- Email: [email protected]
New solutions for difference-difference models of the φ4 equation and nonlinear Klein-Gordon equation in 1 + 1 dimension
Chaos, Solitons
& Fracrals Vol. 7, No. 3, pp. 365536Y. 1996 Copyrtght 0 1996 Elsewer Science Ltd Printed in Great Britain. Ail rights reserved 0960-G779/96 $15.00 - 0.00
0960-0779(95)00075-5
New Solutions for Difference-Difference Models of the #4 Equation and Nonlinear Klein-Gordon Equation in 1 + 1 Dimension KAZUAKI
NARITA
BlOlO, CI Heights, 1-31 Yamada-Nishi, Suita-Shi, Osaka 565, Japan (Accepted 14 July 1995)
Abstract-We present a new solution for Hirota’s difference-difference model of the @4equation in 1 + 1 dimension, extending Hirota’s kink solution. We also present three new solutions for a difference-difference model of the nonlinear Klein-Gordon equation. We discuss asymptotic behaviours of these solutions in small limits of the lattice spacing.
1. INTRODUCTION
As is well known, the discretization of integrable systems must pass the criterion that the discretized equation should also exhibit an N-solution if the original equation exhibits an N-soliton solution. A similar criterion exists for nonintegrable soliton equations. That is, if the original equation exhibits a soliton solution, then the discretization is regarded as proper when the discretized equation also exhibits a soliton solution. For two nonintegrable nonlinear wave equations, i.e. the ti4 equation in 1 + 1 dimension [l]
P2 - (llc*P:l@(x~t) = -ct’a, t) + Pd<& t), (a, B > 01, and the nonlinear Klein-Gordon
(1)
equation in 1 + 1 dimension [2]
PZ - w>~:14+, t) = aa> t) - iw3’ I), cm,B > O),
(2)
two types of discretized equations exist, which are proper in the above meaning. One of them is a wave equation with cubic nonlinearity discretized along one of two light cones [3-51, and the other is the full-discretized +4 equation in 1 + 1 dimension presented in 1979 by Hirota [6] (hn+2.o -I- An,, - bt+l.n+~ - h+l,n-d/A*
= t-~/4N#bl.2,~ +
the full-discretized
cpm,,
wNhn+*.n
+
+
4%n+1.n+1
+
&n+,.n-,I
~rn,n)~m+l.n+l~m+l.n-1
~m+l,n-*)~m+2,n~m.~l (3) nonlinear Klein-Gordon equation in 1 + 1
+
and its generalization, dimension
+
w?l+l,n+l
+
(@m+2,n+ Qm,,,- d~+~,~+l - 4m+l,n-~)/A2 = (d‘Wm+2,n
+ hn
+ <-P/4)K9h+2,n +
(&?l+l,n+l
+
+ (Pm+l,n+l+ &+l,n-l)
+ ~m,n)~m+~,n+l~m+l.n-l
~m+l,n-l)~m+2,n~m,nl.
(4)
In (3) and (4) we assume that m + IZ is even. By putting x = Am, t = An/c and taking the limit A + 0, (3) is reduced to (l), and (4) is reduced to (2). 365
K. NARITA
366
In this paper, we find all particular solutions for (3) and (4) corresponding to the particular solutions found previously for the semi-discretized equation [3-51, and discuss their asymptotic behaviours in small limits of the lattice spacing. Unlike the previously presented semi-discretized equation, (3) and (4) are characterized by two independent subscripts representing space and time participating on equal footings. Its advantage is that the newly found solutions can give more detailed and more accurate information than the previously found solutions for the semi-discretized equation concerning their asymptotic behaviours in small limits of the lattice spacing. In Section 2, we extend the previously obtained Hirota’s kink solution [6] for (3) and present a superposed solution of a kink, a weak plane wave and a weak envelope soliton. The presentation of this solution is considered to provide a new step toward a resolution of the quark-confinement problem [7]. In Section 3, we present three types of the solution for (4), i.e. a superposed solution of an envelope hole, a weak plane wave and a very weak envelope soliton, an envelope hole pair solution, and a superposed solution of a baseband solution, a weak plane wave and a weak envelope soliton. 2. HIROTA’S
DIFFERENCE-DIFFERENCE MODEL DIMENSION
OF THE G4 EQUATION
IN I+
1
In this section, we present a new solution for (3) extending Hirota’s kink solution which is a superposition of a kink, a weak plane wave and a weak envelope soliton. For this purpose, we assume a trial form of the solution given as + (-l)(m’n)‘20 + S]]. 4 mn = (+)‘(~/~)‘~z[(-1)~“‘“~~2~~~~+~tanh [AK(m-(u/c)n) Substituting (5) into (3), we find that the following equation should be satisfied: 2psinh2X[acosh2X
+ b - (-l)(mfn)‘2 c]/[cosh2X + cosh2((-l)(“‘“)‘% x [cosh2X + cosh2(F(-l)(“‘“)‘%
- AK)] - AKU/C)] = 0, (6)
in which X, a, b, c are given as X = dK(m - (u/c)n + 1) + 6,
(7)
a = (A20(/2)(l - ,02 + E*Y), b = (A2a//2)[(1 + p2 + c*?)cosh20
x cosh[A~(l
+ u/c)]cosh[A~(l
- 2cosh20sinh[A~(l
(8)
+ .~-2/~I~sinh28] - u/c)]
+ u/c)]sinh[A~(l
- u/c)],
c = (A*o1/2)[(1 + p2 + ~~?)sinh28 + e*2/~/ocosh20]sinh[A~(l X cosh[AK(l + o/c)] - 2sinh20sinh[AK(l Now we have the equations
(5)
rt u/c)]cosh[AK(l
(9) T ZJ/C)] T u/c)].
a=b=c=O.
(10) (11)
We found that these equations admit the following reduction: 2 tanh [AK(~ + u/c)] tanh [AK(~ - v/c)] = A*a (1 + ~~2) + rsinhl28,
(12)
sinh*28 = A2e2~~sinh2[A~(1
(13)
in which r = 2sinh[A~(l
Ifr u/c)]cosech[A~(l
r u/c)]/[rcosh*[A~(l
r u/c)]sech[A~(l
T u/c)] - A’cu],
+ u/c)]sech[A~(l
- u/c)],
(14)
Difference-difference
367
models
and p = sgn[8(1 - ]ul/c)](l
+ E’#‘.
(15)
Details of this proof are omitted here. A solution for a difference-differential model of (1) corresponding to the above equations has already been reported in Refs [3, 41. By regarding E and A as small quantities having magnitudes of the same order, and using (12)-(Z), we can expand parameters of the solution in powers of E and A as follows: u = (+)“c(l
- ~~/2r?)“’ [l - A*(o(/3)(1 - a/42)/(1
- ~“&/4ti(l
- (~/2K2)
- 01/2$) -t O(A4)],
/8/ = A2~(~/8)1’21~j]~\[[1T (k)“(l
(16)
- CV/~K’)~‘~]~‘~/[~ -C(k)“(l
- a/21?)‘~‘]““](1 + O(A2)), (17)
p = sgn e[l + c22/2 + O(c4)].
(18) By taking the limit E+ 0, A -+ 0 as E/A = const., @,,, is found to give the kink solution for (1) with everywhere roughness @(x,t) = (+)“‘(cr/@‘” tanh [~(x(T)“c(l
3. DIFFERENCE-DIFFERENCE
- a/2K2)li2t) + 61.
(19)
MODEL OF THE NONLINEAR KLEIN’-GORDON EQUATION IN 1 + 1 DIMENSION
In this section, we present three new solutions for (4).
Solution 1 First, we present a solution representing a superposition of an envelope hole, a weak plane wave and a very weak envelope soliton. For this purpose, we assume a trial form of the solution for (4) given as @m.n = (+)‘A(~/ulp)1/2[(-l)(mi-n)/2~(~I + (-l)(mTn)/2 ptanh[AK(m-(u/c)n)
+ (-1)“8
+ S]]. (20)
Substituting (20) into (4), we find that the following equation should be satisfied: C-1) (“‘“)‘22p[acosh2X
+ b - (-l)“csinh2X]/[cosh2(X
+ (-1)“0)
x [cosh2(X - (-1)“0)
+ cosh(A2~)]
+ cosh(A2&)]
= 0, (21)
in which X, a, b, c are given as X = AK(~ - (u/c)n + 1) + 6, a = (A’a/2)[[1
(22)
+ A2(p2 + c22)] cosh28 + A2&* 21XIosinh28] sinh [a~(1 + U/C)]
x cosh[A~(l
T V/C)] - 2cosh28sinh[A~(l
+ u/c)]cosh[A~(l
b = (A20z/4)[1 - A2(p2 - e2$)] sinh[A2K(l c = (A2~/2)[[1 + A2(p2 + e2$)]sinh28
+ u/c)],
+ u/c)] - sinh[A2~(1 T u/c)],
+ A2&*2]~Iocosh28]sinh[A~(1
xcosh [AK(~ + u/c)] - 2sinh28sinh[A~(l
(23)
f u/c)]cosh[A~(l
(24)
T u/c)]
T u/c)].
(25)
K. NARITA
368
Now we have the equations a=b=c=O.
(26)
We found that these equations admit the following reduction: 4 sinh [ A.K(1 T u/c)] cosech [ AK( 1 f v/c)][ 1 + cash (AUK) cash ( A~Ku/c)] = cash (AUK) -I- cash (A~Ku/c) A2cu(1 + Azc2y) * 8 sinh (AUK) sinh (A~Kv/c) cosech [AUK‘(1 t u/c) 1 (27>
x cosech [A2zc(1 i U/C)]sinh2 20, sinh’28 = A4~2~~sinh2[A~(l
T u/c)]/[4[sinh2[AK(1
_t U/C)] + sinh2[AK(1 T v/c)]]
x cosech [A2~(1 + u/c)]coth[A~(l p = fsgn [&(l
- /ul/c)] . (2/AZ)[4(l/a)
T u/c)] - A2a],
(28)
sinh (AUK) sinh (A~Ku/c) tanh [AK(~ + v/c)]
x cosech[A2rc(l + .~/c)][l - cosech’[Ax(l
T ~/c)]sinh~28]]“~.
(29)
Details of this proof are omitted here. By regarding E and A as small quantities having magnitudes of the same order, and using (27)-(29), we can expand parameters of the solution in powers of E and A as follows: u = kc[l
- A201/2+ A4+r/8
- A4e2&/2
- K2/3) - A’%(~/32 - LYI?/~ - 14~“/15)
+ O(A”)],
(30)
+ O(A2h
101= A64~/862)~xI~~I(1
p = sgn 8 * j/2/~1[1 - A2(o//4 + 2/3) + A4(d/32 + ad/12 - 11~“/30) + - A@?/128 + &?/96 - 131a~~/120 - 49@/630)
(31) A2~2y/2
- A4c2(301/8+ K2/6)2 + O(A’)].
(32) By taking the limit E+ 0, A + 0 as c/A = const., @m,nis found to give the envelope hole solution for (1) WJ)
- W’AW//3)‘p1~1
cos[b-/A)(
x i ct)]tanh[K(x
+ c(1 - A2cr/2)t) + 61. (33)
It is noticed that a term representing a very weak envelope soliton is lost in this asymptotic expression.
Solution 2
Next, we present an envelope hole pair solution for (4). For this purpose, we assume a trial form of the solution given as + Ap/[cosh[AK(m - (z&n) + S] + 511. (34) 4 m,n = (+)‘(-1)‘“‘““2(o(lp)“2[~Ix( Substituting (34) into (4) and using an undetermined constant method similar to one used in [5], we find a dispersion relation and expressions of parameters as follows: 4tanh [(A~/2)(1 + u/c)] coth [(A~/2)(1
+
u/c)]
-
A2a(l + ~‘2) = 0,
[ = ( ~)“E(x/ cosh2[(AK/~)( 1 T u/c)]/y , p = +(+)“(l/A)(l in which
+ .~~~)sinh(A~)sinh(A~u/c)/y,
(35) (36) (37)
369
Difference-difference models
y = [[(l + cash (AK) cash (AKu/c))E’~
+ sinh (AK) sinh (AKu/c)]/~]~‘~.
(38)
By taking the limit E+ 0, A + 0 as E/A = a (const.), r$,,, is found to give again the envelope hole pair solution for (2) +(x, t) - (+>‘W/ulp>l’z cos [(?r/A)(x + ct)][a~~~(i)“d/[(a2~
- K2/2)“2
X cosh[rc(x + c(l - A*cr//)t) + 6](~)“alx~]].
(39)
The upper sign of (k)” must be chosen by reason of the regularity. Solution 3 Finally, we present a superposed solution of a baseband solution, a weak plane wave and a weak envelope soliton. For this purpose, we assume a trial form of the solution for (4) given as [(-l)(m+n)‘2~I~I + p/[cosh[AK(m - (u/c)n) + 61 + (-I)‘“‘““*~]]. (40) Gm.n = (k)‘(~#‘~ Substituting (40) into (4) and using an undetermined constant method similar to one used in Ref. [5], we find a dispersion relation and expressions of parameters as follows: 4tanh[(A~/2)(1 + u/c)] tanh[(A~/2)(1 - u/c)] - A2a(l + ~~2) = 0, (41) + u/c)]/z, c = (T)“&( sinh2 [(A~/2)(1 (42) p = (&)“(l
+ ~~2) cosh(A~) cosh(A~~/~)/z,
(43)
in which z = [[(l f sinh (AK) sinh (AKu/c))E~~ + cash (AK) cash (AKu/c)]/~]@. (44) By taking the limit E+ 0, A + 0 as E/A = const., @m,, is found to give the baseband solution [2] for (2) with everywhere roughness @(x, t) = (f)‘(~)“(2a#p
sech[K(x(+)“‘c(l 4.
- a/d)“2t)+
61.
(45)
CONCLUSION
We have presented a new solution for Hirota’s difference-difference model of the (p” equation in 1 f 1 dimension extending Hirota’s kink solution. We have also presented three new solutions for a difference-difference model of the nonlinear Klein-Gordon equation. We have discussed asymptotic behaviours of these solutions in small limits of the lattice spacing. AcknotYledgement-The with this study.
author wishes to thank Mr Yoshibumi Narita of Wakatake School for his continual help
REFERENCES 1. R. F. Dashen, B. Hasslacher and A. Neveu, Particle spectrum in model field theories from semiclassical functional integral techniques, Phys. Rev. Dll, 3424 (1975). 2. V. (3. Makhankov, Dynamics of classical solitons in non-integrable systems, Phys. Rep. 35, 1 (1978). 3. K. Narita, Asymmetric discrete envelope soliton solutions for semi-discrete nonlinear wave equations. J. Phys. sm. Jpn 56,47 (1987). 4. K. Narita, Addendum to two papers on semi-discrete wave equations with cubic nonlinearity, J. Phys. Sm. Jpn 57, 388 (1988). 5. K. Narita, A discrete model for wave equation with cubic nonlinearity, J. Phys. Sot. Jpn 53, 2472 (1984). 6. R. Hirota, private communication (1979). 7. P. Vinciarelli, Temporary quark confinement in field theoretical bags, Nucl. Phys. B89, 463 (1975).
& Fracrals Vol. 7, No. 3, pp. 365536Y. 1996 Copyrtght 0 1996 Elsewer Science Ltd Printed in Great Britain. Ail rights reserved 0960-G779/96 $15.00 - 0.00
0960-0779(95)00075-5
New Solutions for Difference-Difference Models of the #4 Equation and Nonlinear Klein-Gordon Equation in 1 + 1 Dimension KAZUAKI
NARITA
BlOlO, CI Heights, 1-31 Yamada-Nishi, Suita-Shi, Osaka 565, Japan (Accepted 14 July 1995)
Abstract-We present a new solution for Hirota’s difference-difference model of the @4equation in 1 + 1 dimension, extending Hirota’s kink solution. We also present three new solutions for a difference-difference model of the nonlinear Klein-Gordon equation. We discuss asymptotic behaviours of these solutions in small limits of the lattice spacing.
1. INTRODUCTION
As is well known, the discretization of integrable systems must pass the criterion that the discretized equation should also exhibit an N-solution if the original equation exhibits an N-soliton solution. A similar criterion exists for nonintegrable soliton equations. That is, if the original equation exhibits a soliton solution, then the discretization is regarded as proper when the discretized equation also exhibits a soliton solution. For two nonintegrable nonlinear wave equations, i.e. the ti4 equation in 1 + 1 dimension [l]
P2 - (llc*P:l@(x~t) = -ct’a, t) + Pd<& t), (a, B > 01, and the nonlinear Klein-Gordon
(1)
equation in 1 + 1 dimension [2]
PZ - w>~:14+, t) = aa> t) - iw3’
(2)
two types of discretized equations exist, which are proper in the above meaning. One of them is a wave equation with cubic nonlinearity discretized along one of two light cones [3-51, and the other is the full-discretized +4 equation in 1 + 1 dimension presented in 1979 by Hirota [6] (hn+2.o -I- An,, - bt+l.n+~ - h+l,n-d/A*
= t-~/4N#bl.2,~ +
the full-discretized
cpm,,
wNhn+*.n
+
+
4%n+1.n+1
+
&n+,.n-,I
~rn,n)~m+l.n+l~m+l.n-1
~m+l,n-*)~m+2,n~m.~l (3) nonlinear Klein-Gordon equation in 1 + 1
+
and its generalization, dimension
+
w?l+l,n+l
+
(@m+2,n+ Qm,,,- d~+~,~+l - 4m+l,n-~)/A2 = (d‘Wm+2,n
+ hn
+ <-P/4)K9h+2,n +
(&?l+l,n+l
+
+ (Pm+l,n+l+ &+l,n-l)
+ ~m,n)~m+~,n+l~m+l.n-l
~m+l,n-l)~m+2,n~m,nl.
(4)
In (3) and (4) we assume that m + IZ is even. By putting x = Am, t = An/c and taking the limit A + 0, (3) is reduced to (l), and (4) is reduced to (2). 365
K. NARITA
366
In this paper, we find all particular solutions for (3) and (4) corresponding to the particular solutions found previously for the semi-discretized equation [3-51, and discuss their asymptotic behaviours in small limits of the lattice spacing. Unlike the previously presented semi-discretized equation, (3) and (4) are characterized by two independent subscripts representing space and time participating on equal footings. Its advantage is that the newly found solutions can give more detailed and more accurate information than the previously found solutions for the semi-discretized equation concerning their asymptotic behaviours in small limits of the lattice spacing. In Section 2, we extend the previously obtained Hirota’s kink solution [6] for (3) and present a superposed solution of a kink, a weak plane wave and a weak envelope soliton. The presentation of this solution is considered to provide a new step toward a resolution of the quark-confinement problem [7]. In Section 3, we present three types of the solution for (4), i.e. a superposed solution of an envelope hole, a weak plane wave and a very weak envelope soliton, an envelope hole pair solution, and a superposed solution of a baseband solution, a weak plane wave and a weak envelope soliton. 2. HIROTA’S
DIFFERENCE-DIFFERENCE MODEL DIMENSION
OF THE G4 EQUATION
IN I+
1
In this section, we present a new solution for (3) extending Hirota’s kink solution which is a superposition of a kink, a weak plane wave and a weak envelope soliton. For this purpose, we assume a trial form of the solution given as + (-l)(m’n)‘20 + S]]. 4 mn = (+)‘(~/~)‘~z[(-1)~“‘“~~2~~~~+~tanh [AK(m-(u/c)n) Substituting (5) into (3), we find that the following equation should be satisfied: 2psinh2X[acosh2X
+ b - (-l)(mfn)‘2 c]/[cosh2X + cosh2((-l)(“‘“)‘% x [cosh2X + cosh2(F(-l)(“‘“)‘%
- AK)] - AKU/C)] = 0, (6)
in which X, a, b, c are given as X = dK(m - (u/c)n + 1) + 6,
(7)
a = (A20(/2)(l - ,02 + E*Y), b = (A2a//2)[(1 + p2 + c*?)cosh20
x cosh[A~(l
+ u/c)]cosh[A~(l
- 2cosh20sinh[A~(l
(8)
+ .~-2/~I~sinh28] - u/c)]
+ u/c)]sinh[A~(l
- u/c)],
c = (A*o1/2)[(1 + p2 + ~~?)sinh28 + e*2/~/ocosh20]sinh[A~(l X cosh[AK(l + o/c)] - 2sinh20sinh[AK(l Now we have the equations
(5)
rt u/c)]cosh[AK(l
(9) T ZJ/C)] T u/c)].
a=b=c=O.
(10) (11)
We found that these equations admit the following reduction: 2 tanh [AK(~ + u/c)] tanh [AK(~ - v/c)] = A*a (1 + ~~2) + rsinhl28,
(12)
sinh*28 = A2e2~~sinh2[A~(1
(13)
in which r = 2sinh[A~(l
Ifr u/c)]cosech[A~(l
r u/c)]/[rcosh*[A~(l
r u/c)]sech[A~(l
T u/c)] - A’cu],
+ u/c)]sech[A~(l
- u/c)],
(14)
Difference-difference
367
models
and p = sgn[8(1 - ]ul/c)](l
+ E’#‘.
(15)
Details of this proof are omitted here. A solution for a difference-differential model of (1) corresponding to the above equations has already been reported in Refs [3, 41. By regarding E and A as small quantities having magnitudes of the same order, and using (12)-(Z), we can expand parameters of the solution in powers of E and A as follows: u = (+)“c(l
- ~~/2r?)“’ [l - A*(o(/3)(1 - a/42)/(1
- ~“&/4ti(l
- (~/2K2)
- 01/2$) -t O(A4)],
/8/ = A2~(~/8)1’21~j]~\[[1T (k)“(l
(16)
- CV/~K’)~‘~]~‘~/[~ -C(k)“(l
- a/21?)‘~‘]““](1 + O(A2)), (17)
p = sgn e[l + c22/2 + O(c4)].
(18) By taking the limit E+ 0, A -+ 0 as E/A = const., @,,, is found to give the kink solution for (1) with everywhere roughness @(x,t) = (+)“‘(cr/@‘” tanh [~(x(T)“c(l
3. DIFFERENCE-DIFFERENCE
- a/2K2)li2t) + 61.
(19)
MODEL OF THE NONLINEAR KLEIN’-GORDON EQUATION IN 1 + 1 DIMENSION
In this section, we present three new solutions for (4).
Solution 1 First, we present a solution representing a superposition of an envelope hole, a weak plane wave and a very weak envelope soliton. For this purpose, we assume a trial form of the solution for (4) given as @m.n = (+)‘A(~/ulp)1/2[(-l)(mi-n)/2~(~I + (-l)(mTn)/2 ptanh[AK(m-(u/c)n)
+ (-1)“8
+ S]]. (20)
Substituting (20) into (4), we find that the following equation should be satisfied: C-1) (“‘“)‘22p[acosh2X
+ b - (-l)“csinh2X]/[cosh2(X
+ (-1)“0)
x [cosh2(X - (-1)“0)
+ cosh(A2~)]
+ cosh(A2&)]
= 0, (21)
in which X, a, b, c are given as X = AK(~ - (u/c)n + 1) + 6, a = (A’a/2)[[1
(22)
+ A2(p2 + c22)] cosh28 + A2&* 21XIosinh28] sinh [a~(1 + U/C)]
x cosh[A~(l
T V/C)] - 2cosh28sinh[A~(l
+ u/c)]cosh[A~(l
b = (A20z/4)[1 - A2(p2 - e2$)] sinh[A2K(l c = (A2~/2)[[1 + A2(p2 + e2$)]sinh28
+ u/c)],
+ u/c)] - sinh[A2~(1 T u/c)],
+ A2&*2]~Iocosh28]sinh[A~(1
xcosh [AK(~ + u/c)] - 2sinh28sinh[A~(l
(23)
f u/c)]cosh[A~(l
(24)
T u/c)]
T u/c)].
(25)
K. NARITA
368
Now we have the equations a=b=c=O.
(26)
We found that these equations admit the following reduction: 4 sinh [ A.K(1 T u/c)] cosech [ AK( 1 f v/c)][ 1 + cash (AUK) cash ( A~Ku/c)] = cash (AUK) -I- cash (A~Ku/c) A2cu(1 + Azc2y) * 8 sinh (AUK) sinh (A~Kv/c) cosech [AUK‘(1 t u/c) 1 (27>
x cosech [A2zc(1 i U/C)]sinh2 20, sinh’28 = A4~2~~sinh2[A~(l
T u/c)]/[4[sinh2[AK(1
_t U/C)] + sinh2[AK(1 T v/c)]]
x cosech [A2~(1 + u/c)]coth[A~(l p = fsgn [&(l
- /ul/c)] . (2/AZ)[4(l/a)
T u/c)] - A2a],
(28)
sinh (AUK) sinh (A~Ku/c) tanh [AK(~ + v/c)]
x cosech[A2rc(l + .~/c)][l - cosech’[Ax(l
T ~/c)]sinh~28]]“~.
(29)
Details of this proof are omitted here. By regarding E and A as small quantities having magnitudes of the same order, and using (27)-(29), we can expand parameters of the solution in powers of E and A as follows: u = kc[l
- A201/2+ A4+r/8
- A4e2&/2
- K2/3) - A’%(~/32 - LYI?/~ - 14~“/15)
+ O(A”)],
(30)
+ O(A2h
101= A64~/862)~xI~~I(1
p = sgn 8 * j/2/~1[1 - A2(o//4 + 2/3) + A4(d/32 + ad/12 - 11~“/30) + - A@?/128 + &?/96 - 131a~~/120 - 49@/630)
(31) A2~2y/2
- A4c2(301/8+ K2/6)2 + O(A’)].
(32) By taking the limit E+ 0, A + 0 as c/A = const., @m,nis found to give the envelope hole solution for (1) WJ)
- W’AW//3)‘p1~1
cos[b-/A)(
x i ct)]tanh[K(x
+ c(1 - A2cr/2)t) + 61. (33)
It is noticed that a term representing a very weak envelope soliton is lost in this asymptotic expression.
Solution 2
Next, we present an envelope hole pair solution for (4). For this purpose, we assume a trial form of the solution given as + Ap/[cosh[AK(m - (z&n) + S] + 511. (34) 4 m,n = (+)‘(-1)‘“‘““2(o(lp)“2[~Ix( Substituting (34) into (4) and using an undetermined constant method similar to one used in [5], we find a dispersion relation and expressions of parameters as follows: 4tanh [(A~/2)(1 + u/c)] coth [(A~/2)(1
+
u/c)]
-
A2a(l + ~‘2) = 0,
[ = ( ~)“E(x/ cosh2[(AK/~)( 1 T u/c)]/y , p = +(+)“(l/A)(l in which
+ .~~~)sinh(A~)sinh(A~u/c)/y,
(35) (36) (37)
369
Difference-difference models
y = [[(l + cash (AK) cash (AKu/c))E’~
+ sinh (AK) sinh (AKu/c)]/~]~‘~.
(38)
By taking the limit E+ 0, A + 0 as E/A = a (const.), r$,,, is found to give again the envelope hole pair solution for (2) +(x, t) - (+>‘W/ulp>l’z cos [(?r/A)(x + ct)][a~~~(i)“d/[(a2~
- K2/2)“2
X cosh[rc(x + c(l - A*cr//)t) + 6](~)“alx~]].
(39)
The upper sign of (k)” must be chosen by reason of the regularity. Solution 3 Finally, we present a superposed solution of a baseband solution, a weak plane wave and a weak envelope soliton. For this purpose, we assume a trial form of the solution for (4) given as [(-l)(m+n)‘2~I~I + p/[cosh[AK(m - (u/c)n) + 61 + (-I)‘“‘““*~]]. (40) Gm.n = (k)‘(~#‘~ Substituting (40) into (4) and using an undetermined constant method similar to one used in Ref. [5], we find a dispersion relation and expressions of parameters as follows: 4tanh[(A~/2)(1 + u/c)] tanh[(A~/2)(1 - u/c)] - A2a(l + ~~2) = 0, (41) + u/c)]/z, c = (T)“&( sinh2 [(A~/2)(1 (42) p = (&)“(l
+ ~~2) cosh(A~) cosh(A~~/~)/z,
(43)
in which z = [[(l f sinh (AK) sinh (AKu/c))E~~ + cash (AK) cash (AKu/c)]/~]@. (44) By taking the limit E+ 0, A + 0 as E/A = const., @m,, is found to give the baseband solution [2] for (2) with everywhere roughness @(x, t) = (f)‘(~)“(2a#p
sech[K(x(+)“‘c(l 4.
- a/d)“2t)+
61.
(45)
CONCLUSION
We have presented a new solution for Hirota’s difference-difference model of the (p” equation in 1 f 1 dimension extending Hirota’s kink solution. We have also presented three new solutions for a difference-difference model of the nonlinear Klein-Gordon equation. We have discussed asymptotic behaviours of these solutions in small limits of the lattice spacing. AcknotYledgement-The with this study.
author wishes to thank Mr Yoshibumi Narita of Wakatake School for his continual help
REFERENCES 1. R. F. Dashen, B. Hasslacher and A. Neveu, Particle spectrum in model field theories from semiclassical functional integral techniques, Phys. Rev. Dll, 3424 (1975). 2. V. (3. Makhankov, Dynamics of classical solitons in non-integrable systems, Phys. Rep. 35, 1 (1978). 3. K. Narita, Asymmetric discrete envelope soliton solutions for semi-discrete nonlinear wave equations. J. Phys. sm. Jpn 56,47 (1987). 4. K. Narita, Addendum to two papers on semi-discrete wave equations with cubic nonlinearity, J. Phys. Sm. Jpn 57, 388 (1988). 5. K. Narita, A discrete model for wave equation with cubic nonlinearity, J. Phys. Sot. Jpn 53, 2472 (1984). 6. R. Hirota, private communication (1979). 7. P. Vinciarelli, Temporary quark confinement in field theoretical bags, Nucl. Phys. B89, 463 (1975).