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NOTE: Special Solutions to Nonlinear Difference-Differential Equations
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
205, 273]279 Ž1997.
AY965171
NOTE Special Solutions to Nonlinear Difference-Differential Equations Kazuaki Narita B1010 CI Heights, 1-31 Yamada-Nishi, Suita-Shi, Osaka 565, Japan Submitted by William F. Ames Received January 29, 1996
In a previous paper, we presented new difference-differential equations, and studied their N-soliton solution, elliptic function solutions, and 1-soliton solution under a nonvanishing boundary condition at the infinity w1x. In this note, we present new special solutions for the above equations which cannot be obtained from known solutions through the transformation. We also study the corresponding physical solutions for the new equations derived from the above equations through seven different transformations. Recently, Shabat and Yamilov have presented various new solvable difference-differential equations w2]4x. We also make reference to the connection between their equations and our transformed equations. First, we remark that Eqs. Ž1.7., Ž2.3. of w1x are rewritten through the transformation i ª n, u i ª 2 q '2 u n , ¨ i ª Ž1 q ¨ n .r2 in the forms u ˙n s 2 Ž u ny1 y u nq1 . u 2nr Ž u ny1 u n y 1 .
1r2
q Ž u n u nq1 y 1 .
¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . Ž 1 q ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
1r2 2
, Ž 1.
Ž 2.
Equation Ž2.2. of w1x, giving a relation between Ž1. and Ž2., is also rewritten in the form ¨ n2 s u ny1r2 u nq1r2 y 1,
Ž 3.
and consequently Ž1. becomes 2
u ˙n s 2 Ž u ny1 y u nq1 . u 2nr Ž ¨ ny1r2 q ¨ nq1r2 . .
Ž 4.
273 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
274
NOTE
As such types of solutions for Ž1., Ž2. that are underivable from known ones, we present the equations u n s Ž " . 1 Ž cos x q cosh u . rsin Ž kr2 . sinh u ,
Ž 5.
¨ n s Ž " . 2 cos x q cos Ž kr2 . cosh u rsin Ž kr2 . sinh u ,
Ž 6.
x s kn q vt q d,
Ž 7.
where
and u is an arbitrary parameter. We find that Ž5., Ž6. automatically satisfy Ž3.. Substituting Ž5., Ž6. into Ž4., we find the dispersion relation
v s y2 tan Ž kr2 . .
Ž 8.
In the following, we present the new equations derived from Ž1., Ž2. through seven different transformations, and discuss their physical solutions. Transformation 1.
u n ª iu n , ¨ n ª i¨ n .
This transformation converts Ž1., Ž2. into u ˙n s 2 Ž u ny1 y u nq1 . u 2nr Ž 1 q u ny1 u n .
1r2
q Ž 1 q u n u nq1 .
1r2 2
¨˙n s 2 Ž ¨ nq1 y ¨ ny1 . Ž 1 y ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
, Ž 9.
Ž 10 .
Equation Ž9. is also obtained by assuming m s 0, h s '2 in Eqs. Ž18., Ž19. of w2x. The corresponding physical solutions are derived from Ž5., Ž6., Ž8. through the transformation u ª i u to become u n s Ž . . 1 Ž cos x q cos u . rsin Ž kr2 . sin u ,
Ž 11 .
¨ n s Ž . . 2 cos x q cos Ž kr2 . cos u rsin Ž kr2 . sin u ,
Ž 12 .
v s y2 tan Ž kr2 . .
Ž 13 .
Under the assumption u s pr2, Ž11., Ž12. give the solutions
Ž . . 1 u n s Ž . . 2 ¨ n s cos xrsin Ž kr2. . Transformation 2.
Ž 14.
u n ª Žy1. n u n , ¨ n ª Žy1. n ¨ n .
This transformation converts Ž1. into Ž9., and Ž2. into ¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . Ž 1 q ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 15 .
NOTE
275
Equation Ž15. is also obtained by assuming Q Ž qn . s y2 qn2 y 2
Ž 16 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. The corresponding physical solutions are derived from Ž5., Ž6., Ž8. through the transformation k ª k q p to become n
u n s Ž " . 1 cos x q Ž y1 . cosh u rcos Ž kr2 . sinh u , n
Ž 17 .
¨ n s Ž " . 2 cos x y Ž y1 . sin Ž kr2 . cosh u rcos Ž kr2 . sinh u ,
Ž 18 .
v s 2 cot Ž kr2 . .
Ž 19 .
Transformation 3.
u n ª Žy1. n iu n , ¨ n ª Žy1. n i¨ n .
This transformation converts Ž1. into itself and Ž2. into ¨˙n s 2 Ž ¨ nq1 y ¨ ny1 . Ž 1 y ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 20 .
Equation Ž20. is also obtained by assuming Q Ž qn . s y2 qn2 q 2
Ž 21 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. We remark here that Ž20. is derived from the new difference-differential equation w ˙n s Ž wny1 q wnq1 . wn y 4 r Ž wny1 y wnq1 .
Ž 22 .
through the transformation ¨ n s Ž " . 3 Ž wny1r2 wnq1r2 y 1 . .
Ž 23 .
The corresponding physical solutions for u n , ¨ n are derived from Ž5., Ž6., Ž8. through the transformation u ª i u , k ª k q p to become n
u n s Ž . . 1 cos x q Ž y1 . cos u rcos Ž kr2 . sin u , n
Ž 24 .
¨ n s Ž . . 2 cos x y Ž y1 . sin Ž kr2 . cos u rcos Ž kr2 . sin u ,
Ž 25 .
v s 2 cot Ž kr2 . .
Ž 26 .
Under the assumption u s pr2, Ž24., Ž25. give the solutions
Ž . . 1 u n s Ž . . 2 ¨ n s cos xrcos Ž kr2. .
Ž 27.
276
NOTE
The solution for wn under the same assumption reads wn s Ž " . 4'2 cos Ž xr2. r cos Ž kr2 .
1r2
,
Ž 28 .
in which the choice of signs must satisfy the condition
Ž . . 2 Ž " . 31 s 1. Transformation 4.
Ž 29.
u n ª 1ru n , ¨ n ª 1r¨ n .
This transformation converts Ž1., Ž2. into u ˙n s 2 Ž u ny1 y u nq1 . u n r
½
u nq 1 Ž 1 y u ny1 u n .
1r2
q u ny1 Ž 1 y u n u nq1 .
1r2 2
5,
¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 q ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
Ž 30 . Ž 31 .
The corresponding physical solutions are derived from Ž5., Ž6., Ž8. to become u n s Ž " . 1sin Ž kr2 . sinh ur Ž cos x q cosh u . ,
Ž 32 .
¨ n s Ž " . 2sin Ž kr2 . sinh ur cos x q cos Ž kr2 . cosh u ,
Ž 33 .
v s y2 tan Ž kr2 . .
Ž 34 .
In Ž33., k must satisfy the condition < cos Ž kr2 . < ) sech u
Ž 35 .
in order that ¨ n is nonsingular. Transformation 5.
u n ª iru n , ¨ n ª ir¨ n .
This transformation converts Ž1., Ž2. into u ˙n s 2 Ž u ny1 y u nq1 . u n r
½
u nq 1 Ž 1 q u ny1 u n .
1r2
q u ny1 Ž 1 q u n u nq1 .
1r2 2
5,
¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 y ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
Ž 36 . Ž 37 .
Equation Ž37. is also obtained by assuming m s '2 , h s 0 in Eqs. Ž18., Ž19. of w2x.
277
NOTE
One of the corresponding physical solutions is derived from Ž5., Ž6., Ž8. through the transformation k ª i k , v ª i v , d ª i d to become u n s Ž . . 1sinh Ž kr2 . sinh ur Ž cosh x q cosh u . ,
Ž 38 .
¨ n s Ž . . 2sinh Ž kr2 . sinh ur cosh x q cosh Ž kr2 . cosh u ,
Ž 39 .
v s y2 tanh Ž kr2 . .
Ž 40 .
By taking the limit x ª x Ž".5 u , u ª `, Ž38. ] Ž40. give the different solutions u n s Ž . . 1 Ž 1r2 . sinh Ž kr2 . 1 Ž . . 5tanh Ž xr2. ,
Ž 41 .
¨ n s Ž . . 2 Ž 1r2 . tanh Ž kr2 . 1 Ž . . 5tanh
Ž 42 .
Ž x Ž .. 5 f . r2 4 ,
in which
f s log cosh Ž kr2 . .
Ž 43 .
Another physical solution is derived from Ž5., Ž8. through the transformation k ª i k q p , v ª i v , d ª i d , u ª i u to become n
u n s Ž . . 1cosh Ž kr2 . sin ur Ž y1 . cosh x q cos u ,
v s y2 coth Ž kr2 . .
Ž 44 . Ž 45 .
Under the assumption u s pr2, Ž44. gives the solution n
u n s Ž . . 1 Ž y1 . cosh Ž kr2 . sech x. Transformation 6.
Ž 46 .
u n ª Žy1. nru n , ¨ n ª Žy1. nr¨ n .
This transformation converts Ž1. into Ž36., and Ž2. into ¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 q ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 47 .
Equation Ž47. is also obtained by assuming Q Ž qn . s y2 qn4 y 2 qn2
Ž 48 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. The corresponding physical solutions are derived from Ž5., Ž6., Ž8. through the transformation k ª k q p to become n
u n s Ž " . 1cos Ž kr2 . sinh ur cos x q Ž y1 . cosh u , n
Ž 49 .
¨ n s Ž " . 2cos Ž kr2 . sinh ur cos x y Ž y1 . sin Ž kr2 . cosh u ,
Ž 50 .
v s 2 cot Ž kr2 . .
Ž 51 .
278
NOTE
In Ž50., k must satisfy the condition sin Ž < k
Ž 52 .
in order that ¨ n is nonsingular. u n ª Žy1. n iru n , ¨ n ª Žy1. n ir¨ n .
Transformation 7.
This transformation converts Ž1. into Ž30., and Ž2. into ¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 y ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 53 .
Equation Ž53. is also obtained by assuming Q Ž qn . s 2 qn4 y 2 qn2
Ž 54 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. One of the corresponding physical solutions is derived from Ž5., Ž6., Ž8. through the transformation k ª i k , v ª i v , d ª i d to become n
u n s Ž . . 1 Ž y1 . sinh Ž kr2 . sinh ur Ž cosh x q cosh u . , n
Ž 55 .
¨ n s Ž . . 2 Ž y1 . sinh Ž kr2 . sinh ur cosh x q cosh Ž kr2 . cosh u ,
Ž 56 .
v s y2 tanh Ž kr2 . .
Ž 57 .
By taking the limit x ª x Ž". 6 u , u ª `, Ž55. ] Ž57. give the solutions n
u n s Ž . . 1 Ž y1 . Ž 1r2 . sinh Ž kr2 . 1 Ž . . 6tanh Ž xr2. , ¨ h s Ž . . 2 Ž y1 .
n
Ž 58 .
Ž 1r2. tanh Ž kr2. 1 Ž . . 6tanh Ž x Ž . . 6 f . r2 4 , Ž 59.
in which
f s log cosh Ž kr2 . .
Ž 60 .
Another physical solution is derived from Ž5., Ž8. through the transformation k ª i k q p , v ª i v , d ª i d , u ª i u to become n
u n s Ž . . 1cosh Ž kr2 . sin ur cosh x q Ž y1 . cos u ,
v s y2 coth Ž kr2 . .
Ž 61 . Ž 62 .
Under the assumption u s pr2, Ž61. gives the solution u n s Ž . . 1cosh Ž kr2 . sech x.
Ž 63 .
NOTE
279
ACKNOWLEDGMENT The author thanks Mr. Yoshibumi Narita of Wakatake School for his continual aid to the study.
REFERENCES 1. K. Narita, New nonlinear difference-differential equation related to the Volterra equation, J. Math. Anal. Appl. 186 Ž1994., 120]131. 2. R. I. Yamilov, Construction scheme for discrete Miura transformations, J. Phys. A 27 Ž1994., 6839]6851. 3. A. B. Shabat and R. I. Yamilov, Lattice representation of integrable systems, Phys. Lett. 130 Ž1988., 271]275. 4. A. B. Shabat and R. I. Yamilov, Symmetries of nonlinear chains, Leningrad Math. J. 2 Ž1991., 377]400.
205, 273]279 Ž1997.
AY965171
NOTE Special Solutions to Nonlinear Difference-Differential Equations Kazuaki Narita B1010 CI Heights, 1-31 Yamada-Nishi, Suita-Shi, Osaka 565, Japan Submitted by William F. Ames Received January 29, 1996
In a previous paper, we presented new difference-differential equations, and studied their N-soliton solution, elliptic function solutions, and 1-soliton solution under a nonvanishing boundary condition at the infinity w1x. In this note, we present new special solutions for the above equations which cannot be obtained from known solutions through the transformation. We also study the corresponding physical solutions for the new equations derived from the above equations through seven different transformations. Recently, Shabat and Yamilov have presented various new solvable difference-differential equations w2]4x. We also make reference to the connection between their equations and our transformed equations. First, we remark that Eqs. Ž1.7., Ž2.3. of w1x are rewritten through the transformation i ª n, u i ª 2 q '2 u n , ¨ i ª Ž1 q ¨ n .r2 in the forms u ˙n s 2 Ž u ny1 y u nq1 . u 2nr Ž u ny1 u n y 1 .
1r2
q Ž u n u nq1 y 1 .
¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . Ž 1 q ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
1r2 2
, Ž 1.
Ž 2.
Equation Ž2.2. of w1x, giving a relation between Ž1. and Ž2., is also rewritten in the form ¨ n2 s u ny1r2 u nq1r2 y 1,
Ž 3.
and consequently Ž1. becomes 2
u ˙n s 2 Ž u ny1 y u nq1 . u 2nr Ž ¨ ny1r2 q ¨ nq1r2 . .
Ž 4.
273 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
274
NOTE
As such types of solutions for Ž1., Ž2. that are underivable from known ones, we present the equations u n s Ž " . 1 Ž cos x q cosh u . rsin Ž kr2 . sinh u ,
Ž 5.
¨ n s Ž " . 2 cos x q cos Ž kr2 . cosh u rsin Ž kr2 . sinh u ,
Ž 6.
x s kn q vt q d,
Ž 7.
where
and u is an arbitrary parameter. We find that Ž5., Ž6. automatically satisfy Ž3.. Substituting Ž5., Ž6. into Ž4., we find the dispersion relation
v s y2 tan Ž kr2 . .
Ž 8.
In the following, we present the new equations derived from Ž1., Ž2. through seven different transformations, and discuss their physical solutions. Transformation 1.
u n ª iu n , ¨ n ª i¨ n .
This transformation converts Ž1., Ž2. into u ˙n s 2 Ž u ny1 y u nq1 . u 2nr Ž 1 q u ny1 u n .
1r2
q Ž 1 q u n u nq1 .
1r2 2
¨˙n s 2 Ž ¨ nq1 y ¨ ny1 . Ž 1 y ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
, Ž 9.
Ž 10 .
Equation Ž9. is also obtained by assuming m s 0, h s '2 in Eqs. Ž18., Ž19. of w2x. The corresponding physical solutions are derived from Ž5., Ž6., Ž8. through the transformation u ª i u to become u n s Ž . . 1 Ž cos x q cos u . rsin Ž kr2 . sin u ,
Ž 11 .
¨ n s Ž . . 2 cos x q cos Ž kr2 . cos u rsin Ž kr2 . sin u ,
Ž 12 .
v s y2 tan Ž kr2 . .
Ž 13 .
Under the assumption u s pr2, Ž11., Ž12. give the solutions
Ž . . 1 u n s Ž . . 2 ¨ n s cos xrsin Ž kr2. . Transformation 2.
Ž 14.
u n ª Žy1. n u n , ¨ n ª Žy1. n ¨ n .
This transformation converts Ž1. into Ž9., and Ž2. into ¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . Ž 1 q ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 15 .
NOTE
275
Equation Ž15. is also obtained by assuming Q Ž qn . s y2 qn2 y 2
Ž 16 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. The corresponding physical solutions are derived from Ž5., Ž6., Ž8. through the transformation k ª k q p to become n
u n s Ž " . 1 cos x q Ž y1 . cosh u rcos Ž kr2 . sinh u , n
Ž 17 .
¨ n s Ž " . 2 cos x y Ž y1 . sin Ž kr2 . cosh u rcos Ž kr2 . sinh u ,
Ž 18 .
v s 2 cot Ž kr2 . .
Ž 19 .
Transformation 3.
u n ª Žy1. n iu n , ¨ n ª Žy1. n i¨ n .
This transformation converts Ž1. into itself and Ž2. into ¨˙n s 2 Ž ¨ nq1 y ¨ ny1 . Ž 1 y ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 20 .
Equation Ž20. is also obtained by assuming Q Ž qn . s y2 qn2 q 2
Ž 21 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. We remark here that Ž20. is derived from the new difference-differential equation w ˙n s Ž wny1 q wnq1 . wn y 4 r Ž wny1 y wnq1 .
Ž 22 .
through the transformation ¨ n s Ž " . 3 Ž wny1r2 wnq1r2 y 1 . .
Ž 23 .
The corresponding physical solutions for u n , ¨ n are derived from Ž5., Ž6., Ž8. through the transformation u ª i u , k ª k q p to become n
u n s Ž . . 1 cos x q Ž y1 . cos u rcos Ž kr2 . sin u , n
Ž 24 .
¨ n s Ž . . 2 cos x y Ž y1 . sin Ž kr2 . cos u rcos Ž kr2 . sin u ,
Ž 25 .
v s 2 cot Ž kr2 . .
Ž 26 .
Under the assumption u s pr2, Ž24., Ž25. give the solutions
Ž . . 1 u n s Ž . . 2 ¨ n s cos xrcos Ž kr2. .
Ž 27.
276
NOTE
The solution for wn under the same assumption reads wn s Ž " . 4'2 cos Ž xr2. r cos Ž kr2 .
1r2
,
Ž 28 .
in which the choice of signs must satisfy the condition
Ž . . 2 Ž " . 31 s 1. Transformation 4.
Ž 29.
u n ª 1ru n , ¨ n ª 1r¨ n .
This transformation converts Ž1., Ž2. into u ˙n s 2 Ž u ny1 y u nq1 . u n r
½
u nq 1 Ž 1 y u ny1 u n .
1r2
q u ny1 Ž 1 y u n u nq1 .
1r2 2
5,
¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 q ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
Ž 30 . Ž 31 .
The corresponding physical solutions are derived from Ž5., Ž6., Ž8. to become u n s Ž " . 1sin Ž kr2 . sinh ur Ž cos x q cosh u . ,
Ž 32 .
¨ n s Ž " . 2sin Ž kr2 . sinh ur cos x q cos Ž kr2 . cosh u ,
Ž 33 .
v s y2 tan Ž kr2 . .
Ž 34 .
In Ž33., k must satisfy the condition < cos Ž kr2 . < ) sech u
Ž 35 .
in order that ¨ n is nonsingular. Transformation 5.
u n ª iru n , ¨ n ª ir¨ n .
This transformation converts Ž1., Ž2. into u ˙n s 2 Ž u ny1 y u nq1 . u n r
½
u nq 1 Ž 1 q u ny1 u n .
1r2
q u ny1 Ž 1 q u n u nq1 .
1r2 2
5,
¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 y ¨ n2 . r Ž ¨ ny1 q ¨ n . Ž ¨ n q ¨ nq1 . .
Ž 36 . Ž 37 .
Equation Ž37. is also obtained by assuming m s '2 , h s 0 in Eqs. Ž18., Ž19. of w2x.
277
NOTE
One of the corresponding physical solutions is derived from Ž5., Ž6., Ž8. through the transformation k ª i k , v ª i v , d ª i d to become u n s Ž . . 1sinh Ž kr2 . sinh ur Ž cosh x q cosh u . ,
Ž 38 .
¨ n s Ž . . 2sinh Ž kr2 . sinh ur cosh x q cosh Ž kr2 . cosh u ,
Ž 39 .
v s y2 tanh Ž kr2 . .
Ž 40 .
By taking the limit x ª x Ž".5 u , u ª `, Ž38. ] Ž40. give the different solutions u n s Ž . . 1 Ž 1r2 . sinh Ž kr2 . 1 Ž . . 5tanh Ž xr2. ,
Ž 41 .
¨ n s Ž . . 2 Ž 1r2 . tanh Ž kr2 . 1 Ž . . 5tanh
Ž 42 .
Ž x Ž .. 5 f . r2 4 ,
in which
f s log cosh Ž kr2 . .
Ž 43 .
Another physical solution is derived from Ž5., Ž8. through the transformation k ª i k q p , v ª i v , d ª i d , u ª i u to become n
u n s Ž . . 1cosh Ž kr2 . sin ur Ž y1 . cosh x q cos u ,
v s y2 coth Ž kr2 . .
Ž 44 . Ž 45 .
Under the assumption u s pr2, Ž44. gives the solution n
u n s Ž . . 1 Ž y1 . cosh Ž kr2 . sech x. Transformation 6.
Ž 46 .
u n ª Žy1. nru n , ¨ n ª Žy1. nr¨ n .
This transformation converts Ž1. into Ž36., and Ž2. into ¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 q ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 47 .
Equation Ž47. is also obtained by assuming Q Ž qn . s y2 qn4 y 2 qn2
Ž 48 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. The corresponding physical solutions are derived from Ž5., Ž6., Ž8. through the transformation k ª k q p to become n
u n s Ž " . 1cos Ž kr2 . sinh ur cos x q Ž y1 . cosh u , n
Ž 49 .
¨ n s Ž " . 2cos Ž kr2 . sinh ur cos x y Ž y1 . sin Ž kr2 . cosh u ,
Ž 50 .
v s 2 cot Ž kr2 . .
Ž 51 .
278
NOTE
In Ž50., k must satisfy the condition sin Ž < k
Ž 52 .
in order that ¨ n is nonsingular. u n ª Žy1. n iru n , ¨ n ª Žy1. n ir¨ n .
Transformation 7.
This transformation converts Ž1. into Ž30., and Ž2. into ¨˙n s 2 Ž ¨ ny1 y ¨ nq1 . ¨ n2 Ž 1 y ¨ n2 . r Ž ¨ ny1 y ¨ n . Ž ¨ n y ¨ nq1 . .
Ž 53 .
Equation Ž53. is also obtained by assuming Q Ž qn . s 2 qn4 y 2 qn2
Ž 54 .
in Eq. Ž26. of w3x and Eq. IVŽa. of w4x. One of the corresponding physical solutions is derived from Ž5., Ž6., Ž8. through the transformation k ª i k , v ª i v , d ª i d to become n
u n s Ž . . 1 Ž y1 . sinh Ž kr2 . sinh ur Ž cosh x q cosh u . , n
Ž 55 .
¨ n s Ž . . 2 Ž y1 . sinh Ž kr2 . sinh ur cosh x q cosh Ž kr2 . cosh u ,
Ž 56 .
v s y2 tanh Ž kr2 . .
Ž 57 .
By taking the limit x ª x Ž". 6 u , u ª `, Ž55. ] Ž57. give the solutions n
u n s Ž . . 1 Ž y1 . Ž 1r2 . sinh Ž kr2 . 1 Ž . . 6tanh Ž xr2. , ¨ h s Ž . . 2 Ž y1 .
n
Ž 58 .
Ž 1r2. tanh Ž kr2. 1 Ž . . 6tanh Ž x Ž . . 6 f . r2 4 , Ž 59.
in which
f s log cosh Ž kr2 . .
Ž 60 .
Another physical solution is derived from Ž5., Ž8. through the transformation k ª i k q p , v ª i v , d ª i d , u ª i u to become n
u n s Ž . . 1cosh Ž kr2 . sin ur cosh x q Ž y1 . cos u ,
v s y2 coth Ž kr2 . .
Ž 61 . Ž 62 .
Under the assumption u s pr2, Ž61. gives the solution u n s Ž . . 1cosh Ž kr2 . sech x.
Ž 63 .
NOTE
279
ACKNOWLEDGMENT The author thanks Mr. Yoshibumi Narita of Wakatake School for his continual aid to the study.
REFERENCES 1. K. Narita, New nonlinear difference-differential equation related to the Volterra equation, J. Math. Anal. Appl. 186 Ž1994., 120]131. 2. R. I. Yamilov, Construction scheme for discrete Miura transformations, J. Phys. A 27 Ž1994., 6839]6851. 3. A. B. Shabat and R. I. Yamilov, Lattice representation of integrable systems, Phys. Lett. 130 Ž1988., 271]275. 4. A. B. Shabat and R. I. Yamilov, Symmetries of nonlinear chains, Leningrad Math. J. 2 Ž1991., 377]400.