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Another possible model equation for long waves in nonlinear dispersive systems
Volume 61A, number 7
PHYSICS LETTERS
27 June 1977
ANOTHER POSSIBLE MODEL EQUATION FOR LONG WAVES IN NONLINEAR DISPERSiVE SYSTEMS Richard I. JOSEPH and Robert EGRI Department of Electrical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218, USA Received 20 April 1977 In the same spirit in which Benjamin, Bona, and Mahoney modified the Korteweg-de Vries equation (Ux + Ut + UU~+ Uxxx 0) to obtain the so-called BBM equation, U~+ Ut + UU~— ~ = 0, we propose a different modification: U~+ Ut + UUx + Uxtt = 0. The advantages in this equ 1ation are 1) the system is conservative since it can be derived from the Lagrangian density L = + + ~ — ~ where O~ U; 2) for large wavenumbers 1k, the infinitesimal-wave phase speed falls off like 1 / 1k I, in accord with physical intuition; 3) since the equation is of second order in t, both Uand Ut can be independently specified for t = 0. Several conservation laws satisfied by solutions to this equation are given.
The Korteweg-de Vries equation
[1]
equations by means of the Lagrangian density (1)
has been extensively used to study the propagation of unidirectional finite amplitude shallow water waves. For infinitesimal waves, solutions are of the form U(x, t) = U exp2.{ik[x c(k)t] } where the phase speed Contrary to the assumption of foris c(k) = 1 k ward travelling waves, this c(k) becomes negative for k2 > I and, is unbounded for large 1k I. In an attempt to remedy this shortcoming, Benjamin, Bona, and Mahoney [2] pointed out that in the perturbation derivation of the physical equation which is then scaled to the form of equation (1), both the nonlinear (UU~) —
—
and dispersive (U~~~) terms are in fact multiplied by the small expansion parameter introduced in the theory. Hence to lowest order in one has U~= —Ui. They then proposed that in lieu of eq. (1), the (BBM) equation U~+Ut+UU~—~
0
~2j
.~
~
(3)
,
with 8~ U [3]. Miura et al. [4] have shown that the K-deV equation possesses an infinite number of polynomial conservation laws, e.g. densities Q,~such that a~ f_°°ooQn dx = 0.byThe first threeeq. of these simply obtained multiplying (1) byare U’1,most n = 0, 1,2, integrating over all x and making use of the vanishing of U and all of its derivations for lxi giving Q 2,, Q 3 ~U~,respec0 = Hamiltonian U, Q1 = ~U density 2 = for U this equation is tively. The H ~U2 + U3 ~U~conservation of energy corresponds to conservation of the density (Q 1 + Q2). Although not previously recognized, the BBM equation is also conservative since it can be simply obtained from the I.agrangian density L —10 0 + 102 + 103 + 10 0 4 ~-~‘
—
~
~
—
2
x t
—
2
x
6
x
2
xx
Xt’
with 0~ U.~One similarly finds that the densities Q0=U,Q1
be studied. The infinitesimal wave with speedthat for this equa. 2)—1, agreeing for the tion is c(k) = (1 + k K-deV equation for small k, but it always remains
served. The Hamiltonian density this also equation H = ~ U2 + ~U3 conservation of for energy corre-is sponds to conservation of the density (Q
positive and approaches zero for large 1k I. Both equations (1) and (2) are to be solved subject to the conditions U-~0 for lxi 00, U(x, 0) given. Both have similar classical solitary wave2b(x solutions, Ct). e.g. a solution of the U(x, = a sech lossless, or conservative, Eq. form (1) can be t)considered in the sense that it is derivable from the Euler-Lagrange
It is interesting to note in passing that consistent with what BBM [2] did, one couldjust as well have simultaneously replaced UU~by —UU~and ~ by ~ in eq. (1) to get U +U UU U = 0 (5
-~
—
1
—
X
t
+
Q2).
—
t
XXt
Eq. (5) has the same infinitesimal phase speed as eq. 429
Volume 61A, number 7
PHYSICS LETTERS
(2) as well as possessing a classical solitary wave solution. It too is conservative since it can be derived from the Lagrangian density L = 10 2 x0 t
+ 102 2 t
—
103 6 t
+
102 2 Xt
‘
‘6’ “ ‘ 2, ~U
with_12 O~. U. Conserved densities are Q~-~ = U 1312 ‘12 2 U + U ~LT~, Q The Hamiltonian density is H2= U3 U~ ~U~~) U,~ (U U energy conservation just corresponds to conservation of the density Q —
—
-
—~
~
—
~ —
— — ~~
— —
1.
U
27 June 1977
+~
X
+
~
t
X
+ U xtt
=
0
(7 ‘
instead of by eq. (2). Since the equation is of second order U~mustthe be specified, independently,inatt, tboth = 0. Uand Furthermore, infinitesimal wave speed is now given by 2~’2 11/2k2 I = I1(1 + 4k / c(k~ which for small k goes like I k2 while for large lkl, —
—
becomes c(k) l/lkI. Eq. (7) is also conservative since it can be derived from the Lagrangian density 1. —10 0 + 102 + 103 102 9 it
Although eqs. (2) and (5) correspond to infinitesimal wave speedsc(k) which are always positive and vanish for large ki, there are two basic features associated with them which are physically unacceptable. Firstly,once they,[U(x, just like eq.is(1), are firstone order in t. Hence t)] t=0 specified, automatically has no control of [aU(x, t)/atJ~ 0.If U corresponds to a displacement and a U/at to a velocity, from a physical point of view, one would like to have the possibility of their initial values being independent of each other. Secondly, and equally important, is the way in which c(k) behaves as k -÷00• By definition, c(k) w(k)/k. It is known that in physical situations where these equations apply, that as 1k I -÷ 00~w(k) N, the Brunt-Väisälä stability frequency [5]. Hence one physically expects that as 1k -*00 c(k) —~1/1k. Consequently if one is trying to patch up the K-deV equation, it nught not be unreasonable to do this in such a way so as to try and include there two features. Hence, again in the spirit of BBM [2], of replacing U~by — U~to lowest order, we propose that the KdeV equation be replaced by the new equation
—
2 X
t
2
X
6
X
2
xt
with 0~ U. Conserved densities for this equation are 3 + U~U~ + ~ U~. U, Q1 = U~ ~ is HQ2 ~ U+ ~ U3 + ~ The =Hamiltoman density = ~=U2 ~
—
conservation of energy corresponds to conservation of the density (Q1 + Q2). Eq. (7) like the previous three equations, has a classical solitary wave solution. Further study of this equation is now in progress.
References
—~
430
[1]
D.J. Korteweg and G. de Vries, Phil. Mag. 39(1895)422. [2] T.B. Benjamin, J.L. Bona and J.J. Mahoney, Phil. Trans. Roy. Soc. A272 (1972) 47. [3] A.C. Scott, F.Y.F. Chu and D.W. McLaughlin Proc. I.E.E.E. 61(1973)1443. [4] R.M. Miura, C.S. Gardner and M.D. Kruskal, J. Math. Phys. 9 (1968) 1204. [5] O.M. Phillips, The dynamics of the upper ocean, Cambridge University Press, 1966.
PHYSICS LETTERS
27 June 1977
ANOTHER POSSIBLE MODEL EQUATION FOR LONG WAVES IN NONLINEAR DISPERSiVE SYSTEMS Richard I. JOSEPH and Robert EGRI Department of Electrical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218, USA Received 20 April 1977 In the same spirit in which Benjamin, Bona, and Mahoney modified the Korteweg-de Vries equation (Ux + Ut + UU~+ Uxxx 0) to obtain the so-called BBM equation, U~+ Ut + UU~— ~ = 0, we propose a different modification: U~+ Ut + UUx + Uxtt = 0. The advantages in this equ 1ation are 1) the system is conservative since it can be derived from the Lagrangian density L = + + ~ — ~ where O~ U; 2) for large wavenumbers 1k, the infinitesimal-wave phase speed falls off like 1 / 1k I, in accord with physical intuition; 3) since the equation is of second order in t, both Uand Ut can be independently specified for t = 0. Several conservation laws satisfied by solutions to this equation are given.
The Korteweg-de Vries equation
[1]
equations by means of the Lagrangian density (1)
has been extensively used to study the propagation of unidirectional finite amplitude shallow water waves. For infinitesimal waves, solutions are of the form U(x, t) = U exp2.{ik[x c(k)t] } where the phase speed Contrary to the assumption of foris c(k) = 1 k ward travelling waves, this c(k) becomes negative for k2 > I and, is unbounded for large 1k I. In an attempt to remedy this shortcoming, Benjamin, Bona, and Mahoney [2] pointed out that in the perturbation derivation of the physical equation which is then scaled to the form of equation (1), both the nonlinear (UU~) —
—
and dispersive (U~~~) terms are in fact multiplied by the small expansion parameter introduced in the theory. Hence to lowest order in one has U~= —Ui. They then proposed that in lieu of eq. (1), the (BBM) equation U~+Ut+UU~—~
0
~2j
.~
~
(3)
,
with 8~ U [3]. Miura et al. [4] have shown that the K-deV equation possesses an infinite number of polynomial conservation laws, e.g. densities Q,~such that a~ f_°°ooQn dx = 0.byThe first threeeq. of these simply obtained multiplying (1) byare U’1,most n = 0, 1,2, integrating over all x and making use of the vanishing of U and all of its derivations for lxi giving Q 2,, Q 3 ~U~,respec0 = Hamiltonian U, Q1 = ~U density 2 = for U this equation is tively. The H ~U2 + U3 ~U~conservation of energy corresponds to conservation of the density (Q 1 + Q2). Although not previously recognized, the BBM equation is also conservative since it can be simply obtained from the I.agrangian density L —10 0 + 102 + 103 + 10 0 4 ~-~‘
—
~
~
—
2
x t
—
2
x
6
x
2
xx
Xt’
with 0~ U.~One similarly finds that the densities Q0=U,Q1
be studied. The infinitesimal wave with speedthat for this equa. 2)—1, agreeing for the tion is c(k) = (1 + k K-deV equation for small k, but it always remains
served. The Hamiltonian density this also equation H = ~ U2 + ~U3 conservation of for energy corre-is sponds to conservation of the density (Q
positive and approaches zero for large 1k I. Both equations (1) and (2) are to be solved subject to the conditions U-~0 for lxi 00, U(x, 0) given. Both have similar classical solitary wave2b(x solutions, Ct). e.g. a solution of the U(x, = a sech lossless, or conservative, Eq. form (1) can be t)considered in the sense that it is derivable from the Euler-Lagrange
It is interesting to note in passing that consistent with what BBM [2] did, one couldjust as well have simultaneously replaced UU~by —UU~and ~ by ~ in eq. (1) to get U +U UU U = 0 (5
-~
—
1
—
X
t
+
Q2).
—
t
XXt
Eq. (5) has the same infinitesimal phase speed as eq. 429
Volume 61A, number 7
PHYSICS LETTERS
(2) as well as possessing a classical solitary wave solution. It too is conservative since it can be derived from the Lagrangian density L = 10 2 x0 t
+ 102 2 t
—
103 6 t
+
102 2 Xt
‘
‘6’ “ ‘ 2, ~U
with_12 O~. U. Conserved densities are Q~-~ = U 1312 ‘12 2 U + U ~LT~, Q The Hamiltonian density is H2= U3 U~ ~U~~) U,~ (U U energy conservation just corresponds to conservation of the density Q —
—
-
—~
~
—
~ —
— — ~~
— —
1.
U
27 June 1977
+~
X
+
~
t
X
+ U xtt
=
0
(7 ‘
instead of by eq. (2). Since the equation is of second order U~mustthe be specified, independently,inatt, tboth = 0. Uand Furthermore, infinitesimal wave speed is now given by 2~’2 11/2k2 I = I1(1 + 4k / c(k~ which for small k goes like I k2 while for large lkl, —
—
becomes c(k) l/lkI. Eq. (7) is also conservative since it can be derived from the Lagrangian density 1. —10 0 + 102 + 103 102 9 it
Although eqs. (2) and (5) correspond to infinitesimal wave speedsc(k) which are always positive and vanish for large ki, there are two basic features associated with them which are physically unacceptable. Firstly,once they,[U(x, just like eq.is(1), are firstone order in t. Hence t)] t=0 specified, automatically has no control of [aU(x, t)/atJ~ 0.If U corresponds to a displacement and a U/at to a velocity, from a physical point of view, one would like to have the possibility of their initial values being independent of each other. Secondly, and equally important, is the way in which c(k) behaves as k -÷00• By definition, c(k) w(k)/k. It is known that in physical situations where these equations apply, that as 1k I -÷ 00~w(k) N, the Brunt-Väisälä stability frequency [5]. Hence one physically expects that as 1k -*00 c(k) —~1/1k. Consequently if one is trying to patch up the K-deV equation, it nught not be unreasonable to do this in such a way so as to try and include there two features. Hence, again in the spirit of BBM [2], of replacing U~by — U~to lowest order, we propose that the KdeV equation be replaced by the new equation
—
2 X
t
2
X
6
X
2
xt
with 0~ U. Conserved densities for this equation are 3 + U~U~ + ~ U~. U, Q1 = U~ ~ is HQ2 ~ U+ ~ U3 + ~ The =Hamiltoman density = ~=U2 ~
—
conservation of energy corresponds to conservation of the density (Q1 + Q2). Eq. (7) like the previous three equations, has a classical solitary wave solution. Further study of this equation is now in progress.
References
—~
430
[1]
D.J. Korteweg and G. de Vries, Phil. Mag. 39(1895)422. [2] T.B. Benjamin, J.L. Bona and J.J. Mahoney, Phil. Trans. Roy. Soc. A272 (1972) 47. [3] A.C. Scott, F.Y.F. Chu and D.W. McLaughlin Proc. I.E.E.E. 61(1973)1443. [4] R.M. Miura, C.S. Gardner and M.D. Kruskal, J. Math. Phys. 9 (1968) 1204. [5] O.M. Phillips, The dynamics of the upper ocean, Cambridge University Press, 1966.