
S = S(x1 , x2 , t1 , t2 ),
C = C(x1 , x2 , t1 , t2 ).
(12)
Each of them in turn can be split in the obvious manner (4), (5): Vx1 = A(V ; x1, x2 , t1 , t2 ),
(13)
Vt2 = B(V ; x1, x2 , t1 , t2 ),
(14)
with, respectively, Vx2 = −V 2 − S/2 − A,
(15)
Vt1 = CV − (Cx1 + Cx2 )V 1 + (CS + Cx1 x1 + 2Cx1 x2 + Cx2 x2 ) − B (16) 2 from (11), (12) without loss of generality, because (13) and (14) can simultaneously be considered as the intermediate integrals [16] of more common equations like G(Vkx1 , V(k−1)x1 , . . .) = 0, k > 2. The compatibility conditions for (13)–(16) present a set of six non-homogeneous quasilinear PDEs for A and B, and from their form it is naturally to seek solutions primarily among the class of second order polynomials with respect to V . To simplify the resulting conditions and for our later purposes, we will reduce system (13)–(16) by means of the linear transformation 2
V = w1 (x1 , x2 , t1 , t2 )v(x1 , x2 , t1 , t2 ) + w0 (x1 , x2 , t1 , t2 ) to the form where one of the subsystems, say (13), (16), would be of the original type (8), (9) again: vx1 = −v 2 − s/2, vt1 = cv 2 − cx1 v + (cs + cx1 x1 )/2,
(17)
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
vx2 = av 2 − ax1 v + (as + ax1 x1 )/2, vt2 = bv − bx1 v + (bs + bx1 x1 )/2; 2
(18)
here s, c, a and b are functions of x1 , x2 , t1 , t2 . (In so doing, V and S, C are as follows: V = (1 − a)v +
ax1 ax1 + ax2 + , 2 2(1 − a)
(19)
S = s(a − 1)2 + (a − 2)ax1x1 − ax1 x2 (ax1 )2 3 ax1 + ax2 2 − − 2 2 a−1 ax1 x1 + 2ax1x2 + ax2 x2 + ax1 (ax1 + ax2 ) , + a−1 b+c . C= 1−a
(20) (21)
It is necessary to equate the coefficients at the powers of v to zero in (8) and (9) after the substitution.) The above compatibility conditions then take the simple form: st1 + cx1 x1 x1 + 2scx1 + csx1 = 0, st2 + bx1 x1 x1 + 2sbx1 + bsx1 = 0, sx2 + ax1 x1 x1 + 2sax1 + asx1 = 0, at2 + bax1 − abx1 − bx2 = 0, (22)
System (22) always possesses trivial solutions for s, c and a, b such that s, c = const,
Here the phase θ is an arbitrary function depending on x2 , t2 ; and in view of the rest equations (18), θx2 = −ka, θt2 = −kb. It also linked with S and C by (20), (21) such that θx 2 3 θx2 x2 2 θx2 x2 x2 − + , (27) S =s 1+ 2 k 2 k + θx2 k + θx2 kc − θt2 . C= (28) k + θx2
θ (x2 , t2 ) = lim V V±∞ x1 →±∞
=
θx2 x2 ±sign k (k + θx2 ) − , 2 2(k + θx2 )
(29)
see (26). Secondly, if for x2 → −∞ or/and x2 → +∞ the phase θ has the asymptotic nature
at2 − bx2 = 0, ax1 = at1 = bx1 = bt1 = 0.
Let us construct the related solution v from (17), (18) and, respectively, V (19), starting first with the case s 6= 0. Denoting the wavenumber, frequency and phase by k, ω and θ (s = −k 2 /2, c = −ω/k), one has for v from (17) the well-known kink solution in the (x1 , t1 )space and, as a whole, θx2 x2 θx2 , v− V = 1+ k 2(k + θx2 ) kx1 + ωt1 + θ (x2 , t2 ) k , v = tanh (25) 2 2 with |k| lim v = ± . (26) x1 →±∞ 2
The obtained solution (25) for V can be interpreted simultaneously both as the kink deformed and modulated by the perturbation and vise versa. And as a result, it has the following properties. Firstly,
at1 + cax1 − acx1 − cx2 = 0, ct2 + bcx1 − cbx1 − bt1 = 0.
201
(23)
As will be shown below, this explains the existence and properties of solitons in models associated with SMEs compatible with (23), for example, in view of the forms of (20), (21) that do not depend explicitly on x and t: ∂ ∂ , ; S, C = 0. M (24) ∂x ∂t In so doing, the models themselves may be of a quite different type.
θ±∞ =
lim θ = const,
x2 →±∞
then kink (x1 , t1 ) = V±∞
lim V
x2 →±∞
kx1 + ωt1 + θ±∞ k tanh . (30) 2 2 In so doing, s and c should obviously satisfy the SME of the same form as S and C themselves. So, in the different regions the solution can degenerate to solutions (29) or (30) depending only on x1 , t1 or =
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x2 , t2 . This also implies that combinations (29) for θ θ (x, t) always generate the solutions (invariants) V±∞ of the original equation to V (x, t), corresponding to the perturbation evaluating in the absence of the kink. The same is true for (30). (For a concrete SME this can be verified by a direct substitution, of course.) It is necessary to stress here that the above transitions in (25) appear to be permitted by the construction, due to the established separation of the variables. Simply, for an arbitrary substitution, θ would be bounded and not independent on x1 , t1 . Solution (25) particular to (11), (12) and corresponding to their above splitting simultaneously is the multidimensional presentation for solutions of the original system (8), (9). Using this consider the following problem. Assume that we have the initial data (for definiteness, the kink is disposed to the right and k > 0) kink (x, 0), V (x, 0) = F (x − θ+∞ ) + V+∞
θ+∞ → −∞,
(31)
with some function F (x) that is arbitrary but such that limx→∞ F (x) = 0, so that the x-axis is divided into two areas involving either only the localized kink ). perturbation or only the kink (F − k/2 and V+∞ Supposing that they approach each other, what will happen to this kink and localized wave at t → +∞ after their collision? On the strength of the aforesaid we can establish the consistency between the related solution V (x, t) and expression (25) by setting θ (x2 , 0) F (x2 − θ+∞ ) = V−∞
and will consider the related function θ or, more precisely, its asymptotes, in order to answer the question. Integrating (29) twice with limx2 →+∞ θ = θ+∞ as the boundary (linkage) condition, we have " Z Z+∞ F (τ ) dτ θ = −kx2 + ln k exp θ+∞ + 2 !
x2
#
+ kx2 dx2 + C1 , C1 = const .
(32)
It is easy to see that there are only two cases for x2 → −∞ — the first at C1 = 0: lim θ = θ+∞ + 2I,
x2 →−∞
Z+∞ I= F (τ ) dτ,
(33)
−∞
and the second at C1 6= 0: θ ∼ ln C1 − kx2 + x2 → −∞.
exp(θ+∞ + 2I + kx2 ) , C1 (34)
In the last case, however, from (29), k exp(θ+∞ + 2I + kx2 ) , V−∞ ∼ − − 2 2C1 x2 → −∞
(35)
(Note here that for many equations I is a conservation quantity.) What will happen after the collision in these cases? If one assumes that the changes of the asymptotes for the interaction time are negligible or, differently, ta.change tinteraction,
(36)
then the same asymptotes can again be used. (Taking into account the relation on s and c, it is easy, however, to see from (27), (28) that for a number of SMEs, in particular of the form C = M(S, Sx , . . .),
(37)
this assumption is superfluous because ∂θ∞ /∂t2 ≡ 0 in any case.) As a result, for the first case the perturbation θ kink now) will be separated and V−∞ and kink (V+∞ θ = k/2 again. (The linkage condition limx2 →−∞ V+∞ is satisfied identically.) The second case is specific. The situation is that separation after a collision is absent in this case (see the left θ asymptote). However, according to the dispersion relation of the appropriate linearized equation, the related perturbation’s tail (35) must propagate with the same speed as the kink, so that in reality the interaction itself will last infinitely long. (This may appear to be wrong in models with complicated dispersion relations or without such.) Both scenarios are depicted in Figs. 1 and 2 with immovable perturbations for clarity. Fig. 1 with k, ω = 1 and θ =
−8 +4 e x2 + 1
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
Fig. 1. Typical elastic interaction.
Fig. 2. ‘Capture’ case.
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demonstrates the classical elastic interaction, whereas Fig. 2, where and θ = ln ex2 + 1 − x2 −
7 , e x2 + 1 could be described as the ‘kink capture’ because of the absence of separation after the collision. In these examples θ does not change with time. Although this is not necessarily the case in reality, nevertheless they illustrate the main and common features of such interactions. Generally, there takes place the following linkage between the end-states of a perturbation: k, ω = 1
+∞ θ θ = S−∞ Ttinteraction V+∞ . V−∞ +∞ is obvious (The essence of the switch operator S−∞ from (29), and Ttinteraction corresponds to the evolution of θ during an interaction.) The kink itself exerts a phase shift only kink kink (x, t) = V+∞ (x + 1, t), V−∞ θ−∞ − θ+∞ 2I = . 1= (38) k k All the aforesaid can be employed for s = 0 as well, and an analogous analysis can be performed. In this case one has for v in (19) the pole solution
1 , lim v = 0, x1 →∞ x1 + ωt1 + θ (x2 , t2 ) with c = −ω and the relation −θx2 x2 θ (x2 , t2 ) = V∞ 2(1 + θx2 ) v=
Fig. 3. Elastic interaction with the pole. The chain curve indicates its position in the absence of perturbations.
(a = −θx2 and b = −θt2 now), which after two integrations with θ |x2 =+∞ = θ+∞ gives ! Z Z+∞ θ = θ+∞ − x2 +
V∞ dτ dx2 .
exp 2
(39)
x2
From this it follows immediately that an elastic interaction or θ |x2 =−∞ = const is possible only when Z+∞ V∞ dx2 = 0, I= −∞
and there is no phase shift for the pole. Fig. 3, where θ=
ex2 −t2 2[e2(x2−t2 ) + 1]
− 5,
demonstrates such a collision, while Fig. 4 with θ = − ln e−4x2 + 1 corresponds to the opposite situation. (The shelf appears between the pole and perturbation after the collision.) Previously the analysis for the simplest ‘expansion’ (7), V itself, has been made only with the assumption that the SME is of the type shown in (24), and the perturbation is some localized wave approaching the kink for an interaction, with estimation (36) being satisfied by the asymptotes. For other expansions we will have u=
0 X i=m
Wi (θx2 , θt2 , . . .)v i ,
m ∈ N,
(40)
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
205
rect investigation of the asymptotes should be considered instead. (Since one deals with localized waves, use of the related linearizations may be sufficient.) However, if (40) does not depend on θt2 , θx2 t2 , . . . (i.e., C absents in (7) possibly in view of a SME like (37), or when |θt2 | 1) then the asymptotes θx2 → 0 are always associated with localized perturbations. The only question here is the investigation of the possibility of degenerate cases like (34). Moreover, then from (7), (8), u = H (V , Vx , . . .)
(41)
− λ/6), and there (e.g., for the KdV, u = Vx − exists a direct generalization of the above results with regard to these transformations, although Eq. (41) may imply new degenerate cases. Finally, the approach can be generalized to multimanifolds expansions and expansions with non-Riccati ODEs [15] as well. These are important for the analogous descriptions of other types of solitons (e.g., the MKdV soliton is associated with two-manifolds [17]). In conclusion, we note that although our main aim here has been the description of interactions, the expressions found can be used for the direct construction of the solutions, because, in view of invariants (29), formulas (32) and (39) allow one to obtain θ from a perturbation evaluating separately. Also, from (29) it follows that θ θ − V−∞ − k, θx2 = sign k V+∞ V2
Fig. 4. Typical inelastic interaction with the pole. The shelf appears as a result. The chain curve indicates the pole position in the absence of perturbations.
and other expressions similar to (25) and (29), (30), respectively, in view of (25)–(28) or their analogues in the case of a pole. As for the soliton form, this is described by the polynomial Pm (V ) = u|θ=0 and can be either bellshape when Pm (−k/2) = Pm (k/2) (as for the KdV, e.g., u = −2V 2 + (C − 4S)/6, C − S + λ = 0 and λ = const) or kink-shape when Pm (−k/2) 6= Pm (k/2) possibly with additional minima and maxima at the central part. (The latter applies equally to pole solutions.) Respectively, the dependence on θx2 , . . . in (40) determines its deformation during an interaction besides a phase shift. As to the analysis of θ , integrations like (32) or (39) may be possible not always, so the problem of the di-
and we have one of the types of possible superposition laws, when more that two initial solutions are needed to derive a new one [8] (V±∞ are closely related to each other here). In [18] some KdV solutions were found in this manner. Generally, however, (40) may be too complicated. While, taking into account the SME, it can, in principle, be significantly simplified and become an algebraic one in θ or one of its derivatives.
4. The computer simulation of some models In this section a computer simulation is presented for some equations of the above class in order to compare its results with the theory. As has been shown, such equations should have solitonic properties. In particular, in models corresponding to V itself the kink solution should elasti-
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cally interact with localized waves. A number of popular integrable equations belong to this class and have this feature. The simplest of them are the MKdV with the SME, C − S = 0. Here, as other examples, we will consider equations of various types not presented widely yet in the literature and associated with another simple relation between S and C, namely C = λ + αS + βS 2 + Sx ,
λ, α, β = const .
(42)
The related equation for V is of the following general form: Vt + λVx + α V3x − 6V 2 Vx 1 − 4β V Vx2 + V 2 Vxx − V 5 x + Vx2 2x 2 4 + V4x + 4Vx V2x − V 3 3x = 0 (43) 3 and possesses the kink solution (30) with k4 k2 . ω = −k λ − α + β 2 4
(44)
In order to model the above interactions, the implicit finite-difference scheme Vin+1 − Vin (V + + Vx− ) α +λ x + fα − 2βfβ τ 2 2 + 2f = 0, + − − 6V −2 Vx+ − 12V − Vx− V + + V3x fα = V3x
+ 6V −2 Vx− , + + − + 2 V2x − 2V − Vx− V2x fβ = Vx− + V −2 V3x − − + V3x + 4V − V2x + 3Vx−2 − 5V −4 Vx+ − − + 2 V − V3x + 2Vx− V2x − 10V −3 Vx− V + − − − V −2 V3x − 4V − Vx− V2x − Vx−3
+ 15V −4 Vx− ,
+ + + 4 Vx− − V −2 V2x f = V4x − + 4 V2x − 4V − Vx− Vx+ − − − 8 V − V2x + Vx−2 V + + V4x − + 4V −2 V2x + 8V − Vx−2 , ± ± ± , V3x and V4x are the was employed. Here Vx± , V2x standard approximations of the related derivatives on ± ± ± ) or seven points (V3x , V4x ), and the five (Vx± , V2x superscripts ‘−’ and ‘+’ correspond to the nth and
(n + 1)th layers, respectively (e.g., Vx− = (uni−2 − 8uni−1 + 8uni+1 − uni+2 )/12h). This scheme for the mesh function Vin (i = 0, N , n > 0 such that t = τ n and x = hi) of the accuracy O(τ 2 ) + O(h4 ) is supplemented by the appropriate boundary conditions u|x=0, u|x=L = const, ux |x=0,L = uxx |x=0,L = 0, approximated also with the order of accuracy O(h4 ). The problem is then reduced to the set of the algebraic equations n n n n + bin Vi−2 + cin Vi−1 + din Vin + ein Vi+1 ain Vi−3 n n + fin Vi+2 + gin Vi+3 = hni
with a band matrix simple for solving. The scheme proposed appears to be economical, accurate and simple in its implementation. For β = 0 it is absolutely stable. However, the term S 2 in (42) introduces an instability, and without the damping due to Sx its simulation could be impossible. In the general case for β 6= 0 of order unity there is some value τ∗ depending on h, β and the disturbance amplitude such that for τ < τ∗ the scheme is stable. Before proceeding to the description of the experiments, we note only that the accuracy in the test calculations with various time and space intervals is in agreement with the estimates. Most of the results obtained were performed for h = 0.1 and τ = 0.005 with long double precision (18 figures), that gives accurate enough data. The simulations that were performed fully confirm the existence of solitonic properties for the kink solution in such models. Moreover, characteristic features of such interactions, namely a phase shift for the kink and reflection of the wave according to (29) (see the first addent there) after passage through the former are in agreement with the predictions and correspond to the scenario demonstrated by Fig. 1. The computations were carried out with perturbations of various forms and type (sign) and for various parameters. (The parameter λ determines the coordinate system velocity and was introduced for convenience.) The most significant of them are demonstrated in Figs. 5–7 (k = 2). Figs. 5 and 6 correspond to the choices (λ, α, β) = (1/2, 1, 0) with F (x) = −6/(ex + 1 + e−2x ) (31) and (−4, 0, 1) with F = −5/(e4x + 1 + e−2x ), respectively (h = 0.1, τ =
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
207
Fig. 5. Interaction of the kink with the localized wave (λ, α, β) = (1/2, 1, 0).
Fig. 6. Interaction of the kink with the localized wave (λ, α, β) = (−4, 0, 1).
0.00005 for the latter). The λ were chosen such that the kink itself is stationary in view of (44). As a result, both the phase shift and its asymptotic form after the interaction are easily seen here. Fig. 7 for (λ, α, β) = (0, 0, 0) with F = 5/(e2x + e−x ) demonstrates one highly instructive case. The situation corresponds to a purely dissipative system. Both the kink and perturbation themselves are immovable. There is no direct collision by itself, and the interaction with the perturbation’s tail lasts for all the time, and the superposition nature of the process is obvious.
Since I in (33) is conserved for Eq. (43), so an initial disturbance can be used for its calculation, and the SME is of the type (37), in the first two cases we also have the possibility to compare the theoretical values (38) for the phase shift with the experimental data. For the experiments in Figs. 5 and 6, one finally has the errors ε ≈ 0.15% (1theor = −5.5306) and ε ≈ 0.2% (1theor = −2.3044), respectively. These results are in good agreement with the simulation within the accuracy of the scheme and the not ideal boundary and linkage conditions on finite time and space intervals.
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Fig. 7. Interaction between the immovable kink and immovable dissipating perturbation (λ, α, β) = (0, 0, 0).
In the third case (38) is not applicable, because a direct collision is absent. Here, for the whole perturbation, 2I /k = 6.0460 and 1 = 2.8667. This indicates that about 50% of the perturbation nevertheless passed through the kink as expected from the form and location of the former and seen from Fig. 7.
5. Conclusion Previously, the concept of multidimensional superposition has been introduced for PDEs, and the existence of soliton/kink solutions that interact elastically with localized waves has been shown for models of a certain class. The same pertains equally to a pole solution if a perturbation satisfies some additional condition; otherwise a shelf appears. In so doing, the kink experiences only a phase shift, and the perturbation transfers from one state to another; whereas in an elastic pole–perturbation interaction both waves remain unchanged, and the behaviour of the pole is analogous to lumps in 2D systems.
Acknowledgements The author is grateful to Professor Alan Jeffrey for reading of the manuscript and Professor Robert
M. Miura for encouragement. He would also like to thank the referee for the useful remarks. References [1] C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Phys. Rev. Lett. 19 (1967) 1095. [2] M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. [3] R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 1424. [4] M. Wadati, K. Sawada, J. Phys. Soc. Jpn. 48 (1980) 312. [5] P.D. Gerber, IBM J. Res. Develop. 17 (1973) 314. [6] J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys. 24 (1983) 522. [7] O.V. Kaptsov, Nonlinear Anal. 19 (1992) 753. [8] R.L. Anderson, J. Harnad, P. Winternitz, Physica D 4 (1982) 164. [9] A.A. Alexeyev, J. Phys. A: Math. Gen. 32 (1999) 4419. [10] O.V. Kaptsov, Eur. J. Appl. Math. 6 (1995) 265. [11] E.L. Mansfield, Differential Gröbner bases, Ph.D. thesis, University of Sydney, 1992. [12] G.J. Reid, A.D. Wittkopf, A. Boulton, Eur. J. Appl. Math. 7 (1996) 604. [13] R. Conte, Phys. Lett. A 140 (1989) 383. [14] M. Musette, R. Conte, J. Math. Phys. 32 (1991) 1450. [15] A.A. Alexeyev, J. Phys. A: Math. Gen. 33 (2000) 1873. [16] A.M. Vinogradov, I.S. Krasilshchik, V.V. Lychagin, Introduction to Geometry of Nonlinear Differential Equations, Nauka, Moscow, 1986 (in Russian). [17] P.R. Gordoa, P.G. Estévez, Teor. Mat. Fiz. 99 (1994) 370. [18] A.A. Alexeyev, A multidimensional superposition principle: classical solitons I, Preprint, Karlsruhe University, Evolve-L Archive 99-00010.
Physics Letters A 278 (2001) 198–208 www.elsevier.nl/locate/pla
A multidimensional superposition principle: classical solitons I Alexander A. Alexeyev Laboratory of Computer Physics and Mathematical Simulation, Research Division, Room 247, Faculty of Physics–Mathematics and Natural Sciences, Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya str., Moscow 117198, Russia Received 5 August 1999; received in revised form 22 May 2000; accepted 20 November 2000 Communicated by A.R. Bishop
Abstract A concept that easily explains both classical solitonic and more complex wave interactions is proposed for differential equations. Nonlinear PDEs associated with the Riccati equations via ‘truncated expansions’ are considered, and the existence of the KdV-type soliton/kink is shown. Other interactions, including inelastic ones, are indicated. 2001 Elsevier Science B.V. All rights reserved. PACS: 02.30; 03.40.K Keywords: Soliton; Nonlinear interaction; Superposition; Nonlinear PDE
1. Introduction Owing to the extensive studies of integrable nonlinear models performed during the previous decades, especially for such famous equations as the KdV, MKdV and SG equations, nowadays there exists the deep understanding of soliton behaviour in such systems. In many respects we owe this to the inverse scattering transform [1]. Although for an arbitrary potential the related equations and algebra are too difficult to handle, so ultimately asymptotic analysis is applied [2], in reality this is not essential. The theory gives a comprehensive description of the main features, and computer simulation can be used for refinements. What is more important, the approach itself is rather restrictive, and finding a suitable spectral problem and generalization to (n + 1) cases are also open questions at the present time. Because of this, a number of direct
E-mail address: [email protected] (A.A. Alexeyev).
techniques have been invented (see, e.g., [3–6]). They may be effective for the construction of solutions even when the ITS is not applicable. But they do not explain the nature of solitonic interactions, while in the IST solitons are embedded and emerge from the discrete spectrum. Thus the present Letter is an attempt to propose a direct concept of solitons in some sense alternative to the IST in which the notion of solitons would be incorporated by definition and be, perhaps, more tractable. In addition, a different view to solitonic interactions could indicate novel aspects of nonlinear dynamics as a whole.
2. Theoretical background Let us suppose that there is some nonlinear PDE, assumed to be in (1 + 1) dimensions, and for the simplicity not depending explicitly on the independent
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 7 5 - 1
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
variables, ∂ ∂ , ; u(x, t) = 0, E ∂x ∂t
(1)
and construct its (2 + 2)-dimensional adjoint equation by making the formal replacement ∂ ∂ ∂ ∂ + , + ; u(x1 , x2 , t1 , t2 ) = 0. (2) E ∂x1 ∂x2 ∂t1 ∂t2 Keeping in mind that u(x, t) = u(x1 , x2 , t1 , t2 )|x1 =x2 =x, t1 =t2 =t
(3)
we could investigate the latter equation instead of the original one, and because of non-uniqueness of map (3) any subset of the related solutions of (2) with a suitable representation in terms of x1 , x2 , t1 , t2 could be used. In particular, we can restrict ourselves to solutions of (2) associated with additional compatible differential constraints (see, e.g., [7]) such that the equations only with the derivatives with respect to x1 , t1 , ∂ ∂ , ; u(x1, x2 , t1 , t2 ) = 0, i = 1, n1 , G1i ∂x1 ∂t1 (4) and/or x2 , t2 , ∂ ∂ , ; u(x1, x2 , t1 , t2 ) = 0, j = 1, n2 , G2j ∂x2 ∂t2 (5) can be isolated from them, possibly along with other relations ∂ ∂ ∂ ∂ , , , ; u(x1, x2 , t1 , t2 ) = 0, G3l ∂x1 ∂x2 ∂t1 ∂t2 l = 1, n3 ,
n1 , n2 , n3 ∈ N.
(6)
We lose nothing here from the viewpoint of u(x, t) in (3). However, in view of (4)–(6) the variables x1 , t1 and x2 , t2 appear to be separated, the full solution (process) u(x1 , x2 , t1 , t2 ) and, respectively, u(x, t) can be presented as superposition of the two separated solutions (processes) proceeding in the different (x1 , t1 )and (x2 , t2 )-spaces and could be obtained by consecutive solving with respect to x1 , t1 and x2 , t2 . Such a paradigm will be called a multidimensional superposition principle in contrast to a conventional algebraic superposition for ODEs [8]. In so doing, an observer in our (1 + 1) real world sees only a projection of events occurring in that (2 + 2)-dimensional
199
world; only the section x1 = x2 , t1 = t2 is accessible, and these processes are not distinguishable for him. Although the consecutive integration mentioned can really simplify the construction of solutions in a number of cases, the properties of such multidimensional systems themselves and their reflection in the features of the projected solution u(x, t) can appear to be more essential from the viewpoint of understanding and describing nonlinear phenomena and their applications. For instance, if u corresponds to an elastic interaction of two localized waves, then their mutual modulations and deformations are determined by the associated parts of (4)–(6), and the asymptotic states before and after the interaction, u(x1 , x2 , t1 , t2 ) → ubefore, after (x1 , t1 ), u(x1 , x2 , t1 , t2 ) → ubefore, after (x2 , t2 ), are described by the related degeneracies of (4)– (6). From these overdetermined systems one can judge what classes of waves could take part in such interactions and what changes of them are possible in principle. As for the former, several nontrivial situations are possible. Firstly, one degenerate system describes a special solution, and another is reduced to the only equation of the initial type (1) (a conventional soliton case). Secondly, both such systems describe some special solutions. As a result, ‘solitons’ will interact with one perturbations elastically and inelastically with others. Thirdly, both systems are reduced to equations of the form (1). Linear equations are trivially shown to be among such models. As for changes in the waves, an interaction switches a wave from one state to another, in the simplest cases differing by the values of its parameters. Obviously, classical soliton interactions corresponds to changing only a trivial parameter, namely the phase. While in [9] the case with the wavenumbers and velocities changes is adduced. Note in conclusion that all the aforesaid can be extended to Eqs. (1), (2) and (4)–(6) with explicit dependence on the independent variables. In these cases, however, it is necessary in addition to take into account known functions (coefficients) and their properties in (4)–(6) and (1), (2) (for (4)–(6) to reveal the degenerations, and for (1), (2) to chose corresponding conditions directly in the form f (x1 , x2 , t1 ,
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t2 )|x1 =x2 =x, t1 =t2 =t = f (x, t) or when possible through the introduction of auxiliary differential equations that are to be satisfied).
3. The multidimensional superposition and singular manifold technique Both finding sets of differential constraints possessed by an equation itself [7,10], and the investigation of overdetermined systems [11,12] in particular are still open questions for contemporary mathematics as a whole, and frequently possible in full volume only for simple enough cases. While within the framework of the invariant version of the so-called singular manifold approach [13] a number of nonlinear PDEs of various kinds can be reduced by use of the series u(x, t) =
0 X
wi (S, C, Sx , Cx , St , Ct , . . .)V i (x, t),
i=m
m ∈ N,
(7)
to the system of the ODEs Vx = −V 2 − S/2, 1 Vt = CV 2 − Cx V + (CS + Cxx ), 2 S = S(x, t), C = C(x, t),
(8) (9)
with the formal compatibility condition St + Cxxx + 2SCx + CSx = 0 and some extra restriction to the functions S and C, ∂ ∂ , ; S, C; x, t = 0 M (10) ∂x ∂t (a ‘singular manifold equation’), which and determines the type of an equation under consideration. Further study of those system may arrive at (auto)Bäcklund transformations, Lax pairs, etc. for the initial equations [13,14]. Obviously, various NPDEs may be associated with one and the same SME via different expansions (7). (The simplest one of them is an equation in V itself. When an SME is of the form C = M(S, Sx , St , . . . ; x, t), it directly follows from (9). In other cases, set (8)–(10) can in principle [11,12] be transformed to isolate an equation only in V .) By this means the properties of all these equations appear to closely connected, and the explicit maps (see the famous Miura one) can be derived using (7)–(10). As
was shown in [15], all of this technique itself is of an algebraic nature, and more complicated structures than (7)–(9) should frequently be used for a successful outcome. Here our attention will be concentrated on the above case, and the function V with (8), (9) will be considered from the viewpoint proposed in the previous section. The adjoint equations (2) for (8), (9) are as follows: Vx1 + Vx2 = −V 2 − S/2,
(11)
Vt1 + Vt2 = CV − (Cx1 + Cx2 )V 1 + (CS + Cx1 x1 + 2Cx1 x2 + Cx2 x2 ), 2 2
S = S(x1 , x2 , t1 , t2 ),
C = C(x1 , x2 , t1 , t2 ).
(12)
Each of them in turn can be split in the obvious manner (4), (5): Vx1 = A(V ; x1, x2 , t1 , t2 ),
(13)
Vt2 = B(V ; x1, x2 , t1 , t2 ),
(14)
with, respectively, Vx2 = −V 2 − S/2 − A,
(15)
Vt1 = CV − (Cx1 + Cx2 )V 1 + (CS + Cx1 x1 + 2Cx1 x2 + Cx2 x2 ) − B (16) 2 from (11), (12) without loss of generality, because (13) and (14) can simultaneously be considered as the intermediate integrals [16] of more common equations like G(Vkx1 , V(k−1)x1 , . . .) = 0, k > 2. The compatibility conditions for (13)–(16) present a set of six non-homogeneous quasilinear PDEs for A and B, and from their form it is naturally to seek solutions primarily among the class of second order polynomials with respect to V . To simplify the resulting conditions and for our later purposes, we will reduce system (13)–(16) by means of the linear transformation 2
V = w1 (x1 , x2 , t1 , t2 )v(x1 , x2 , t1 , t2 ) + w0 (x1 , x2 , t1 , t2 ) to the form where one of the subsystems, say (13), (16), would be of the original type (8), (9) again: vx1 = −v 2 − s/2, vt1 = cv 2 − cx1 v + (cs + cx1 x1 )/2,
(17)
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
vx2 = av 2 − ax1 v + (as + ax1 x1 )/2, vt2 = bv − bx1 v + (bs + bx1 x1 )/2; 2
(18)
here s, c, a and b are functions of x1 , x2 , t1 , t2 . (In so doing, V and S, C are as follows: V = (1 − a)v +
ax1 ax1 + ax2 + , 2 2(1 − a)
(19)
S = s(a − 1)2 + (a − 2)ax1x1 − ax1 x2 (ax1 )2 3 ax1 + ax2 2 − − 2 2 a−1 ax1 x1 + 2ax1x2 + ax2 x2 + ax1 (ax1 + ax2 ) , + a−1 b+c . C= 1−a
(20) (21)
It is necessary to equate the coefficients at the powers of v to zero in (8) and (9) after the substitution.) The above compatibility conditions then take the simple form: st1 + cx1 x1 x1 + 2scx1 + csx1 = 0, st2 + bx1 x1 x1 + 2sbx1 + bsx1 = 0, sx2 + ax1 x1 x1 + 2sax1 + asx1 = 0, at2 + bax1 − abx1 − bx2 = 0, (22)
System (22) always possesses trivial solutions for s, c and a, b such that s, c = const,
Here the phase θ is an arbitrary function depending on x2 , t2 ; and in view of the rest equations (18), θx2 = −ka, θt2 = −kb. It also linked with S and C by (20), (21) such that θx 2 3 θx2 x2 2 θx2 x2 x2 − + , (27) S =s 1+ 2 k 2 k + θx2 k + θx2 kc − θt2 . C= (28) k + θx2
θ (x2 , t2 ) = lim V V±∞ x1 →±∞
=
θx2 x2 ±sign k (k + θx2 ) − , 2 2(k + θx2 )
(29)
see (26). Secondly, if for x2 → −∞ or/and x2 → +∞ the phase θ has the asymptotic nature
at2 − bx2 = 0, ax1 = at1 = bx1 = bt1 = 0.
Let us construct the related solution v from (17), (18) and, respectively, V (19), starting first with the case s 6= 0. Denoting the wavenumber, frequency and phase by k, ω and θ (s = −k 2 /2, c = −ω/k), one has for v from (17) the well-known kink solution in the (x1 , t1 )space and, as a whole, θx2 x2 θx2 , v− V = 1+ k 2(k + θx2 ) kx1 + ωt1 + θ (x2 , t2 ) k , v = tanh (25) 2 2 with |k| lim v = ± . (26) x1 →±∞ 2
The obtained solution (25) for V can be interpreted simultaneously both as the kink deformed and modulated by the perturbation and vise versa. And as a result, it has the following properties. Firstly,
at1 + cax1 − acx1 − cx2 = 0, ct2 + bcx1 − cbx1 − bt1 = 0.
201
(23)
As will be shown below, this explains the existence and properties of solitons in models associated with SMEs compatible with (23), for example, in view of the forms of (20), (21) that do not depend explicitly on x and t: ∂ ∂ , ; S, C = 0. M (24) ∂x ∂t In so doing, the models themselves may be of a quite different type.
θ±∞ =
lim θ = const,
x2 →±∞
then kink (x1 , t1 ) = V±∞
lim V
x2 →±∞
kx1 + ωt1 + θ±∞ k tanh . (30) 2 2 In so doing, s and c should obviously satisfy the SME of the same form as S and C themselves. So, in the different regions the solution can degenerate to solutions (29) or (30) depending only on x1 , t1 or =
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x2 , t2 . This also implies that combinations (29) for θ θ (x, t) always generate the solutions (invariants) V±∞ of the original equation to V (x, t), corresponding to the perturbation evaluating in the absence of the kink. The same is true for (30). (For a concrete SME this can be verified by a direct substitution, of course.) It is necessary to stress here that the above transitions in (25) appear to be permitted by the construction, due to the established separation of the variables. Simply, for an arbitrary substitution, θ would be bounded and not independent on x1 , t1 . Solution (25) particular to (11), (12) and corresponding to their above splitting simultaneously is the multidimensional presentation for solutions of the original system (8), (9). Using this consider the following problem. Assume that we have the initial data (for definiteness, the kink is disposed to the right and k > 0) kink (x, 0), V (x, 0) = F (x − θ+∞ ) + V+∞
θ+∞ → −∞,
(31)
with some function F (x) that is arbitrary but such that limx→∞ F (x) = 0, so that the x-axis is divided into two areas involving either only the localized kink ). perturbation or only the kink (F − k/2 and V+∞ Supposing that they approach each other, what will happen to this kink and localized wave at t → +∞ after their collision? On the strength of the aforesaid we can establish the consistency between the related solution V (x, t) and expression (25) by setting θ (x2 , 0) F (x2 − θ+∞ ) = V−∞
and will consider the related function θ or, more precisely, its asymptotes, in order to answer the question. Integrating (29) twice with limx2 →+∞ θ = θ+∞ as the boundary (linkage) condition, we have " Z Z+∞ F (τ ) dτ θ = −kx2 + ln k exp θ+∞ + 2 !
x2
#
+ kx2 dx2 + C1 , C1 = const .
(32)
It is easy to see that there are only two cases for x2 → −∞ — the first at C1 = 0: lim θ = θ+∞ + 2I,
x2 →−∞
Z+∞ I= F (τ ) dτ,
(33)
−∞
and the second at C1 6= 0: θ ∼ ln C1 − kx2 + x2 → −∞.
exp(θ+∞ + 2I + kx2 ) , C1 (34)
In the last case, however, from (29), k exp(θ+∞ + 2I + kx2 ) , V−∞ ∼ − − 2 2C1 x2 → −∞
(35)
(Note here that for many equations I is a conservation quantity.) What will happen after the collision in these cases? If one assumes that the changes of the asymptotes for the interaction time are negligible or, differently, ta.change tinteraction,
(36)
then the same asymptotes can again be used. (Taking into account the relation on s and c, it is easy, however, to see from (27), (28) that for a number of SMEs, in particular of the form C = M(S, Sx , . . .),
(37)
this assumption is superfluous because ∂θ∞ /∂t2 ≡ 0 in any case.) As a result, for the first case the perturbation θ kink now) will be separated and V−∞ and kink (V+∞ θ = k/2 again. (The linkage condition limx2 →−∞ V+∞ is satisfied identically.) The second case is specific. The situation is that separation after a collision is absent in this case (see the left θ asymptote). However, according to the dispersion relation of the appropriate linearized equation, the related perturbation’s tail (35) must propagate with the same speed as the kink, so that in reality the interaction itself will last infinitely long. (This may appear to be wrong in models with complicated dispersion relations or without such.) Both scenarios are depicted in Figs. 1 and 2 with immovable perturbations for clarity. Fig. 1 with k, ω = 1 and θ =
−8 +4 e x2 + 1
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
Fig. 1. Typical elastic interaction.
Fig. 2. ‘Capture’ case.
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A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
demonstrates the classical elastic interaction, whereas Fig. 2, where and θ = ln ex2 + 1 − x2 −
7 , e x2 + 1 could be described as the ‘kink capture’ because of the absence of separation after the collision. In these examples θ does not change with time. Although this is not necessarily the case in reality, nevertheless they illustrate the main and common features of such interactions. Generally, there takes place the following linkage between the end-states of a perturbation: k, ω = 1
+∞ θ θ = S−∞ Ttinteraction V+∞ . V−∞ +∞ is obvious (The essence of the switch operator S−∞ from (29), and Ttinteraction corresponds to the evolution of θ during an interaction.) The kink itself exerts a phase shift only kink kink (x, t) = V+∞ (x + 1, t), V−∞ θ−∞ − θ+∞ 2I = . 1= (38) k k All the aforesaid can be employed for s = 0 as well, and an analogous analysis can be performed. In this case one has for v in (19) the pole solution
1 , lim v = 0, x1 →∞ x1 + ωt1 + θ (x2 , t2 ) with c = −ω and the relation −θx2 x2 θ (x2 , t2 ) = V∞ 2(1 + θx2 ) v=
Fig. 3. Elastic interaction with the pole. The chain curve indicates its position in the absence of perturbations.
(a = −θx2 and b = −θt2 now), which after two integrations with θ |x2 =+∞ = θ+∞ gives ! Z Z+∞ θ = θ+∞ − x2 +
V∞ dτ dx2 .
exp 2
(39)
x2
From this it follows immediately that an elastic interaction or θ |x2 =−∞ = const is possible only when Z+∞ V∞ dx2 = 0, I= −∞
and there is no phase shift for the pole. Fig. 3, where θ=
ex2 −t2 2[e2(x2−t2 ) + 1]
− 5,
demonstrates such a collision, while Fig. 4 with θ = − ln e−4x2 + 1 corresponds to the opposite situation. (The shelf appears between the pole and perturbation after the collision.) Previously the analysis for the simplest ‘expansion’ (7), V itself, has been made only with the assumption that the SME is of the type shown in (24), and the perturbation is some localized wave approaching the kink for an interaction, with estimation (36) being satisfied by the asymptotes. For other expansions we will have u=
0 X i=m
Wi (θx2 , θt2 , . . .)v i ,
m ∈ N,
(40)
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
205
rect investigation of the asymptotes should be considered instead. (Since one deals with localized waves, use of the related linearizations may be sufficient.) However, if (40) does not depend on θt2 , θx2 t2 , . . . (i.e., C absents in (7) possibly in view of a SME like (37), or when |θt2 | 1) then the asymptotes θx2 → 0 are always associated with localized perturbations. The only question here is the investigation of the possibility of degenerate cases like (34). Moreover, then from (7), (8), u = H (V , Vx , . . .)
(41)
− λ/6), and there (e.g., for the KdV, u = Vx − exists a direct generalization of the above results with regard to these transformations, although Eq. (41) may imply new degenerate cases. Finally, the approach can be generalized to multimanifolds expansions and expansions with non-Riccati ODEs [15] as well. These are important for the analogous descriptions of other types of solitons (e.g., the MKdV soliton is associated with two-manifolds [17]). In conclusion, we note that although our main aim here has been the description of interactions, the expressions found can be used for the direct construction of the solutions, because, in view of invariants (29), formulas (32) and (39) allow one to obtain θ from a perturbation evaluating separately. Also, from (29) it follows that θ θ − V−∞ − k, θx2 = sign k V+∞ V2
Fig. 4. Typical inelastic interaction with the pole. The shelf appears as a result. The chain curve indicates the pole position in the absence of perturbations.
and other expressions similar to (25) and (29), (30), respectively, in view of (25)–(28) or their analogues in the case of a pole. As for the soliton form, this is described by the polynomial Pm (V ) = u|θ=0 and can be either bellshape when Pm (−k/2) = Pm (k/2) (as for the KdV, e.g., u = −2V 2 + (C − 4S)/6, C − S + λ = 0 and λ = const) or kink-shape when Pm (−k/2) 6= Pm (k/2) possibly with additional minima and maxima at the central part. (The latter applies equally to pole solutions.) Respectively, the dependence on θx2 , . . . in (40) determines its deformation during an interaction besides a phase shift. As to the analysis of θ , integrations like (32) or (39) may be possible not always, so the problem of the di-
and we have one of the types of possible superposition laws, when more that two initial solutions are needed to derive a new one [8] (V±∞ are closely related to each other here). In [18] some KdV solutions were found in this manner. Generally, however, (40) may be too complicated. While, taking into account the SME, it can, in principle, be significantly simplified and become an algebraic one in θ or one of its derivatives.
4. The computer simulation of some models In this section a computer simulation is presented for some equations of the above class in order to compare its results with the theory. As has been shown, such equations should have solitonic properties. In particular, in models corresponding to V itself the kink solution should elasti-
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cally interact with localized waves. A number of popular integrable equations belong to this class and have this feature. The simplest of them are the MKdV with the SME, C − S = 0. Here, as other examples, we will consider equations of various types not presented widely yet in the literature and associated with another simple relation between S and C, namely C = λ + αS + βS 2 + Sx ,
λ, α, β = const .
(42)
The related equation for V is of the following general form: Vt + λVx + α V3x − 6V 2 Vx 1 − 4β V Vx2 + V 2 Vxx − V 5 x + Vx2 2x 2 4 + V4x + 4Vx V2x − V 3 3x = 0 (43) 3 and possesses the kink solution (30) with k4 k2 . ω = −k λ − α + β 2 4
(44)
In order to model the above interactions, the implicit finite-difference scheme Vin+1 − Vin (V + + Vx− ) α +λ x + fα − 2βfβ τ 2 2 + 2f = 0, + − − 6V −2 Vx+ − 12V − Vx− V + + V3x fα = V3x
+ 6V −2 Vx− , + + − + 2 V2x − 2V − Vx− V2x fβ = Vx− + V −2 V3x − − + V3x + 4V − V2x + 3Vx−2 − 5V −4 Vx+ − − + 2 V − V3x + 2Vx− V2x − 10V −3 Vx− V + − − − V −2 V3x − 4V − Vx− V2x − Vx−3
+ 15V −4 Vx− ,
+ + + 4 Vx− − V −2 V2x f = V4x − + 4 V2x − 4V − Vx− Vx+ − − − 8 V − V2x + Vx−2 V + + V4x − + 4V −2 V2x + 8V − Vx−2 , ± ± ± , V3x and V4x are the was employed. Here Vx± , V2x standard approximations of the related derivatives on ± ± ± ) or seven points (V3x , V4x ), and the five (Vx± , V2x superscripts ‘−’ and ‘+’ correspond to the nth and
(n + 1)th layers, respectively (e.g., Vx− = (uni−2 − 8uni−1 + 8uni+1 − uni+2 )/12h). This scheme for the mesh function Vin (i = 0, N , n > 0 such that t = τ n and x = hi) of the accuracy O(τ 2 ) + O(h4 ) is supplemented by the appropriate boundary conditions u|x=0, u|x=L = const, ux |x=0,L = uxx |x=0,L = 0, approximated also with the order of accuracy O(h4 ). The problem is then reduced to the set of the algebraic equations n n n n + bin Vi−2 + cin Vi−1 + din Vin + ein Vi+1 ain Vi−3 n n + fin Vi+2 + gin Vi+3 = hni
with a band matrix simple for solving. The scheme proposed appears to be economical, accurate and simple in its implementation. For β = 0 it is absolutely stable. However, the term S 2 in (42) introduces an instability, and without the damping due to Sx its simulation could be impossible. In the general case for β 6= 0 of order unity there is some value τ∗ depending on h, β and the disturbance amplitude such that for τ < τ∗ the scheme is stable. Before proceeding to the description of the experiments, we note only that the accuracy in the test calculations with various time and space intervals is in agreement with the estimates. Most of the results obtained were performed for h = 0.1 and τ = 0.005 with long double precision (18 figures), that gives accurate enough data. The simulations that were performed fully confirm the existence of solitonic properties for the kink solution in such models. Moreover, characteristic features of such interactions, namely a phase shift for the kink and reflection of the wave according to (29) (see the first addent there) after passage through the former are in agreement with the predictions and correspond to the scenario demonstrated by Fig. 1. The computations were carried out with perturbations of various forms and type (sign) and for various parameters. (The parameter λ determines the coordinate system velocity and was introduced for convenience.) The most significant of them are demonstrated in Figs. 5–7 (k = 2). Figs. 5 and 6 correspond to the choices (λ, α, β) = (1/2, 1, 0) with F (x) = −6/(ex + 1 + e−2x ) (31) and (−4, 0, 1) with F = −5/(e4x + 1 + e−2x ), respectively (h = 0.1, τ =
A.A. Alexeyev / Physics Letters A 278 (2001) 198–208
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Fig. 5. Interaction of the kink with the localized wave (λ, α, β) = (1/2, 1, 0).
Fig. 6. Interaction of the kink with the localized wave (λ, α, β) = (−4, 0, 1).
0.00005 for the latter). The λ were chosen such that the kink itself is stationary in view of (44). As a result, both the phase shift and its asymptotic form after the interaction are easily seen here. Fig. 7 for (λ, α, β) = (0, 0, 0) with F = 5/(e2x + e−x ) demonstrates one highly instructive case. The situation corresponds to a purely dissipative system. Both the kink and perturbation themselves are immovable. There is no direct collision by itself, and the interaction with the perturbation’s tail lasts for all the time, and the superposition nature of the process is obvious.
Since I in (33) is conserved for Eq. (43), so an initial disturbance can be used for its calculation, and the SME is of the type (37), in the first two cases we also have the possibility to compare the theoretical values (38) for the phase shift with the experimental data. For the experiments in Figs. 5 and 6, one finally has the errors ε ≈ 0.15% (1theor = −5.5306) and ε ≈ 0.2% (1theor = −2.3044), respectively. These results are in good agreement with the simulation within the accuracy of the scheme and the not ideal boundary and linkage conditions on finite time and space intervals.
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Fig. 7. Interaction between the immovable kink and immovable dissipating perturbation (λ, α, β) = (0, 0, 0).
In the third case (38) is not applicable, because a direct collision is absent. Here, for the whole perturbation, 2I /k = 6.0460 and 1 = 2.8667. This indicates that about 50% of the perturbation nevertheless passed through the kink as expected from the form and location of the former and seen from Fig. 7.
5. Conclusion Previously, the concept of multidimensional superposition has been introduced for PDEs, and the existence of soliton/kink solutions that interact elastically with localized waves has been shown for models of a certain class. The same pertains equally to a pole solution if a perturbation satisfies some additional condition; otherwise a shelf appears. In so doing, the kink experiences only a phase shift, and the perturbation transfers from one state to another; whereas in an elastic pole–perturbation interaction both waves remain unchanged, and the behaviour of the pole is analogous to lumps in 2D systems.
Acknowledgements The author is grateful to Professor Alan Jeffrey for reading of the manuscript and Professor Robert
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