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Uncertainty principle for the nonlinear waves of the Korteweg–de Vries equation
Chaos, Solitons and Fractals 32 (2007) 431–444 www.elsevier.com/locate/chaos
Uncertainty principle for the nonlinear waves of the Korteweg–de Vries equation Mikhail Kovalyov Department of Mathematics, University of Alberta, Edmonton, Alta., Canada T6G 2G1 Accepted 5 June 2006
Abstract The solutions of KdV studied in Kovalyov [Kovalyov M. On a class of solutions of KdV. J Differ Equations 1951;15:309–60; Kovalyov M. Basic motions of the Korteweg–de Vries equation, Nonlinear Anal Theory Methods Appl 1998;31(5/6):599–619; Kovalyov M. Modulating properties of harmonic breather solutions of KdV. J Phys A: Math Gen 1998;31:5117–28.] exhibit wave–particle behavior albeit somewhat different from the wave–particle behavior of quantum mechanics. Yet the balance between the wave and particle properties of these solutions is governed by an uncertainty principle remarkably similar to that of quantum mechanics. Ó 2006 Published by Elsevier Ltd.
1. Introduction The wave–particle duality of certain dynamical processes has been known for centuries. Some properties of light, for example, are best described if light is viewed as a sequence of waves, while others are easier to understand if it is viewed as a large collection of particles. The analytical mechanics of Hamilton and Lagrange provided the mathematical machinery for describing the motion of both particles and waves by the same equations. It was subsequently adapted for use in quantum mechanics where the motions of elementary particles exhibit both wave- and particle-like properties, [5,7] and further extended to some generalizations of quantum mechanics, [18]. The balancing act between wave and particle components of a quantum-mechanical motion is governed by the Bohr–Heisenberg uncertainty principle, which essentially states that the more wave-like a motion is the less particle-like it is and vice versa. The mathematical proof for the uncertainty principle is based on Fourier transform and as such is valid for any linear motion. It is, however, restricted to linear waves only. We will show in this paper that there is a class of nonlinear waves which satisfy an uncertainty principle whose mathematical formulation is practically identical to that of the Bohr–Heisenberg uncertainty principle, yet the very meaning of the parameters is considerably different. These nonlinear waves appear as solutions of the Korteweg–de Vries equation ut þ 6uux þ uxxx ¼ 0: A single such wave is given by
E-mail address: [email protected]. 0960-0779/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.06.050
ðKdVÞ
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
d2 sin 2kðc 4k2 t xÞ 2 ln p 12k t x 2k dx2 " #2 2 8k sin 2C cos 2C 1 2 ¼ þ 8k ; 2kðp 12k2 t xÞ sin 2C 2kðp 12k2 t xÞ sin 2C
uðt; xÞ ¼ 2
ð1:1Þ
where C = k(c 4k2t x), and constants k, c, p are correspondingly the spectral parameter, phase and displacement of the wave. These waves have been studied in a number of articles, e.g. [1–3,8–17,19–21]. Fig. 1.1 shows the structure of (1.1), with k = 1.0, c = 0.0 at time t = 0.1. The graph of (1.1) can be divided into three parts: a neighborhood of the always present singular point shown on the graph by the dotted line and two decaying oscillatory tails on both sides of the neighborhood of the singularity shown on the graph by solid lines. While in the neighborhood of the singularity (1.1) does not exhibit much of wave-like behavior, the two oscillatory tails appear to be very much wave-like. Nonlinear superposition of such nonlinear waves is given by the formula [10], 8 sin 2Cn 2 > > ; Cn ¼ kn ðcn 4k2n t xÞ; n ¼ m; < pn 12kn t x 2 d 2kn u ¼ 2 2 ln det D; Dnm ¼ ð1:2Þ > dx sinðCn Cm Þ sinðCn þ Cm Þ > : ; n 6¼ m; kn km kn þ km where n, m = 1, 2, 3, . . . , N, pn’s, cn’s and kn’s are constants. The derivation of (1.2) is rather long and thus is omitted here. The interested reader is referred to [10] for details. It is easy to show that function (1.2) has exactly N singularities. Indeed, let rank(x) be the dimension of the largest linear space on which the bilinear form N X
Dnm nm nn ¼
n;m¼1
N N N X X X sinðCn Cm Þ sinðCn þ Cm Þ ðpn 12k2n t xÞn2n þ nm nn þ nm nn : k k kn þ km n m n¼1 n;m¼1 n;m¼1 n6¼m
is negative definite. Since for large values of jxj N X
Dnm nm nn x
n;m¼1
N X
n2n ;
n¼1
rank(x) = 0 for large negative x and rank(x) = N for large positive x. Due to the easily verifiable inequality !2 N N X d X Dnm nm nn ¼ 2 nn sin Cn 6 0; dx n;m¼1 n¼1 rank(x) is an increasing function of x assuming integer values from a subset of 0, 1, 2, 3, . . . , N. The points of jumps of rank(x) are exactly the poles of u(t, x) and there are at least one and at most N of them. Sufficiently far away from all singularities the function u(t, x) given by (1.2) behaves like uðt; xÞ
N X n¼1
4kn sin 2Cn : pn 12k2n t x
ð1:3aÞ
If all pn’s are sufficiently large and have the same sign and x, t are restricted to a bounded set, say jxj, jtj 6 1, (1.3a) simplifies to uðt; xÞ
N X 4kn sin 2Cn : pn n¼1
ð1:3bÞ
Fig. 1.1. Graph of function (1.1) with k = 1.0, p = c = 0.0 at t = 0.1, x is measured along the horizontal axis, u along the vertical axis. For other values of k, p, c and t the graph has a similar structure.
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
Formula (1.2) is a nonlinear analogue of the Fourier sum solutions X uk cos 2kk ½cðkk Þ x 4k2k t
433
ð1:4aÞ
over appropriate subset of N
of the linear Korteweg–de Vries equation ut þ uxxx ¼ 0:
ðlKdVÞ
Just like appropriate limits of Riemann sum solutions (1.4a) of l KdV lead to integral type solutions of lKdV of the form Z ^uðkÞ cos 2k½cðkÞ x 4k2 t dk; ð1:4bÞ over appropriate subset of R
the appropriately taken limits of (1.2) lead to nonlinear analogues of (1.4b), [10]. As such limits are taken the singularities of (1.2) move to either 1 or +1 and the nonlinear superposition of the oscillatory tails forms miscellaneous solutions of KdV, just like the linear superposition of cos 2kk ½cðkk Þ x 4k2k t forms miscellaneous solutions of lKdV. The existence of the integrals in the linear case is assured by the phenomenon of linear interference, which essentially states that if A1 cos 2k1 ½c1 x 4k21 t
and
A2 cos 2k1 ½c1 x 4k21 t
ð1:5aÞ
are two solutions of lKdV then so is their linear superposition A12 cos 2k1 ½c1 x 4k21 t;
ð1:5bÞ
A12 ¼ A1 þ A2 :
ð1:5cÞ
where
Taking linear superposition of two solutions (1.5a) of lKdV with the same frequencies and phases simply amounts to the addition of their amplitudes. Fig. 1.2 illustrates linear superposition of two linear waves (1.5a). Just like linear interference is behind the mathematics of the linear case, a similar phenomenon of nonlinear interference is behind the mathematics of the nonlinear case, [10]. Since, due to vanishing denominators, it is impossible to simply take (1.2) with N = 2 and k2 = k1, c2 = c1, let us take (1.2) with N = 2 d2 sin 2k1 ðc1 4k21 t xÞ sin 2k2 ðc2 4k22 t xÞ 2 2 p2 12k2 t x uðt; xÞ ¼ 2 2 ln p1 12k1 t x dx 2k1 2k2 2 ) 2 2 2 2 sinðk1 ðc1 4k1 t xÞ k2 ðc2 4k2 t xÞÞ sinðk1 ðc1 4k1 t xÞ þ k2 ðc2 4k2 t xÞÞ ; ð1:6Þ k1 k2 k1 þ k2 and additional assumptions that c1 = c(k1), c2 = c(k2), where c(k) is a differentiable function of k, and pass to the limit as k2 ! k1. As k2 ! k1, formula (1.6) degenerates into the formula d2 sin 2k1 ðc1 4k21 t xÞ ; ð1:7Þ u12 ðt; xÞ ¼ 2 2 ln p12 12k21 t x dx 2k1 where the parameter p12 satisfies
Fig. 1.2. Two linear waves of type (1.5a) with k1 = k2 = 1.0, c1 = c2 = 0.0, A1 = 0.2, A2 = 0.1 at t = 0.0 are shown in thin solid and broken black; their linear superposition (1.5b) is shown in solid gray.
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4k1
¼
4k1
4k1
; ð1:8Þ p12 p1 p2 oðkcÞ ok k1 k1 k1 and oðkcÞ denotes the value of oðkcÞ at k = k1. Function (1.7) can be viewed as a nonlinear superposition of two nonlinear ok ok k1 waves d2 sin 2k1 ðc1 4k21 t xÞ 2 uj ðt; xÞ ¼ 2 2 ln pj 12k1 t x ; j ¼ 1; 2: ð1:9Þ dx 2k1 oðkcÞ ok
þ
oðkcÞ ok
Sufficiently far away from the singularities, each nonlinear wave (1.9) satisfies uj ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ ; pj oðkcÞ 12k21 t x ok
j ¼ 1; 2;
ð1:10aÞ
k¼k1
while their nonlinear superposition (1.7) satisfies u12 ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ : p12 oðkcÞ 12k21 t x ok
ð1:10bÞ
k¼k1
If p1, p2, p12 are sufficiently large and x, t are restricted to a finite domain, say jxj, jtj < 1, the estimates (1.10) simplify to uj ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ ; pj oðkcÞ ok
j ¼ 1; 2;
ð1:11aÞ
k¼k1
and u12 ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ : p12 oðkcÞ ok
ð1:11bÞ
k¼k1
Due to (1.11) we call expressions
4k1 oðkcÞ p1 ok j
, k1
4k1 oðkcÞ p2 ok j
, k1
4k1 oðkcÞ p1;2 ok j
asymptotic amplitudes of the nonlinear waves (1.9) and k1
(1.7). The argument just provided suggests that formulas (1.3) for the asymptotic behavior of (1.2) be replaced with uðt; xÞ
N X n¼1
4kn sin 2Cn pn 12k2n t x oðkcÞ ok
kn
and uðt; xÞ
N X 4kn sin 2Cn oðkcÞ n¼1 p n ok kn
in order to be consistent with nonlinear interference. For a more detailed discussion of nonlinear interference the reader is referred to [10]. Fig. 1.3 illustrates the nonlinear superposition of two nonlinear waves (1.2).
Fig. 1.3. Two nonlinear waves of type (1.2) with k1 = k2 = 1.0, c1 = c2 = 0.0, c(k) = 0.0, p1 = 30, p2 = 20 at t = 0.0 are shown in a thin solid and broken black; their nonlinear superposition is shown in a thick solid gray.
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
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We see that sufficiently far from the singularity, the oscillatory tails of (1.1) exhibit wave-like behavior similar to that of (1.5a), while on the other hand, the singular points of (1.1) behave just like point particles and their motion is that of point particles. One may conjecture that the further away from the singularity the more wave-like is the behavior of (1.1), but the closer to the singularity the more particle-like is the behavior of (1.1). The functions (1.1) contain within themselves both wave and particle motions, and in this aspect they are somewhat similar to wave–particles of quantum mechanics. Yet the very combination of the two motions is fundamentally different from that of modern quantum mechanicsR where wave–particles are described by R functions f(t, x) R and their Fourier transforms 1 f~ ðt; -Þ ¼ Rn f ðt; xÞei-x dx, appropriately normalized, i.e. Rn jf ðt; xÞj2 dx ¼ ð2pÞ jf~ ðt; -Þj2 d- ¼ 1. The balancing n Rn act between the wave and particle aspects of quantum-mechanical motion is governed by the Bohr–Heisenberg uncertainty principle 1 DxD- P ; 2 where ðDxÞ2 ¼
Z
jx xj2 jf ðt; xÞj2 dx; x ¼ Rn Z 1 j- -j2 jf~ ðt; -Þj2 d-; ðD-Þ2 ¼ 2p Rn
ð1:12aÞ Z
xjf ðt; xÞj2 dx; Z 1 -¼ -jf~ ðt; -Þj2 d2p Rn
ð1:12bÞ
Rn
ð1:12cÞ
and n is the dimensionality of the problem, which for our purpose, may be taken as equal to one. It is worthwhile to notice that the original Bohr–Heisenberg uncertainty principle was formulated by Bohr and Heisenberg in the form DxD- ¼ Oð1Þ
ð1:13Þ
based on physical arguments, some of which may be found in [6,7]. The nice mathematical definitions (1.12) were introduced considerably later. Assuming, without loss of generality, that f(t, x) is real and continuously differentiable and x ¼ - ¼ limx!1 xf 2 ðt; xÞ ¼ 0, the proof of (1.12a) is essentially given by the following string of formulas: Z þ1 Z þ1 1 jxj2 jf ðt; xÞj2 dx j-j2 jf~ ðt; -Þj2 dðDxD-Þ2 ¼ 2p 1 1 Z þ1 2 Z þ1 Z þ1 of ðt; xÞ2 of ðt; xÞ dx P dx jxj2 jf ðt; xÞj2 dx xf ðt; xÞ ¼ ox ox 1 1 1 Z þ1 2 Z Z 2 þ1 þ1 1 of 2 ðt; xÞ 1 o o 2 ¼ x x; dx ¼ ðxf 2 ðt; xÞÞ dx þ f ðt; xÞ dx 4 1 ox 4 1 ox ox 1 2 Z þ1 1 2 1 ¼ xf ðt; xÞjþ1 f 2 ðt; xÞ dx ¼ ; 1 þ 4 4 1 o where x; ox is the commutator of x and oxo . A more general and detailed proof of (1.12a) follows along these lines and may be found in [4,5]. According to (1.12), at each point x all frequencies - are present and D- is the average deviation of - from its average -, whereas Dx is the average deviation of x from its average x. The particle aspect of the motion is measured by the spread Dx from x. The smaller the spread Dx, the more particle-like is the motion. The wave aspect of the motion is measured by the spread D- which determines by how much the frequencies of the components ei-x of f(t, x) deviate from -. The smaller the spread D-, the more wave-like is the motion.
Fig. 1.4. Graph of function (1.1) with k = 1.0, p = c = 0.0 at t = 0.1; x is measured the along horizontal axis, u along the vertical axis. For other values of k, p, c and t the graph has a similar structure. The vertical dashed line indicates the singularity of the function.
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
The situation is somewhat different for (1.1) as shown in Fig. 1.4. At any given time t and at each point x only one frequency is present, but it varies with x and t and approaches 2k as x ! ± 1. The location of the singularity is also strictly defined. The uncertainty principle for (1.1) must provide an estimate for ð1:14Þ
DxDl;
where Dx is a measure of how far point P(t, x) is removed from the singularity and Dl is the measure of by how much the local frequency l(P) differs from 2k = limx!±1l(P).
2. The uncertainty principle for the Korteweg–de Vries equation in space-frequency variables To get a rough estimate on DxDl, rewrite the expression (1.1) as follows: uðt; xÞ ¼
32k2 ½sin C kðp 12k2 t xÞ cos C sin C ½2kðp 12k2 t xÞ sin 2C2
;
ð2:1Þ
where C = k(c 4k2t x). Let {. . ., x4, x2, x0, x2, x4, . . .} denote the solutions of sin C ¼ 0 and {. . ., x5, x3, x1, x1, x3, x5, . . .} denote the solutions of sin C kðp 12k2 t xÞ cos C ¼ 0: Clearly x2n ¼ c 4k2 t þ
pn : k
The elements of {. . ., x5, x3, x1, x1, x3, x5, . . .} are the solutions of cot kðx2nþ1 þ 4k2 t cÞ ¼
1 ; kðx2nþ1 þ 12k2 t pÞ
and for large jx2n+1j, satisfy x2nþ1 c 4k2 t þ
pn p 1 þ : k 2k k2 ðx2nþ1 þ 12k2 t pÞ
For large n, the frequency l between two adjacent zeros x2n and x2n+1 of u satisfies " # p p 2k 2 l p ; 2k 1 þ 1 2 jx2nþ1 x2n j 2k 1 pkðx þ12k k2 ðx þ12k pkðx2nþ1 þ 12k2 t pÞ 2 2 tpÞ tpÞ 2nþ1
2nþ1
which implies jl 2kj
4 pjx2nþ1 þ 12k2 t pj
and consequently 4 jl 2kjjx2nþ1 þ 12k2 t pj : p
ð2:2Þ
The same argument shows that between zeros x2n+1 and x2n+2, frequency l also satisfies (2.2). Eq. (2.2) provides a rough form of the uncertainty principle for (1.1). It is more convenient to formulate and consider the uncertainty principle for KdV in the frame of reference moving with velocity 12k2t, which is achieved by replacing x with n ¼ x þ 12k2 t p:
ð2:3Þ
By an appropriate shift in time variable t the constant c p can be made zero, so for the rest of this section it will be assumed to vanish. In the new variables function (1.1) becomes
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
( ) 2 8k2 sin 2C cos 2C 1 kn sin 2C þ 1 cos 2C ¼ 16k2 þ 8k2 2kn þ sin 2C 2kn þ sin 2C ð2kn þ sin 2CÞ2 ( ) kn sin 2G þ cos 2G 1 ¼ 16k2 ; ð2kn sin 2GÞ2
437
uðt; nÞ ¼
ð2:4Þ
where G = C = k(n 8k2t). In the coordinate system of n and t the singularity of (2.4) moves from n ¼ 2k1 to n ¼ 2k1 , disappearing upon reaching n ¼ 2k1 and at the same moment re-emerging at n ¼ 2k1 . The oscillatory tails move to the right sandwiched between two envelopes as shown in Fig. 2.1. Let envu(n), and envl(n) denote correspondingly the upper and lower envelopes; their existence, exact form and derivation will3 be discussed at the end of this section. In the system of coordinates x, t both envelopes move with velocity 12k2 ¼ dð2kÞ . This formula looks like group velocity in linear theory, only here just one frequency 2k is present. dð2kÞ Define if uðt; nÞ P 0; envu ðnÞ; ð2:5aÞ Aðt; nÞ ¼ envl ðnÞ; if uðt; nÞ 6 0: and
8 uðt;nÞ uðt; nÞ uðt; nÞ þ juðt; nÞj uðt; nÞ juðt; nÞj < envu ðnÞ ; sðt; nÞ ¼ þ ¼ ¼ : uðt;nÞ Aðt; nÞ 2envu ðnÞ 2envl ðnÞ ; envl ðnÞ
if uðt; nÞ P 0; if uðt; nÞ 6 0:
ð2:5bÞ
The function (2.4) then can be written as uðt; nÞ ¼ Aðt; nÞsðt; nÞ:
ð2:5cÞ
1 , 2k
If jnj > then js(t, n)j 6 1 and thus A(t, n) and s(t, n) can be viewed correspondingly as the amplitude and oscillatory factor of (2.4). For jnj 6 2k1 , however, js(t, n)j might become greater than one and thus it loses its interpretation as the oscillatory factor. The shape of s(t, n) is shown in Fig. 2.2. Define the frequency l(t, n), Dl and Dn of u(t, n) to be correspondingly
Fig. 2.1. Graph of function (2.4) together with two envelopes, plotted with k = 1.0, c = 0.0 at t = 0.2; n is measured along the horizontal axis, u, envu(n), envl(n) along the vertical axis. The function itself is shown in thick solid gray, the upper envelope in solid black and the lower envelope in dotted black. Note that the upper envelope consists of three disjoint parts defined correspondingly on 1; 2k1 , 2k1 ; 2k1 , and 2k1 ; þ1 : For other values of k, c and t, the graph has a similar structure.
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
Fig. 2.2. The graph of s(t, n) plotted with k = 1.0, c = 0.0 at t = 0.7, n is measured along the horizontal axis, s(t, n) along the vertical axis. For jnj P 2k1 ; js(t, n)j 6 1. For jnj 6 2k1 ; s(t, n) might become >1. For other values of k, c and t the graph has a similar structure.
o lðt; nÞ ¼ arcsin sðt; nÞ; on
ð2:6aÞ
Dl ¼ jlðt; nÞ 2kj;
ð2:6bÞ
sin 2G sin 2kðn 8k2 tÞ ¼n : Dn ¼ n 2k 2k
ð2:6cÞ
The product DnDl oscillates and due to its rather complicated form does not appear to be of much interest. What is of
3R p 3R p p 3 R more interest are the averaged quantities 4kp 04k3 jDnj dt 4kp 04k3 jDlj dt and 4kp 04k3 jDnDlj dt whose graphs are shown in Figs. 2.3 and 2.4. Changing the value of k scales the graphs horizontally but not vertically. Based on the graphs the following conclusion can be drawn.
3R p 3 R p 3 3 Fig. 2.3. Graph of 4kp 04k jDnj dt 4kp 04k jDlj dt plotted for k = 1.0; n is measured along the horizontal axis, the value of
3R p 3R p 4k 4k 4k3 4k3 0 jDnj dt 0 jDlj dt along the vertical axis. For other values of k the graph scales horizontally but not vertically. p p
p p 3 R 3 3 R 3 Fig. 2.4. Graph of 4kp 04k jDnDlj dt plotted for k = 1.0, n is measured along horizontal, the value of 4kp 04k jDnDlj dt along vertical. For other values of k the graph scales horizontally but not vertically.
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
439
Uncertainty principle for KdV in space-frequency variables Let Dl and Dn be defined by (2.6). Then there exist positive constants U1, U2 such that 3Z p
3Z p
4k3 4k3 4k 4k jDnj dt jDlj dt P U 1 p 0 p 0
ð2:7aÞ
and 4k3 p
Z
p 4k3
jDnDlj dt P U 2 :
ð2:7bÞ
0
Moreover as n approaches infinity, 3 Z p
3Z p
3 Z p
3Z p
4k3 4k3 4k3 4k3 4k 4k 4k 4k 4 jDnj dt jDlj dt ¼ lim jDnj dt jDlj dt ¼ lim n!þ1 n!1 p p 0 p 0 p 0 p 0
ð2:8aÞ
and lim
n!þ1
4k3 p
Z
p 4k3
jDnDlj dt ¼ lim
n!1
0
4k3 p
Z
p 4k3
0
4 jDnDlj dt ¼ : p
ð2:8bÞ
It is somewhat remarkable that the inequalities (2.7) hold not only for jnj P 2k1 but also for jnj < 2k1 ; where s(t, n) > 1 and arcsin s(t, n) is complex. The exact numerical values of the constants U1, U2 are not known. Numerical computations yield U1 0.7488 (attained at n 0.6371) and U2 0.7894 (attained at n 0.6598). The conclusions (2.7) are drawn based on the graphs in Figs. 2.3 and 2.4 and the fact that changing the value of k in (1.1) and (2.4) results in horizontal re-scaling of the graphs in Figs. 2.3 and 2.4 with no vertical re-scaling. No attempt was made to prove (2.7) analytically, although with due diligence an analytical proof should not be difficult. The main difficulty in obtaining an analytical proof of (2.7) is due to a rather complicated analytical form of the function s(t, n), which itself is due to rather complicated formulas arising in the computation of the envelopes. Formulas (2.8) can be either deduced from the graphs in Figs. 2.3 and 2.4 or obtained analytically from (2.2). The graphs in Figs. 2.3 and 2.4 were obtained using independently Maple and Matlab, the former being unexpectedly much faster while the latter provided better quality graphs. Computation of envelopes Since u(t, n) contains n in the combination kn only, it suffices, without loss of generality, to obtain envelopes for k = 1. The general case then is recovered by multiplying the final answer by k2 and replacing n with kn. The envelopes of u(t, n) are obtained by eliminating t from the system 8 16ðn sin 2G þ cos 2G 1Þ > > ; > < uðt; nÞ ¼ ð2n sin 2GÞ2 ð2:9Þ > ou 2n2 cos 2G þ n cos 2G sin 2G 2n sin 2G þ cos2 2G 2 cos 2G þ 1 > > ¼ 0; ¼ : oG 2ð2n sin 2GÞ3 where the second equation is due to the identity the system to 8 2 < u ¼ 8ðn cot 2G1Þðcot Gþ1Þ ; ðn½cot Gþ1cot GÞ2 : cot4 G cot3 G 3 cot G þ n
n
2 n2
ou ot
2
ou G1 2 cot G ¼ 8 oG . Substitutions cos 2G ¼ cot and sin 2G ¼ cot 2 Gþ1 simplify cot2 Gþ1
1 ¼ 0:
ð2:10Þ
Solving the second equation for cotG and substituting the answer into the first equation one obtains envelopes of (2.4). The second equation is a fourth order equation in cotG equivalent to a fourth order equation in the standard form in the variable ðcot G 4n1 Þ:
4
2
1 3 1 1 3 1 5 3 2 cot G þ 1 ¼ 0: cot G þ 2 cot G 4n 4n 4n 8n 8n3 n 4n 256n4 The cubic resolvent in variable z for this equation is, [22],
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
2
3 5 3 1 3 2 z z þ þ 4 þ þ ¼0 4n2 n2 16n4 8n3 n
whose reduced form in variable z þ 4n12 is
3
1 5 1 8 2 z þ 2 þ 2 þ 4 ¼ 0: zþ 2 þ 4 2 4n n 4n n n z3 þ
Following the standard procedure outlined in [22], the roots of the cubic resolvent are obtained in the form vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 1u 108 27 720 2196 273 81 2 t 2 4 þ 3 192 2 þ 4 þ 6 þ 8 z1 ¼ 3 n n n n n n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s u 3 1u 108 27 720 2196 273 81 1 2 t 2 4 3 192 2 þ 4 þ 6 þ 8 2 ; þ 3 n n n n n n 4n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi!2 u 3 1 1 i 3 u 108 27 720 2196 273 81 2 t þ 2 4 þ 3 192 2 þ 4 þ 6 þ 8 z2 ¼ 3 2 2 n n n n n n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ! pffiffiffi u 3 1 1 i 3 u 108 27 720 2196 273 81 1 2 t þ 2 4 3 192 2 þ 4 þ 6 þ 8 2 ; þ 3 2 2 n n n n n n 4n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi!u 3 1 1 i 3 u 108 27 720 2196 273 81 2 t þ 2 4 þ 3 192 2 þ 4 þ 6 þ 8 z3 ¼ 3 2 2 n n n n n n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ! u pffiffiffi 2 3 1 1 i 3 u 108 27 720 2196 273 81 1 2 t þ 2 4 3 192 2 þ 4 þ 6 þ 8 2 ; þ 3 2 2 n n n n n n 4n which provide four possible choices for the value of cot G: p ffiffiffiffi p ffiffiffiffi pffiffiffiffi 2 z1 þ 2 z2 þ 2 z3 1 signðnÞ þ ; c1 ¼ 4n 2 p ffiffiffiffi p ffiffiffiffi pffiffiffiffi 2 z1 2 z2 2 z3 1 c2 ¼ signðnÞ þ ; 2 4n pffiffiffiffi pffiffiffiffi pffiffiffiffi 2 z1 þ 2 z2 2 z3 1 signðnÞ þ ; c3 ¼ 4n 2 pffiffiffiffi pffiffiffiffi pffiffiffiffi 2 z1 2 z2 þ 2 z3 1 c4 ¼ signðnÞ þ ; 4n 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n where signðnÞ ¼ jnj and 3 expression; 2 expression denote the roots of the corresponding order of the expression whose 2
cot G1Þðcot Gþ1Þ arguments are closest to that of the expression. Substitution of c1 and c2 for cot G in u ¼ 8ðn yields corðn½cot2 Gþ1cot GÞ2
respondingly the upper and lower envelopes. Quantities c3 and c4 are redundant and should be discarded.
3. The uncertainty principle for the Korteweg–de Vries equation in time-frequency variables In the previous section we obtained a nonlinear uncertainty principle relating the local spacial frequency l of (1.1) at a point x at time t and the distance from x to the singularity at the same time. In this section we discuss a nonlinear uncertainty principle relating the local time-frequency x of (1.1) at a point x at time t and the length of time required for the singularity to arrive at this point. Just like in the previous section we first get a rough estimate for DtDx. Recall that from (2.1): uðt; xÞ ¼
32k2 ½sin C kðp 12k2 t xÞ cos C sin C ½2kðp 12k2 t xÞ sin 2C2
;
where C = k(c 4k2t x). Let {. . ., t4, t2, t0, t2, t4, . . .} denote the roots of
ð3:1Þ
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
441
sin C ¼ sin kðc 4k2 t xÞ ¼ 0 and {. . ., t5, t3, t1, t1, t3, t5, . . .} denote the roots of sin C kðp 12k2 t xÞ cos C ¼ sin kðc 4k2 t xÞ kðp 12k2 t xÞ cos kðc 4k2 t xÞ ¼ 0: Clearly 4k2 t2n ¼ c x þ
pn : k
The elements of {. . ., t5, t3, t1, t1, t3, t5, . . .} are the solutions of cot kðx þ 4k2 t2nþ1 cÞ ¼
1 ; kðx þ 12k2 t2nþ1 pÞ
and for large jt2n+1j, satisfy 4k2 t2nþ1 c x þ
pn p 1 : þ k 2k k2 ðx þ 12k2 t2nþ1 pÞ
For large n, the frequency x between two adjacent zeros t2n and t2n+1 of u satisfies " # p 4k2 p 8k3 2 3 x p ; 8k 1 þ 1 jt2nþ1 t2n j 2k k2 ðx þ12k 1 pkðxþ12k22 t pÞ pkðx þ 12k2 t2nþ1 pÞ 2 tpÞ 2nþ1
2nþ1
which implies jx 8k3 j
16k2 pjx þ 12k2 t2nþ1 pj
and consequently x p 4 : þ t jx 8k3 j 2nþ1 3p 2 12k
ð3:2Þ
The same argument shows that between zeros t2n+1 and t2n+2 frequency s also satisfies (3.2). Just like as in the case with the space-frequency uncertainty principle obtained in the previous section, the time-frequency uncertainty principle is better visualized in modified coordinates x and xp s¼tþ : ð3:3Þ 12k2 By an appropriate shift in space variable x the constant c p3 can be made zero and so for the rest of this section it will be assumed to vanish. In the new variables, function (1.1) becomes ( ) 3 2 12k s sin 2 þ cos 2 1 uðs; xÞ ¼ 16k ; ð3:4Þ ð24k3 s sin 2 Þ2 where ¼ C ¼ k 23 x þ 4k2 s : The last term in (3.4) has the same form as the last term in (2.4). It can be obtained form (2.4) by replacing n and G correspondingly with 12k3s and . Thus the two envelopes for (3.4) can be obtained by simply replacing n with 12k3s in envu(n), and envl(n). Let us denote the upper and lower envelopes of (3.4) correspondingly by envu(s), and envl(s). Just as in §2, define the oscillatory factor 8 < uðs;xÞ ; if uðs; xÞ P 0; envu ðsÞ rðs; xÞ ¼ ð3:5Þ : uðs;xÞ ; if uðs; xÞ 6 0; envl ðsÞ Define the frequency x(s, x), Dx and D s of u(s, x) to be correspondingly o xðs; xÞ ¼ arcsin rðs; xÞ; os Dx ¼ jxðs; xÞ 8k3 j; Ds ¼ s
sin 2 ¼s 24k3
sin 2k 23 x þ 4k2 s 24k3
ð3:6aÞ ð3:6bÞ
;
ð3:6cÞ
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
and, similarly to §2, consider the averaged quantities
2k 3p
R 3p 2k
0
R 3p R 3p 2k 2k 2k 2k jDsj dx 3p jDxj dx and 3p jDxDxj dx whose graphs 0 0
are shown in Figs. 3.1 and 3.2. Changing the value of k scales the graphs horizontally but not vertically. Based on the graphs the following conclusion can be drawn. Uncertainty principle for KdV in time-frequency variables Let Dx and Dx be defined by (3.6). Then there exist positive constants U3, U4 such that ! ! Z 3p Z 3p 2k 2k 2k 2k jDsj dx jDxj dx P U 3 3p 0 3p 0
ð3:7aÞ
and 2k 3p
Z
3p 2k
jDsDxj dx P U 4 :
ð3:7bÞ
0
Moreover as s approaches infinity, " ! !# " ! Z 3p Z 3p Z 3p 2k 2k 2k 2k 2k 2k lim jDsj dx jDxj dx ¼ lim jDsj dx s!þ1 s!1 3p 0 3p 0 3p 0
2k 3p
Z 0
3p 2k
!# jDxj dx
¼
4 3p
ð3:8aÞ
and lim
s!þ1
2k 3p
Z 0
3p 2k
jDsDxj dx ¼ lim
s!1
4k3 p
Z 0
p 4k3
jDsDxj dx ¼
4 : 3p
ð3:8bÞ
The exact numerical values of the constants U3, U4 are not known. Numerical computations yield U3 0.2816 (attained at s 0.1030) and U4 0.2729 (attained at s 0.1049).
R 3p R 3p 2k 2k 2k 2k Fig. 3.1. Graph of 3p jDsj dx 3p jDxj dx plotted for k = 1.0; s is measured along the horizontal axis, the value of 0 0
R 3p R 3p 2k 2k 2k 2k jDsj dx 3p jDxj dx along the vertical axis. For other values of k the graph scales horizontally but not vertically. 0 3p 0
R 3p R 3p 2k 2k 2k 2k Fig. 3.2. Graph of 3p jDsDxj dx plotted for k = 1.0, s is measured along the horizontal axis, the value of 3p jDsDxj dx along the 0 0 vertical axis. For other values of k the graph scales horizontally but not vertically.
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
443
4. Conclusion The physical uncertainty principle was obtained by N. Bohr and W. Heisenberg in the context of quantum mechanics based on physical arguments some of which are described in [6,7]. As shown in [6,7] it exhibits itself in different forms and even almost a century since its discovery physicists are still divided on whether it is a fundamental law of nature or just a reflection of our limitation in measurements. A few years after the discovery of the physical uncertainty principle, formulas (1.12) were obtained and interpreted as the mathematical derivation of the uncertainty principle. Although some hoped that the derivation of (1.12) would put to an end the controversy on the philosophical nature of the uncertainty principle, the effect was quite the opposite. Numerous discussions can be found on the internet, some quite interesting. Meanwhile the uncertainty principle has been extended from quantum mechanics to statistics, engineering, finance and many other fields. As essentially a property of the Fourier transform, formulas (1.12) appear wherever the Fourier transform is, and that means practically in all fields described by linear partial differential equations. The uncertainty principle (2.7) described in this paper adds another facet to both the controversy and applications of the general uncertainty principle. Being obtained for a nonlinear equation, it shows that even in mathematical formulas the uncertainty principle appears in forms other than (1.12). Although in many aspects fundamentally different from the uncertainty principle (1.12), the uncertainty principle (2.7) nevertheless serves the same purpose: to describe the balancing act between wave and particle aspects of a single motion that exhibits both wave and particle properties. The uncertainty principle (1.12) is global in the space variable x for it relates the average deviations D - and Dx from their corresponding averages - and x, the averaging being over the whole space. The uncertainty principle (2.7) is local, in the sense that it relates the deviations Dl and Dn of the local frequency and position from correspondingly the internal frequency 2k and the singularity. Can the uncertainty principle (2.7) affect our understanding of quantum mechanics and physical aspects of the general uncertainty principle? Hopefully so, although not in the form formulated in this paper since it is derived for KdV which is not a fundamental equation of quantum mechanics. But just like KdV is a nonlinear generalization of its linear part, there most likely exists an appropriate nonlinear generalization of the Schrodinger equation that not only incorporates all quantum-mechanical aspects of the original Schrodinger equation but also extends it to describe new phenomena. An extension of the uncertainty principle (2.7) to such an equation would be a viable alternative to (1.12) as a mathematical derivation of the physical uncertainty principle.
References [1] Arkadiev VA, Pogrebkov AK, Polivanov MK. Theor Math Phys 1983;53:1051. Also preprint Singulayrniye resheniyea uravneniya KdV i metod obratnoy zadachi [in Russian]. [2] Beutler R. J Math Phys 1993;34:3098. [3] Beutler R, Stahlhofen A, Matveev VB. What do solitons, breathers and positons have in common? Phys Scr 1994;50:9–12. [4] Dym H, McKean HP. Fourier series and integrals. New York: Academic Press; 1972. [5] Fermi E. Notes on quantum mechanics. 2nd ed. University of Chicago Press; 1995. [6] Feynman R, Leighton R, Sands M. The Feynman lectures on physics, V. I. Addison-Wesley Publishing Company; 1977. [7] Feynman RP, Hibbs AR. Quantum mechanics and path integrals. McGraw-Hill Companies; 1965. [8] Gelfand IM, Levitan BM. Izv Akad Nauk USSR, Se Mat 1951;15:309–60. [9] Jaworski M. Breather-like solution of the Korteweg–de Vries equation. Phys Lett A 1984;104(5):245–7. [10] Kovalyov M. On a class of solutions of KdV. J Differ Equations 2005;213:1–80. [11] Kovalyov M. Basic motions of the Korteweg–de Vries equation. Nonlinear Anal Theory Methods Appl 1998;31(5/6): 599–619. [12] Kovalyov M. Modulating properties of harmonic breather solutions of KdV. J Phys A: Math Gen 1998;31:5117–28. [13] Ma WX. Complexiton solutions to the Korteweg–de Vries equation. Phys Lett A 2002;301(1–2):35–44. [14] Matveev VB. Asymptotics of the multipositon–soliton s-function of the Korteweg–de Vries equation and the supertransparency. J Math Phys 1994;35:2955. [15] Matveev VB. Phys Lett A 1992;135:2009–12. [16] Matveev VB. Trivial S-matrices, Wigner–Von Neumann resonances and positon solutions of the integrable systems. In: Boutet de Monvel A, Marchenko V, editors. Algebraic and geometric methods in mathematical physics, Proceedings of international workshop. Kluwer Academic Publishers; 1996. p. 343–55. [17] Matveev VB, Salle MA. In: Darboux transformations and solitons. Springer series in nonlinear dynamics. Berlin, Heidelberg: Springer-Verlag; 1991.
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[18] El Naschie MS. Fredholm operators and the wave–particle duality in cantorian space. Chaos, Solitons & Fractals 1998;9(6): 975. [19] Novikov R, Khenkin G. Oscillating weakly localized solutions of the Korteweg–de Vries equation. Teor Matemat Fizika 1984;61(2):199–213 [in Russian]. [20] Stahlhofen AA. On completely transparent potentials of the Shro¨dinger equation. Phys Rev A 1995;51:934. [21] Stahlhofen AA, Maisch H. Dynamical properties of positons. Phys Scr 1995;52:228. [22] van der Waerden BL. Algebra. New York: Springer-Verlag; 1991.
Uncertainty principle for the nonlinear waves of the Korteweg–de Vries equation Mikhail Kovalyov Department of Mathematics, University of Alberta, Edmonton, Alta., Canada T6G 2G1 Accepted 5 June 2006
Abstract The solutions of KdV studied in Kovalyov [Kovalyov M. On a class of solutions of KdV. J Differ Equations 1951;15:309–60; Kovalyov M. Basic motions of the Korteweg–de Vries equation, Nonlinear Anal Theory Methods Appl 1998;31(5/6):599–619; Kovalyov M. Modulating properties of harmonic breather solutions of KdV. J Phys A: Math Gen 1998;31:5117–28.] exhibit wave–particle behavior albeit somewhat different from the wave–particle behavior of quantum mechanics. Yet the balance between the wave and particle properties of these solutions is governed by an uncertainty principle remarkably similar to that of quantum mechanics. Ó 2006 Published by Elsevier Ltd.
1. Introduction The wave–particle duality of certain dynamical processes has been known for centuries. Some properties of light, for example, are best described if light is viewed as a sequence of waves, while others are easier to understand if it is viewed as a large collection of particles. The analytical mechanics of Hamilton and Lagrange provided the mathematical machinery for describing the motion of both particles and waves by the same equations. It was subsequently adapted for use in quantum mechanics where the motions of elementary particles exhibit both wave- and particle-like properties, [5,7] and further extended to some generalizations of quantum mechanics, [18]. The balancing act between wave and particle components of a quantum-mechanical motion is governed by the Bohr–Heisenberg uncertainty principle, which essentially states that the more wave-like a motion is the less particle-like it is and vice versa. The mathematical proof for the uncertainty principle is based on Fourier transform and as such is valid for any linear motion. It is, however, restricted to linear waves only. We will show in this paper that there is a class of nonlinear waves which satisfy an uncertainty principle whose mathematical formulation is practically identical to that of the Bohr–Heisenberg uncertainty principle, yet the very meaning of the parameters is considerably different. These nonlinear waves appear as solutions of the Korteweg–de Vries equation ut þ 6uux þ uxxx ¼ 0: A single such wave is given by
E-mail address: [email protected]. 0960-0779/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.06.050
ðKdVÞ
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
d2 sin 2kðc 4k2 t xÞ 2 ln p 12k t x 2k dx2 " #2 2 8k sin 2C cos 2C 1 2 ¼ þ 8k ; 2kðp 12k2 t xÞ sin 2C 2kðp 12k2 t xÞ sin 2C
uðt; xÞ ¼ 2
ð1:1Þ
where C = k(c 4k2t x), and constants k, c, p are correspondingly the spectral parameter, phase and displacement of the wave. These waves have been studied in a number of articles, e.g. [1–3,8–17,19–21]. Fig. 1.1 shows the structure of (1.1), with k = 1.0, c = 0.0 at time t = 0.1. The graph of (1.1) can be divided into three parts: a neighborhood of the always present singular point shown on the graph by the dotted line and two decaying oscillatory tails on both sides of the neighborhood of the singularity shown on the graph by solid lines. While in the neighborhood of the singularity (1.1) does not exhibit much of wave-like behavior, the two oscillatory tails appear to be very much wave-like. Nonlinear superposition of such nonlinear waves is given by the formula [10], 8 sin 2Cn 2 > > ; Cn ¼ kn ðcn 4k2n t xÞ; n ¼ m; < pn 12kn t x 2 d 2kn u ¼ 2 2 ln det D; Dnm ¼ ð1:2Þ > dx sinðCn Cm Þ sinðCn þ Cm Þ > : ; n 6¼ m; kn km kn þ km where n, m = 1, 2, 3, . . . , N, pn’s, cn’s and kn’s are constants. The derivation of (1.2) is rather long and thus is omitted here. The interested reader is referred to [10] for details. It is easy to show that function (1.2) has exactly N singularities. Indeed, let rank(x) be the dimension of the largest linear space on which the bilinear form N X
Dnm nm nn ¼
n;m¼1
N N N X X X sinðCn Cm Þ sinðCn þ Cm Þ ðpn 12k2n t xÞn2n þ nm nn þ nm nn : k k kn þ km n m n¼1 n;m¼1 n;m¼1 n6¼m
is negative definite. Since for large values of jxj N X
Dnm nm nn x
n;m¼1
N X
n2n ;
n¼1
rank(x) = 0 for large negative x and rank(x) = N for large positive x. Due to the easily verifiable inequality !2 N N X d X Dnm nm nn ¼ 2 nn sin Cn 6 0; dx n;m¼1 n¼1 rank(x) is an increasing function of x assuming integer values from a subset of 0, 1, 2, 3, . . . , N. The points of jumps of rank(x) are exactly the poles of u(t, x) and there are at least one and at most N of them. Sufficiently far away from all singularities the function u(t, x) given by (1.2) behaves like uðt; xÞ
N X n¼1
4kn sin 2Cn : pn 12k2n t x
ð1:3aÞ
If all pn’s are sufficiently large and have the same sign and x, t are restricted to a bounded set, say jxj, jtj 6 1, (1.3a) simplifies to uðt; xÞ
N X 4kn sin 2Cn : pn n¼1
ð1:3bÞ
Fig. 1.1. Graph of function (1.1) with k = 1.0, p = c = 0.0 at t = 0.1, x is measured along the horizontal axis, u along the vertical axis. For other values of k, p, c and t the graph has a similar structure.
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
Formula (1.2) is a nonlinear analogue of the Fourier sum solutions X uk cos 2kk ½cðkk Þ x 4k2k t
433
ð1:4aÞ
over appropriate subset of N
of the linear Korteweg–de Vries equation ut þ uxxx ¼ 0:
ðlKdVÞ
Just like appropriate limits of Riemann sum solutions (1.4a) of l KdV lead to integral type solutions of lKdV of the form Z ^uðkÞ cos 2k½cðkÞ x 4k2 t dk; ð1:4bÞ over appropriate subset of R
the appropriately taken limits of (1.2) lead to nonlinear analogues of (1.4b), [10]. As such limits are taken the singularities of (1.2) move to either 1 or +1 and the nonlinear superposition of the oscillatory tails forms miscellaneous solutions of KdV, just like the linear superposition of cos 2kk ½cðkk Þ x 4k2k t forms miscellaneous solutions of lKdV. The existence of the integrals in the linear case is assured by the phenomenon of linear interference, which essentially states that if A1 cos 2k1 ½c1 x 4k21 t
and
A2 cos 2k1 ½c1 x 4k21 t
ð1:5aÞ
are two solutions of lKdV then so is their linear superposition A12 cos 2k1 ½c1 x 4k21 t;
ð1:5bÞ
A12 ¼ A1 þ A2 :
ð1:5cÞ
where
Taking linear superposition of two solutions (1.5a) of lKdV with the same frequencies and phases simply amounts to the addition of their amplitudes. Fig. 1.2 illustrates linear superposition of two linear waves (1.5a). Just like linear interference is behind the mathematics of the linear case, a similar phenomenon of nonlinear interference is behind the mathematics of the nonlinear case, [10]. Since, due to vanishing denominators, it is impossible to simply take (1.2) with N = 2 and k2 = k1, c2 = c1, let us take (1.2) with N = 2 d2 sin 2k1 ðc1 4k21 t xÞ sin 2k2 ðc2 4k22 t xÞ 2 2 p2 12k2 t x uðt; xÞ ¼ 2 2 ln p1 12k1 t x dx 2k1 2k2 2 ) 2 2 2 2 sinðk1 ðc1 4k1 t xÞ k2 ðc2 4k2 t xÞÞ sinðk1 ðc1 4k1 t xÞ þ k2 ðc2 4k2 t xÞÞ ; ð1:6Þ k1 k2 k1 þ k2 and additional assumptions that c1 = c(k1), c2 = c(k2), where c(k) is a differentiable function of k, and pass to the limit as k2 ! k1. As k2 ! k1, formula (1.6) degenerates into the formula d2 sin 2k1 ðc1 4k21 t xÞ ; ð1:7Þ u12 ðt; xÞ ¼ 2 2 ln p12 12k21 t x dx 2k1 where the parameter p12 satisfies
Fig. 1.2. Two linear waves of type (1.5a) with k1 = k2 = 1.0, c1 = c2 = 0.0, A1 = 0.2, A2 = 0.1 at t = 0.0 are shown in thin solid and broken black; their linear superposition (1.5b) is shown in solid gray.
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
4k1
¼
4k1
4k1
; ð1:8Þ p12 p1 p2 oðkcÞ ok k1 k1 k1 and oðkcÞ denotes the value of oðkcÞ at k = k1. Function (1.7) can be viewed as a nonlinear superposition of two nonlinear ok ok k1 waves d2 sin 2k1 ðc1 4k21 t xÞ 2 uj ðt; xÞ ¼ 2 2 ln pj 12k1 t x ; j ¼ 1; 2: ð1:9Þ dx 2k1 oðkcÞ ok
þ
oðkcÞ ok
Sufficiently far away from the singularities, each nonlinear wave (1.9) satisfies uj ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ ; pj oðkcÞ 12k21 t x ok
j ¼ 1; 2;
ð1:10aÞ
k¼k1
while their nonlinear superposition (1.7) satisfies u12 ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ : p12 oðkcÞ 12k21 t x ok
ð1:10bÞ
k¼k1
If p1, p2, p12 are sufficiently large and x, t are restricted to a finite domain, say jxj, jtj < 1, the estimates (1.10) simplify to uj ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ ; pj oðkcÞ ok
j ¼ 1; 2;
ð1:11aÞ
k¼k1
and u12 ðt; xÞ
4k1 sin 2k1 ðc1 4k21 t xÞ : p12 oðkcÞ ok
ð1:11bÞ
k¼k1
Due to (1.11) we call expressions
4k1 oðkcÞ p1 ok j
, k1
4k1 oðkcÞ p2 ok j
, k1
4k1 oðkcÞ p1;2 ok j
asymptotic amplitudes of the nonlinear waves (1.9) and k1
(1.7). The argument just provided suggests that formulas (1.3) for the asymptotic behavior of (1.2) be replaced with uðt; xÞ
N X n¼1
4kn sin 2Cn pn 12k2n t x oðkcÞ ok
kn
and uðt; xÞ
N X 4kn sin 2Cn oðkcÞ n¼1 p n ok kn
in order to be consistent with nonlinear interference. For a more detailed discussion of nonlinear interference the reader is referred to [10]. Fig. 1.3 illustrates the nonlinear superposition of two nonlinear waves (1.2).
Fig. 1.3. Two nonlinear waves of type (1.2) with k1 = k2 = 1.0, c1 = c2 = 0.0, c(k) = 0.0, p1 = 30, p2 = 20 at t = 0.0 are shown in a thin solid and broken black; their nonlinear superposition is shown in a thick solid gray.
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
435
We see that sufficiently far from the singularity, the oscillatory tails of (1.1) exhibit wave-like behavior similar to that of (1.5a), while on the other hand, the singular points of (1.1) behave just like point particles and their motion is that of point particles. One may conjecture that the further away from the singularity the more wave-like is the behavior of (1.1), but the closer to the singularity the more particle-like is the behavior of (1.1). The functions (1.1) contain within themselves both wave and particle motions, and in this aspect they are somewhat similar to wave–particles of quantum mechanics. Yet the very combination of the two motions is fundamentally different from that of modern quantum mechanicsR where wave–particles are described by R functions f(t, x) R and their Fourier transforms 1 f~ ðt; -Þ ¼ Rn f ðt; xÞei-x dx, appropriately normalized, i.e. Rn jf ðt; xÞj2 dx ¼ ð2pÞ jf~ ðt; -Þj2 d- ¼ 1. The balancing n Rn act between the wave and particle aspects of quantum-mechanical motion is governed by the Bohr–Heisenberg uncertainty principle 1 DxD- P ; 2 where ðDxÞ2 ¼
Z
jx xj2 jf ðt; xÞj2 dx; x ¼ Rn Z 1 j- -j2 jf~ ðt; -Þj2 d-; ðD-Þ2 ¼ 2p Rn
ð1:12aÞ Z
xjf ðt; xÞj2 dx; Z 1 -¼ -jf~ ðt; -Þj2 d2p Rn
ð1:12bÞ
Rn
ð1:12cÞ
and n is the dimensionality of the problem, which for our purpose, may be taken as equal to one. It is worthwhile to notice that the original Bohr–Heisenberg uncertainty principle was formulated by Bohr and Heisenberg in the form DxD- ¼ Oð1Þ
ð1:13Þ
based on physical arguments, some of which may be found in [6,7]. The nice mathematical definitions (1.12) were introduced considerably later. Assuming, without loss of generality, that f(t, x) is real and continuously differentiable and x ¼ - ¼ limx!1 xf 2 ðt; xÞ ¼ 0, the proof of (1.12a) is essentially given by the following string of formulas: Z þ1 Z þ1 1 jxj2 jf ðt; xÞj2 dx j-j2 jf~ ðt; -Þj2 dðDxD-Þ2 ¼ 2p 1 1 Z þ1 2 Z þ1 Z þ1 of ðt; xÞ2 of ðt; xÞ dx P dx jxj2 jf ðt; xÞj2 dx xf ðt; xÞ ¼ ox ox 1 1 1 Z þ1 2 Z Z 2 þ1 þ1 1 of 2 ðt; xÞ 1 o o 2 ¼ x x; dx ¼ ðxf 2 ðt; xÞÞ dx þ f ðt; xÞ dx 4 1 ox 4 1 ox ox 1 2 Z þ1 1 2 1 ¼ xf ðt; xÞjþ1 f 2 ðt; xÞ dx ¼ ; 1 þ 4 4 1 o where x; ox is the commutator of x and oxo . A more general and detailed proof of (1.12a) follows along these lines and may be found in [4,5]. According to (1.12), at each point x all frequencies - are present and D- is the average deviation of - from its average -, whereas Dx is the average deviation of x from its average x. The particle aspect of the motion is measured by the spread Dx from x. The smaller the spread Dx, the more particle-like is the motion. The wave aspect of the motion is measured by the spread D- which determines by how much the frequencies of the components ei-x of f(t, x) deviate from -. The smaller the spread D-, the more wave-like is the motion.
Fig. 1.4. Graph of function (1.1) with k = 1.0, p = c = 0.0 at t = 0.1; x is measured the along horizontal axis, u along the vertical axis. For other values of k, p, c and t the graph has a similar structure. The vertical dashed line indicates the singularity of the function.
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
The situation is somewhat different for (1.1) as shown in Fig. 1.4. At any given time t and at each point x only one frequency is present, but it varies with x and t and approaches 2k as x ! ± 1. The location of the singularity is also strictly defined. The uncertainty principle for (1.1) must provide an estimate for ð1:14Þ
DxDl;
where Dx is a measure of how far point P(t, x) is removed from the singularity and Dl is the measure of by how much the local frequency l(P) differs from 2k = limx!±1l(P).
2. The uncertainty principle for the Korteweg–de Vries equation in space-frequency variables To get a rough estimate on DxDl, rewrite the expression (1.1) as follows: uðt; xÞ ¼
32k2 ½sin C kðp 12k2 t xÞ cos C sin C ½2kðp 12k2 t xÞ sin 2C2
;
ð2:1Þ
where C = k(c 4k2t x). Let {. . ., x4, x2, x0, x2, x4, . . .} denote the solutions of sin C ¼ 0 and {. . ., x5, x3, x1, x1, x3, x5, . . .} denote the solutions of sin C kðp 12k2 t xÞ cos C ¼ 0: Clearly x2n ¼ c 4k2 t þ
pn : k
The elements of {. . ., x5, x3, x1, x1, x3, x5, . . .} are the solutions of cot kðx2nþ1 þ 4k2 t cÞ ¼
1 ; kðx2nþ1 þ 12k2 t pÞ
and for large jx2n+1j, satisfy x2nþ1 c 4k2 t þ
pn p 1 þ : k 2k k2 ðx2nþ1 þ 12k2 t pÞ
For large n, the frequency l between two adjacent zeros x2n and x2n+1 of u satisfies " # p p 2k 2 l p ; 2k 1 þ 1 2 jx2nþ1 x2n j 2k 1 pkðx þ12k k2 ðx þ12k pkðx2nþ1 þ 12k2 t pÞ 2 2 tpÞ tpÞ 2nþ1
2nþ1
which implies jl 2kj
4 pjx2nþ1 þ 12k2 t pj
and consequently 4 jl 2kjjx2nþ1 þ 12k2 t pj : p
ð2:2Þ
The same argument shows that between zeros x2n+1 and x2n+2, frequency l also satisfies (2.2). Eq. (2.2) provides a rough form of the uncertainty principle for (1.1). It is more convenient to formulate and consider the uncertainty principle for KdV in the frame of reference moving with velocity 12k2t, which is achieved by replacing x with n ¼ x þ 12k2 t p:
ð2:3Þ
By an appropriate shift in time variable t the constant c p can be made zero, so for the rest of this section it will be assumed to vanish. In the new variables function (1.1) becomes
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
( ) 2 8k2 sin 2C cos 2C 1 kn sin 2C þ 1 cos 2C ¼ 16k2 þ 8k2 2kn þ sin 2C 2kn þ sin 2C ð2kn þ sin 2CÞ2 ( ) kn sin 2G þ cos 2G 1 ¼ 16k2 ; ð2kn sin 2GÞ2
437
uðt; nÞ ¼
ð2:4Þ
where G = C = k(n 8k2t). In the coordinate system of n and t the singularity of (2.4) moves from n ¼ 2k1 to n ¼ 2k1 , disappearing upon reaching n ¼ 2k1 and at the same moment re-emerging at n ¼ 2k1 . The oscillatory tails move to the right sandwiched between two envelopes as shown in Fig. 2.1. Let envu(n), and envl(n) denote correspondingly the upper and lower envelopes; their existence, exact form and derivation will3 be discussed at the end of this section. In the system of coordinates x, t both envelopes move with velocity 12k2 ¼ dð2kÞ . This formula looks like group velocity in linear theory, only here just one frequency 2k is present. dð2kÞ Define if uðt; nÞ P 0; envu ðnÞ; ð2:5aÞ Aðt; nÞ ¼ envl ðnÞ; if uðt; nÞ 6 0: and
8 uðt;nÞ uðt; nÞ uðt; nÞ þ juðt; nÞj uðt; nÞ juðt; nÞj < envu ðnÞ ; sðt; nÞ ¼ þ ¼ ¼ : uðt;nÞ Aðt; nÞ 2envu ðnÞ 2envl ðnÞ ; envl ðnÞ
if uðt; nÞ P 0; if uðt; nÞ 6 0:
ð2:5bÞ
The function (2.4) then can be written as uðt; nÞ ¼ Aðt; nÞsðt; nÞ:
ð2:5cÞ
1 , 2k
If jnj > then js(t, n)j 6 1 and thus A(t, n) and s(t, n) can be viewed correspondingly as the amplitude and oscillatory factor of (2.4). For jnj 6 2k1 , however, js(t, n)j might become greater than one and thus it loses its interpretation as the oscillatory factor. The shape of s(t, n) is shown in Fig. 2.2. Define the frequency l(t, n), Dl and Dn of u(t, n) to be correspondingly
Fig. 2.1. Graph of function (2.4) together with two envelopes, plotted with k = 1.0, c = 0.0 at t = 0.2; n is measured along the horizontal axis, u, envu(n), envl(n) along the vertical axis. The function itself is shown in thick solid gray, the upper envelope in solid black and the lower envelope in dotted black. Note that the upper envelope consists of three disjoint parts defined correspondingly on 1; 2k1 , 2k1 ; 2k1 , and 2k1 ; þ1 : For other values of k, c and t, the graph has a similar structure.
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
Fig. 2.2. The graph of s(t, n) plotted with k = 1.0, c = 0.0 at t = 0.7, n is measured along the horizontal axis, s(t, n) along the vertical axis. For jnj P 2k1 ; js(t, n)j 6 1. For jnj 6 2k1 ; s(t, n) might become >1. For other values of k, c and t the graph has a similar structure.
o lðt; nÞ ¼ arcsin sðt; nÞ; on
ð2:6aÞ
Dl ¼ jlðt; nÞ 2kj;
ð2:6bÞ
sin 2G sin 2kðn 8k2 tÞ ¼n : Dn ¼ n 2k 2k
ð2:6cÞ
The product DnDl oscillates and due to its rather complicated form does not appear to be of much interest. What is of
3R p 3R p p 3 R more interest are the averaged quantities 4kp 04k3 jDnj dt 4kp 04k3 jDlj dt and 4kp 04k3 jDnDlj dt whose graphs are shown in Figs. 2.3 and 2.4. Changing the value of k scales the graphs horizontally but not vertically. Based on the graphs the following conclusion can be drawn.
3R p 3 R p 3 3 Fig. 2.3. Graph of 4kp 04k jDnj dt 4kp 04k jDlj dt plotted for k = 1.0; n is measured along the horizontal axis, the value of
3R p 3R p 4k 4k 4k3 4k3 0 jDnj dt 0 jDlj dt along the vertical axis. For other values of k the graph scales horizontally but not vertically. p p
p p 3 R 3 3 R 3 Fig. 2.4. Graph of 4kp 04k jDnDlj dt plotted for k = 1.0, n is measured along horizontal, the value of 4kp 04k jDnDlj dt along vertical. For other values of k the graph scales horizontally but not vertically.
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
439
Uncertainty principle for KdV in space-frequency variables Let Dl and Dn be defined by (2.6). Then there exist positive constants U1, U2 such that 3Z p
3Z p
4k3 4k3 4k 4k jDnj dt jDlj dt P U 1 p 0 p 0
ð2:7aÞ
and 4k3 p
Z
p 4k3
jDnDlj dt P U 2 :
ð2:7bÞ
0
Moreover as n approaches infinity, 3 Z p
3Z p
3 Z p
3Z p
4k3 4k3 4k3 4k3 4k 4k 4k 4k 4 jDnj dt jDlj dt ¼ lim jDnj dt jDlj dt ¼ lim n!þ1 n!1 p p 0 p 0 p 0 p 0
ð2:8aÞ
and lim
n!þ1
4k3 p
Z
p 4k3
jDnDlj dt ¼ lim
n!1
0
4k3 p
Z
p 4k3
0
4 jDnDlj dt ¼ : p
ð2:8bÞ
It is somewhat remarkable that the inequalities (2.7) hold not only for jnj P 2k1 but also for jnj < 2k1 ; where s(t, n) > 1 and arcsin s(t, n) is complex. The exact numerical values of the constants U1, U2 are not known. Numerical computations yield U1 0.7488 (attained at n 0.6371) and U2 0.7894 (attained at n 0.6598). The conclusions (2.7) are drawn based on the graphs in Figs. 2.3 and 2.4 and the fact that changing the value of k in (1.1) and (2.4) results in horizontal re-scaling of the graphs in Figs. 2.3 and 2.4 with no vertical re-scaling. No attempt was made to prove (2.7) analytically, although with due diligence an analytical proof should not be difficult. The main difficulty in obtaining an analytical proof of (2.7) is due to a rather complicated analytical form of the function s(t, n), which itself is due to rather complicated formulas arising in the computation of the envelopes. Formulas (2.8) can be either deduced from the graphs in Figs. 2.3 and 2.4 or obtained analytically from (2.2). The graphs in Figs. 2.3 and 2.4 were obtained using independently Maple and Matlab, the former being unexpectedly much faster while the latter provided better quality graphs. Computation of envelopes Since u(t, n) contains n in the combination kn only, it suffices, without loss of generality, to obtain envelopes for k = 1. The general case then is recovered by multiplying the final answer by k2 and replacing n with kn. The envelopes of u(t, n) are obtained by eliminating t from the system 8 16ðn sin 2G þ cos 2G 1Þ > > ; > < uðt; nÞ ¼ ð2n sin 2GÞ2 ð2:9Þ > ou 2n2 cos 2G þ n cos 2G sin 2G 2n sin 2G þ cos2 2G 2 cos 2G þ 1 > > ¼ 0; ¼ : oG 2ð2n sin 2GÞ3 where the second equation is due to the identity the system to 8 2 < u ¼ 8ðn cot 2G1Þðcot Gþ1Þ ; ðn½cot Gþ1cot GÞ2 : cot4 G cot3 G 3 cot G þ n
n
2 n2
ou ot
2
ou G1 2 cot G ¼ 8 oG . Substitutions cos 2G ¼ cot and sin 2G ¼ cot 2 Gþ1 simplify cot2 Gþ1
1 ¼ 0:
ð2:10Þ
Solving the second equation for cotG and substituting the answer into the first equation one obtains envelopes of (2.4). The second equation is a fourth order equation in cotG equivalent to a fourth order equation in the standard form in the variable ðcot G 4n1 Þ:
4
2
1 3 1 1 3 1 5 3 2 cot G þ 1 ¼ 0: cot G þ 2 cot G 4n 4n 4n 8n 8n3 n 4n 256n4 The cubic resolvent in variable z for this equation is, [22],
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
2
3 5 3 1 3 2 z z þ þ 4 þ þ ¼0 4n2 n2 16n4 8n3 n
whose reduced form in variable z þ 4n12 is
3
1 5 1 8 2 z þ 2 þ 2 þ 4 ¼ 0: zþ 2 þ 4 2 4n n 4n n n z3 þ
Following the standard procedure outlined in [22], the roots of the cubic resolvent are obtained in the form vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 1u 108 27 720 2196 273 81 2 t 2 4 þ 3 192 2 þ 4 þ 6 þ 8 z1 ¼ 3 n n n n n n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s u 3 1u 108 27 720 2196 273 81 1 2 t 2 4 3 192 2 þ 4 þ 6 þ 8 2 ; þ 3 n n n n n n 4n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi!2 u 3 1 1 i 3 u 108 27 720 2196 273 81 2 t þ 2 4 þ 3 192 2 þ 4 þ 6 þ 8 z2 ¼ 3 2 2 n n n n n n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ! pffiffiffi u 3 1 1 i 3 u 108 27 720 2196 273 81 1 2 t þ 2 4 3 192 2 þ 4 þ 6 þ 8 2 ; þ 3 2 2 n n n n n n 4n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi!u 3 1 1 i 3 u 108 27 720 2196 273 81 2 t þ 2 4 þ 3 192 2 þ 4 þ 6 þ 8 z3 ¼ 3 2 2 n n n n n n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ! u pffiffiffi 2 3 1 1 i 3 u 108 27 720 2196 273 81 1 2 t þ 2 4 3 192 2 þ 4 þ 6 þ 8 2 ; þ 3 2 2 n n n n n n 4n which provide four possible choices for the value of cot G: p ffiffiffiffi p ffiffiffiffi pffiffiffiffi 2 z1 þ 2 z2 þ 2 z3 1 signðnÞ þ ; c1 ¼ 4n 2 p ffiffiffiffi p ffiffiffiffi pffiffiffiffi 2 z1 2 z2 2 z3 1 c2 ¼ signðnÞ þ ; 2 4n pffiffiffiffi pffiffiffiffi pffiffiffiffi 2 z1 þ 2 z2 2 z3 1 signðnÞ þ ; c3 ¼ 4n 2 pffiffiffiffi pffiffiffiffi pffiffiffiffi 2 z1 2 z2 þ 2 z3 1 c4 ¼ signðnÞ þ ; 4n 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n where signðnÞ ¼ jnj and 3 expression; 2 expression denote the roots of the corresponding order of the expression whose 2
cot G1Þðcot Gþ1Þ arguments are closest to that of the expression. Substitution of c1 and c2 for cot G in u ¼ 8ðn yields corðn½cot2 Gþ1cot GÞ2
respondingly the upper and lower envelopes. Quantities c3 and c4 are redundant and should be discarded.
3. The uncertainty principle for the Korteweg–de Vries equation in time-frequency variables In the previous section we obtained a nonlinear uncertainty principle relating the local spacial frequency l of (1.1) at a point x at time t and the distance from x to the singularity at the same time. In this section we discuss a nonlinear uncertainty principle relating the local time-frequency x of (1.1) at a point x at time t and the length of time required for the singularity to arrive at this point. Just like in the previous section we first get a rough estimate for DtDx. Recall that from (2.1): uðt; xÞ ¼
32k2 ½sin C kðp 12k2 t xÞ cos C sin C ½2kðp 12k2 t xÞ sin 2C2
;
where C = k(c 4k2t x). Let {. . ., t4, t2, t0, t2, t4, . . .} denote the roots of
ð3:1Þ
M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
441
sin C ¼ sin kðc 4k2 t xÞ ¼ 0 and {. . ., t5, t3, t1, t1, t3, t5, . . .} denote the roots of sin C kðp 12k2 t xÞ cos C ¼ sin kðc 4k2 t xÞ kðp 12k2 t xÞ cos kðc 4k2 t xÞ ¼ 0: Clearly 4k2 t2n ¼ c x þ
pn : k
The elements of {. . ., t5, t3, t1, t1, t3, t5, . . .} are the solutions of cot kðx þ 4k2 t2nþ1 cÞ ¼
1 ; kðx þ 12k2 t2nþ1 pÞ
and for large jt2n+1j, satisfy 4k2 t2nþ1 c x þ
pn p 1 : þ k 2k k2 ðx þ 12k2 t2nþ1 pÞ
For large n, the frequency x between two adjacent zeros t2n and t2n+1 of u satisfies " # p 4k2 p 8k3 2 3 x p ; 8k 1 þ 1 jt2nþ1 t2n j 2k k2 ðx þ12k 1 pkðxþ12k22 t pÞ pkðx þ 12k2 t2nþ1 pÞ 2 tpÞ 2nþ1
2nþ1
which implies jx 8k3 j
16k2 pjx þ 12k2 t2nþ1 pj
and consequently x p 4 : þ t jx 8k3 j 2nþ1 3p 2 12k
ð3:2Þ
The same argument shows that between zeros t2n+1 and t2n+2 frequency s also satisfies (3.2). Just like as in the case with the space-frequency uncertainty principle obtained in the previous section, the time-frequency uncertainty principle is better visualized in modified coordinates x and xp s¼tþ : ð3:3Þ 12k2 By an appropriate shift in space variable x the constant c p3 can be made zero and so for the rest of this section it will be assumed to vanish. In the new variables, function (1.1) becomes ( ) 3 2 12k s sin 2 þ cos 2 1 uðs; xÞ ¼ 16k ; ð3:4Þ ð24k3 s sin 2 Þ2 where ¼ C ¼ k 23 x þ 4k2 s : The last term in (3.4) has the same form as the last term in (2.4). It can be obtained form (2.4) by replacing n and G correspondingly with 12k3s and . Thus the two envelopes for (3.4) can be obtained by simply replacing n with 12k3s in envu(n), and envl(n). Let us denote the upper and lower envelopes of (3.4) correspondingly by envu(s), and envl(s). Just as in §2, define the oscillatory factor 8 < uðs;xÞ ; if uðs; xÞ P 0; envu ðsÞ rðs; xÞ ¼ ð3:5Þ : uðs;xÞ ; if uðs; xÞ 6 0; envl ðsÞ Define the frequency x(s, x), Dx and D s of u(s, x) to be correspondingly o xðs; xÞ ¼ arcsin rðs; xÞ; os Dx ¼ jxðs; xÞ 8k3 j; Ds ¼ s
sin 2 ¼s 24k3
sin 2k 23 x þ 4k2 s 24k3
ð3:6aÞ ð3:6bÞ
;
ð3:6cÞ
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M. Kovalyov / Chaos, Solitons and Fractals 32 (2007) 431–444
and, similarly to §2, consider the averaged quantities
2k 3p
R 3p 2k
0
R 3p R 3p 2k 2k 2k 2k jDsj dx 3p jDxj dx and 3p jDxDxj dx whose graphs 0 0
are shown in Figs. 3.1 and 3.2. Changing the value of k scales the graphs horizontally but not vertically. Based on the graphs the following conclusion can be drawn. Uncertainty principle for KdV in time-frequency variables Let Dx and Dx be defined by (3.6). Then there exist positive constants U3, U4 such that ! ! Z 3p Z 3p 2k 2k 2k 2k jDsj dx jDxj dx P U 3 3p 0 3p 0
ð3:7aÞ
and 2k 3p
Z
3p 2k
jDsDxj dx P U 4 :
ð3:7bÞ
0
Moreover as s approaches infinity, " ! !# " ! Z 3p Z 3p Z 3p 2k 2k 2k 2k 2k 2k lim jDsj dx jDxj dx ¼ lim jDsj dx s!þ1 s!1 3p 0 3p 0 3p 0
2k 3p
Z 0
3p 2k
!# jDxj dx
¼
4 3p
ð3:8aÞ
and lim
s!þ1
2k 3p
Z 0
3p 2k
jDsDxj dx ¼ lim
s!1
4k3 p
Z 0
p 4k3
jDsDxj dx ¼
4 : 3p
ð3:8bÞ
The exact numerical values of the constants U3, U4 are not known. Numerical computations yield U3 0.2816 (attained at s 0.1030) and U4 0.2729 (attained at s 0.1049).
R 3p R 3p 2k 2k 2k 2k Fig. 3.1. Graph of 3p jDsj dx 3p jDxj dx plotted for k = 1.0; s is measured along the horizontal axis, the value of 0 0
R 3p R 3p 2k 2k 2k 2k jDsj dx 3p jDxj dx along the vertical axis. For other values of k the graph scales horizontally but not vertically. 0 3p 0
R 3p R 3p 2k 2k 2k 2k Fig. 3.2. Graph of 3p jDsDxj dx plotted for k = 1.0, s is measured along the horizontal axis, the value of 3p jDsDxj dx along the 0 0 vertical axis. For other values of k the graph scales horizontally but not vertically.
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4. Conclusion The physical uncertainty principle was obtained by N. Bohr and W. Heisenberg in the context of quantum mechanics based on physical arguments some of which are described in [6,7]. As shown in [6,7] it exhibits itself in different forms and even almost a century since its discovery physicists are still divided on whether it is a fundamental law of nature or just a reflection of our limitation in measurements. A few years after the discovery of the physical uncertainty principle, formulas (1.12) were obtained and interpreted as the mathematical derivation of the uncertainty principle. Although some hoped that the derivation of (1.12) would put to an end the controversy on the philosophical nature of the uncertainty principle, the effect was quite the opposite. Numerous discussions can be found on the internet, some quite interesting. Meanwhile the uncertainty principle has been extended from quantum mechanics to statistics, engineering, finance and many other fields. As essentially a property of the Fourier transform, formulas (1.12) appear wherever the Fourier transform is, and that means practically in all fields described by linear partial differential equations. The uncertainty principle (2.7) described in this paper adds another facet to both the controversy and applications of the general uncertainty principle. Being obtained for a nonlinear equation, it shows that even in mathematical formulas the uncertainty principle appears in forms other than (1.12). Although in many aspects fundamentally different from the uncertainty principle (1.12), the uncertainty principle (2.7) nevertheless serves the same purpose: to describe the balancing act between wave and particle aspects of a single motion that exhibits both wave and particle properties. The uncertainty principle (1.12) is global in the space variable x for it relates the average deviations D - and Dx from their corresponding averages - and x, the averaging being over the whole space. The uncertainty principle (2.7) is local, in the sense that it relates the deviations Dl and Dn of the local frequency and position from correspondingly the internal frequency 2k and the singularity. Can the uncertainty principle (2.7) affect our understanding of quantum mechanics and physical aspects of the general uncertainty principle? Hopefully so, although not in the form formulated in this paper since it is derived for KdV which is not a fundamental equation of quantum mechanics. But just like KdV is a nonlinear generalization of its linear part, there most likely exists an appropriate nonlinear generalization of the Schrodinger equation that not only incorporates all quantum-mechanical aspects of the original Schrodinger equation but also extends it to describe new phenomena. An extension of the uncertainty principle (2.7) to such an equation would be a viable alternative to (1.12) as a mathematical derivation of the physical uncertainty principle.
References [1] Arkadiev VA, Pogrebkov AK, Polivanov MK. Theor Math Phys 1983;53:1051. Also preprint Singulayrniye resheniyea uravneniya KdV i metod obratnoy zadachi [in Russian]. [2] Beutler R. J Math Phys 1993;34:3098. [3] Beutler R, Stahlhofen A, Matveev VB. What do solitons, breathers and positons have in common? Phys Scr 1994;50:9–12. [4] Dym H, McKean HP. Fourier series and integrals. New York: Academic Press; 1972. [5] Fermi E. Notes on quantum mechanics. 2nd ed. University of Chicago Press; 1995. [6] Feynman R, Leighton R, Sands M. The Feynman lectures on physics, V. I. Addison-Wesley Publishing Company; 1977. [7] Feynman RP, Hibbs AR. Quantum mechanics and path integrals. McGraw-Hill Companies; 1965. [8] Gelfand IM, Levitan BM. Izv Akad Nauk USSR, Se Mat 1951;15:309–60. [9] Jaworski M. Breather-like solution of the Korteweg–de Vries equation. Phys Lett A 1984;104(5):245–7. [10] Kovalyov M. On a class of solutions of KdV. J Differ Equations 2005;213:1–80. [11] Kovalyov M. Basic motions of the Korteweg–de Vries equation. Nonlinear Anal Theory Methods Appl 1998;31(5/6): 599–619. [12] Kovalyov M. Modulating properties of harmonic breather solutions of KdV. J Phys A: Math Gen 1998;31:5117–28. [13] Ma WX. Complexiton solutions to the Korteweg–de Vries equation. Phys Lett A 2002;301(1–2):35–44. [14] Matveev VB. Asymptotics of the multipositon–soliton s-function of the Korteweg–de Vries equation and the supertransparency. J Math Phys 1994;35:2955. [15] Matveev VB. Phys Lett A 1992;135:2009–12. [16] Matveev VB. Trivial S-matrices, Wigner–Von Neumann resonances and positon solutions of the integrable systems. In: Boutet de Monvel A, Marchenko V, editors. Algebraic and geometric methods in mathematical physics, Proceedings of international workshop. Kluwer Academic Publishers; 1996. p. 343–55. [17] Matveev VB, Salle MA. In: Darboux transformations and solitons. Springer series in nonlinear dynamics. Berlin, Heidelberg: Springer-Verlag; 1991.
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[18] El Naschie MS. Fredholm operators and the wave–particle duality in cantorian space. Chaos, Solitons & Fractals 1998;9(6): 975. [19] Novikov R, Khenkin G. Oscillating weakly localized solutions of the Korteweg–de Vries equation. Teor Matemat Fizika 1984;61(2):199–213 [in Russian]. [20] Stahlhofen AA. On completely transparent potentials of the Shro¨dinger equation. Phys Rev A 1995;51:934. [21] Stahlhofen AA, Maisch H. Dynamical properties of positons. Phys Scr 1995;52:228. [22] van der Waerden BL. Algebra. New York: Springer-Verlag; 1991.