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Exact solutions of nonlinear generalizations of the wave equation
JOURNAL
OF MATHEMATICAL
Exact
ANALYSIS
AND
(1980)
Solutions of Nonlinear Generalizations of the Wave Equation PHILIP
of Physics and Astronomy,
Department
73, 49-51
APPLICATIONS
B.
Clemson
Submitted
&lRT
University,
Clemson,
South
Carolina
29631
by W, F. Ames
Exact solutions of nonlinear generalizations of the wave equation are constructed. In some cases these solutions are solitary waves or solitions. Thus, by explicit construction solitons or solitary waves are shown to exist in dispersionless systems. In contrast to previous solitary wave solutions, these solutions are limiting cases of solutions of nonlinear partial differential equations with dispersion.
Several nonlinear generalizations of the Klein Gordon equation have exact The solutions which, in special cases, are solitary waves or solitons (l-3). examples previously studied are both nonlinear and dispersive. Tn this paper a class of nonlinear generalizations of the wave equation is shown to have exact solutions exhibiting solitary wave or soliton behavior. These solutions are examples of dispersion free solitary waves or solitons. A nonlinear partial differential equation with possible applications in quantum field theory is
32 2% -g f
+ ay2p $- 2) k-z”+1 + 2ab(p + 2) Z”+’ 2bc(--p
+ 2) z-J+1
+ cy-2p
+ 2) z-s”+1
= 0,
(P f
0)
(1)
where b2 + 2ac 7 0
(2)
ac > 0.
(3)
and
This is a nonlinear generalization of the wave equation, a partial differential an equation which is dispersion free. As may be verified by differentiation, exact solution of Eq. (1) is z = [(l - n”) (-b
+ d + np(b + d))-l2c]l’”
(4)
49 0022-247X/80/01004903$02.00/0 Copyright 8 1980 by AcademicPress, Inc. All rights of reproduction in any form reserved.
50
PHILIP
B.
BURT
where d = (b2 -
4~2~)~‘~ # 0
(5)
and n = 1 TZ,exp a,. w,t i r
i
kyixi
i=l
= C N,.(x, , t) 7
(6)
with n, arbitrary constantsand
w,’ -
f
k,“, # 0
(‘3)
i=l
for all r, s. The sum on Y is over any collection (w, , k, ,..., k,,). The conditions contained in Eqs. (7)-(g) are similar to conditions obtained previously for solutionsof nonlinear, dispersivep.d.e. They describe nonlinear superpositionsof wavesreferred to ascollapsons.As has been shown previously (3), Eqs. (7)-(g) restrict the collection (wr , k,J in the following ways: (I)
either w,2 - Cr=r Rri > 0, all P or w,2 - CT=, k,i < 0, all r;
(2) (% 9hi) # -(Ws , ki) (3) Zr=, k,& # 0, any r, s. Furthermore, if all the vectors (k,, ,..., k,,) are parallel or if w,1-fkfi>O i=l
the sum of functions Nr(xi , t) collapsesto a singleexponential. These conditions define collapsonsolutions of a nonlinear p.d.e. Finally, for the solutions of the p.d.e. with nonlinearity and dispersion,Whitham has shown that if
a coordinate systemcan be found in which all (kT1 ,..., k,,) are coplanar(4). The proof is also valid for the solutions given in Eqs. (4)-(g). Consequently, these solutions are effectively 2 + 1 dimensional. The solutions given in Eqs. (4)-(g) are, in specialcases,solitary waves. Let n = 1 and Y = 1 so that 2
=
[(I
_
&+"wt-""')(&J
+
d +
(b
+
d)A~e~(Wt-""')-12C]1/P
(9)
with w2 -
k2 = -2d2.
(10)
SOLUTIONS
OF NONLINEAR
WAVE
51
EQUATIONS
Since the phase speed eujk is constant 2 will be a solitary wave in the usual sense (5) if it is finite everywhere. Thus, it is sufficient to choose Ap < 0 and 0 < d < b to make 2 a solitary wave. For other choices of the set (w,. , kri) and arbitrary n the solutions are multisolitons. The argument deponstrating the soliton behavior is the same as for the dispersive solutions and will not be repeated here (3). The important new property exhibited by the solutions in Eqs. (4)-(8) is that nonlinearity alone is sufficient for a system to have solitary wave or soliton solutions. A second important property of these solutions is that they are limiting cases of solutions of nonlinear, dispersive p.d.e. If the condition in Eq. (2) is removed the solutions given in Eqs. (4)-(8) become solutions of + 2(b2 + 24 + (2bc) (-p
Y + a2(2p + 2) Y2p+l + 2ub(p + 2) Yn+l
+ 2) Y--p+1 + ~7-2~
+ 2) y--2p+l=
0
(P z 0).
(11)
The term 2(b2 + 2ac) Y makes this a nonlinear generalization of the Klein Gordon equation and provides dispersion. Thus, in contrast to the usual situation, solutions of this nonlinear, dispersive p.d.e. continue to exhibit soliton properties even when the dispersion vanishes.
REFERENCES 1. P. B. BURT AND J. L. REID, Exact solutions to a nonlinear Klein-Gordon equation, J. Math. Anal. Appl. 55 (1976), 43-45. 2. P. B. BURT, Exact solutions of nonlinear generalizations of the Klein Gordon and Schrtidinger equations, j. Math. Anal. Appl. 66 (1978), 135-142. 3. P. B. BURT, Exact, multiple soliton solutions of the double sine-Gordon equation, Proc. Roy. Sot. London Ser. A 359 (1978), 479-495. 4. G. B. WHITHAM, Comments on some recent multisoliton solutions, jour. Phys. A12, (1979), l-3. 5. G. B. WHITHAM, “Linear and Nonlinear Waves,” Wiley, New York, 1974.
OF MATHEMATICAL
Exact
ANALYSIS
AND
(1980)
Solutions of Nonlinear Generalizations of the Wave Equation PHILIP
of Physics and Astronomy,
Department
73, 49-51
APPLICATIONS
B.
Clemson
Submitted
&lRT
University,
Clemson,
South
Carolina
29631
by W, F. Ames
Exact solutions of nonlinear generalizations of the wave equation are constructed. In some cases these solutions are solitary waves or solitions. Thus, by explicit construction solitons or solitary waves are shown to exist in dispersionless systems. In contrast to previous solitary wave solutions, these solutions are limiting cases of solutions of nonlinear partial differential equations with dispersion.
Several nonlinear generalizations of the Klein Gordon equation have exact The solutions which, in special cases, are solitary waves or solitons (l-3). examples previously studied are both nonlinear and dispersive. Tn this paper a class of nonlinear generalizations of the wave equation is shown to have exact solutions exhibiting solitary wave or soliton behavior. These solutions are examples of dispersion free solitary waves or solitons. A nonlinear partial differential equation with possible applications in quantum field theory is
32 2% -g f
+ ay2p $- 2) k-z”+1 + 2ab(p + 2) Z”+’ 2bc(--p
+ 2) z-J+1
+ cy-2p
+ 2) z-s”+1
= 0,
(P f
0)
(1)
where b2 + 2ac 7 0
(2)
ac > 0.
(3)
and
This is a nonlinear generalization of the wave equation, a partial differential an equation which is dispersion free. As may be verified by differentiation, exact solution of Eq. (1) is z = [(l - n”) (-b
+ d + np(b + d))-l2c]l’”
(4)
49 0022-247X/80/01004903$02.00/0 Copyright 8 1980 by AcademicPress, Inc. All rights of reproduction in any form reserved.
50
PHILIP
B.
BURT
where d = (b2 -
4~2~)~‘~ # 0
(5)
and n = 1 TZ,exp a,. w,t i r
i
kyixi
i=l
= C N,.(x, , t) 7
(6)
with n, arbitrary constantsand
w,’ -
f
k,“, # 0
(‘3)
i=l
for all r, s. The sum on Y is over any collection (w, , k, ,..., k,,). The conditions contained in Eqs. (7)-(g) are similar to conditions obtained previously for solutionsof nonlinear, dispersivep.d.e. They describe nonlinear superpositionsof wavesreferred to ascollapsons.As has been shown previously (3), Eqs. (7)-(g) restrict the collection (wr , k,J in the following ways: (I)
either w,2 - Cr=r Rri > 0, all P or w,2 - CT=, k,i < 0, all r;
(2) (% 9hi) # -(Ws , ki) (3) Zr=, k,& # 0, any r, s. Furthermore, if all the vectors (k,, ,..., k,,) are parallel or if w,1-fkfi>O i=l
the sum of functions Nr(xi , t) collapsesto a singleexponential. These conditions define collapsonsolutions of a nonlinear p.d.e. Finally, for the solutions of the p.d.e. with nonlinearity and dispersion,Whitham has shown that if
a coordinate systemcan be found in which all (kT1 ,..., k,,) are coplanar(4). The proof is also valid for the solutions given in Eqs. (4)-(g). Consequently, these solutions are effectively 2 + 1 dimensional. The solutions given in Eqs. (4)-(g) are, in specialcases,solitary waves. Let n = 1 and Y = 1 so that 2
=
[(I
_
&+"wt-""')(&J
+
d +
(b
+
d)A~e~(Wt-""')-12C]1/P
(9)
with w2 -
k2 = -2d2.
(10)
SOLUTIONS
OF NONLINEAR
WAVE
51
EQUATIONS
Since the phase speed eujk is constant 2 will be a solitary wave in the usual sense (5) if it is finite everywhere. Thus, it is sufficient to choose Ap < 0 and 0 < d < b to make 2 a solitary wave. For other choices of the set (w,. , kri) and arbitrary n the solutions are multisolitons. The argument deponstrating the soliton behavior is the same as for the dispersive solutions and will not be repeated here (3). The important new property exhibited by the solutions in Eqs. (4)-(8) is that nonlinearity alone is sufficient for a system to have solitary wave or soliton solutions. A second important property of these solutions is that they are limiting cases of solutions of nonlinear, dispersive p.d.e. If the condition in Eq. (2) is removed the solutions given in Eqs. (4)-(8) become solutions of + 2(b2 + 24 + (2bc) (-p
Y + a2(2p + 2) Y2p+l + 2ub(p + 2) Yn+l
+ 2) Y--p+1 + ~7-2~
+ 2) y--2p+l=
0
(P z 0).
(11)
The term 2(b2 + 2ac) Y makes this a nonlinear generalization of the Klein Gordon equation and provides dispersion. Thus, in contrast to the usual situation, solutions of this nonlinear, dispersive p.d.e. continue to exhibit soliton properties even when the dispersion vanishes.
REFERENCES 1. P. B. BURT AND J. L. REID, Exact solutions to a nonlinear Klein-Gordon equation, J. Math. Anal. Appl. 55 (1976), 43-45. 2. P. B. BURT, Exact solutions of nonlinear generalizations of the Klein Gordon and Schrtidinger equations, j. Math. Anal. Appl. 66 (1978), 135-142. 3. P. B. BURT, Exact, multiple soliton solutions of the double sine-Gordon equation, Proc. Roy. Sot. London Ser. A 359 (1978), 479-495. 4. G. B. WHITHAM, Comments on some recent multisoliton solutions, jour. Phys. A12, (1979), l-3. 5. G. B. WHITHAM, “Linear and Nonlinear Waves,” Wiley, New York, 1974.