Boussinesq Equation and Its Modifications

10.2.1. Boussinesq Equation and Its Modifications

Boussinesq equation in canonical form . This equation arises in several physical applications: propagation of long waves in shallow water, one-dimensional nonlinear lattice-waves, vibrations in

a nonlinear string, and ion sound waves in a plasma.

References : Boussinesq (1872), M. Toda (1975), A. C. Scott (1975).

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions

w 1 2 = 2 C 1 w(C 1 x+C 2 , ☞ C 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . Solutions:

2 w(x, t) = 2C 2

1 x − 2C 1 t + C 2 t+C 3 ,

w(x, t) = (C 1 t+C 2 ) x−

4 2 w(x, t) = − 2 + + C ( t+C ) , ( t+C 2 ) 2 t+C 2

w(x, t) = − 2 + C 1 t x−

w(x, t) = −

( t+C 2 ) 2

( x+C 1 ) 2

2 −2 w(x, t) = −3λ 1 cos

2 λ(x ☞ λt) + C 1 ,

where C 1 , ...,C 4 and λ are arbitrary constants.

3 ◦ . Traveling-wave solution (generalizes the last solution of Item 2 ◦ ):

w = w(ζ),

ζ = x + λt,

where the function w(ζ) is determined by the second-order ordinary differential equation (C 1 and C 2 are arbitrary constants)

2 w 2 ′′

ζζ + w +2 λ w+C 1 ζ+C 2 = 0.

For ✡✂☛ C 1 = 0, this equation is integrable by quadrature.

References : T. Nishitani and M. Tajiri (1982), G. R. W. Quispel, F. W. Nijhoff, and H. W. Capel (1982).

4 ◦ . Self-similar solution:

1 x w= U (z), z= √ ,

where the function U = U (z) is determined by the ordinary differential equation

U ′′′′ +( UU ′ z ) ′ z + 4 z U zzzz ′′ zz + 4 zU ′ z +2 U = 0.

Reference : T. Nishitani and M. Tajiri (1982).

5 ◦ . Degenerate solution (generalizes the first four solutions of Item 2 ◦ ):

w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t),

where the functions ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are determined by the autonomous system of ordinary differential equations

tt ′′ = −6 ϕ , ψ tt ′′ = −6 ϕψ,

tt = −2 ϕχ − ψ .

ξ=x−C 1 t 2 − C 2 t, where the function f (ξ) is determined by the third-order ordinary differential equation

2 w = f (ξ) − 4C 2

1 t −4 C 1 C 2 t,

(1) ✌✂✍ and C 1 , C 2 , and C 3 are arbitrary constants. Equation (1) is reduced to the second Painlev ´e equation.

1 f 2 ξξξ + ff ξ + C 2 f ξ −2 C f = 8C 1 ξ+C 3 ,

References : T. Nishitani and M. Tajiri (1982), G. R. W. Quispel, F. W. Nijhoff, and H. W. Capel (1982), P. A. Clarkson and M. D. Kruskal (1989).

7 ◦ . Generalized separable solution (generalizes the penultimate solution of Item 2 ◦ ):

w = (x + C 1 ) u(t) −

( x+C 1 ) 2

where the function u = u(t) is determined by the second-order autonomous ordinary differential equation

tt ′′ = −6 u .

The function ✌✂✍ u(t) is representable in terms of the Weierstrass elliptic function.

Reference : P. A. Clarkson and M. D. Kruskal (1989).

C is an arbitrary constant and the function F = F (z) is determined by the fourth-order ordinary differential equation

F zzzz ′′′′

3 3 9 +( 2 FF

z ′ ) z ′ + 4 zF z ′ + 2 F− 8 z = 0.

✌✂✍ Its solutions are expressed via solutions of the fourth Painlev ´e equation.

Reference : P. A. Clarkson and M. D. Kruskal (1989).

9 ◦ . Solution:

1 w(x, t) = (a 1

Here, a 1 , a 0 , b 1 , and b 0 are arbitrary constants, and the function U = U (z) is determined by the second-order ordinary differential equation

(2) where c 1 and c 2 are arbitrary constants. For c 1 = 0, the general solution of equation (2) can be written

zz ′′ + 2 U = c 1 z+c 2 ,

out in implicit form. If ✌✂✍ c 1 ≠ 0, the equation is reduced to the first Painlev´e equation.

Reference : P. A. Clarkson and M. D. Kruskal (1989).

10 ◦ . Solution:

2 a 1 x + λ(a 1 t+a 0 ) + a 1 b 1

w(x, t) = (a 1 t+a 0 ) U (z) −

Here, a 1 , a 0 , b 1 , and b 0 are arbitrary constants, and the function U = U (z) is determined by the third-order ordinary differential equation

(3) where ✌✂✍ c is an arbitrary constant. Equation (3) is reduced to the second Painlev ´e equation.

U zzz ′′′ + UU z ′

+5 2 λU = 50λ z + c,

Reference : P. A. Clarkson and M. D. Kruskal (1989).

t ′ ( t) + ψ ′ t ( t) , z = ϕ(t)x + ψ(t). Here, the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the autonomous system of second-order

w(x, t) = ϕ ( t)U (z) −

xϕ

( t)

ordinary differential equations

A is an arbitrary constant and the function U = U (z) is determined by the fourth-order ordinary differential equation

U zzzz ′′′′ + UU zz ′′

2 2 +( 2 U

z ′ ) + AzU z ′ +2 AU = 2A z .

A first integral of equation (4) is given by

( ϕ ′ 2 1 t 6 ) = 3 Aϕ + B,

where

B is an arbitrary constant. The general solution of this equation can be expressed in terms of Jacobi elliptic functions. The general solution of equation (5) can be expressed in terms of ϕ = ϕ(t) by

dt

ψ=C 1 ϕ(t) + C 2

ϕ(t)

ϕ 2 ( t)

where ✎✂✏ C 1 and C 2 are arbitrary constants.

References : P. A. Clarkson and M. D. Kruskal (1989), P. A. Clarkson, D. K. Ludlow, and T. J. Priestley (1997).

12 ◦ . The Boussinesq equation is solved by the inverse scattering method. Any rapidly decaying function

F = F (x, y; t) as x → +∞ and satisfying simultaneously the two linear equations

generates a solution of the Boussinesq equation in the form

w = 12

K(x, x; t),

dx where K(x, y; t) is a solution of the linear Gel’fand–Levitan–Marchenko integral equation Z ∞

K(x, y; t) + F (x, y; t) +

K(x, s; t)F (s, y; t) ds = 0.

Time ✎✂✏ t appears here as a parameter.

References : V. E. Zakharov (1973), M. J. Ablowitz and H. Segur (1981), J. Weiss (1984).

Unnormalized Boussinesq equation.

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions

1 = C 1 w(C 1 x+C 2 , ✑ C 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

w(x, t) = 2C 1 2 x + 2aC 2 1 t + C 2 t+C 3 ,

w(x, t) = (C 1 t+C 2 ) x+

2 ( C 1 t+C 2 ) + C 3 t+C 4 12 ,

( 2 x+C 1 )

w(x, t) =

w(x, t) = 2 + C 1 t x+

w(x, t) =

a(t + C

a(x + C 1 ) 2

w(x, t) =

cosh √ ( x ✒ λt) + C 1 a ,

2 b where C 1 , ...,C 4 and λ are arbitrary constants.

3 ◦ . Traveling-wave solution (generalizes the last solution of Item 2 ◦ ):

w = u(ζ),

ζ = x + λt,

where the function u = u(ζ) is determined by the second-order ordinary differential equation ( C 1 and C 2 are arbitrary constants)

2 bu 2 ′′

ζζ + au −2 λ u+C 1 ζ+C 2 = 0.

For C 1 = 0, this equation is integrable by quadrature.

4 ◦ . Self-similar solution:

1 x w= U (z), z= √ ,

where the function U = U (z) is determined by the ordinary differential equation

7 1 2 2 U+

4 zU z ′ + 4 z U zz ′′ = a(U U z ′ ) ′ z + bU zzzz ′′′′ .

5 ◦ . Degenerate solution (generalizes the first four solutions of Item 2 ◦ ):

w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t),

where the functions ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are determined by the autonomous system of ordinary differential equations

tt =6 aϕ , ψ ′′ tt =6 aϕψ,

tt ′′ =2 aϕχ + aψ .

6 ◦ . Solution:

ξ = x + aC 1 t + aC 2 t, where the function f (ξ) is determined by the third-order ordinary differential equation

and C 1 , C 2 , and C 3 are arbitrary constants.

7 ◦ . Solution (generalizes the penultimate solution of Item 2 ◦ ):

2 12 b

w = (x + C 1 ) u(t) −

a(x + C 1 ) 2

where the function u = u(t) is determined by the second-order autonomous ordinary differential equation

u ′′

2 tt =6 au .

The function u(t) is expressible in terms of the Weierstrass elliptic function.

C is an arbitrary constant and the function F = F (z) is determined by the ordinary differential equation

a(F F z ) z + bF zzzz = zF z ′ + F+

9 ◦ . See also equation 10.2.1.3, Item 6 ◦ .

References for equation 10.2.1.2: T. Nishitani and M. Tajiri (1982), G. R. W. Quispel, F. W. Nijhoff, and H. W. Capel (1982), P. A. Clarkson and M. D. Kruskal (1989).

Solutions of this equation can be represented in the form

(1) where the function u = u(x, t) is determined by the bilinear equation

w(x, t) = 2

1 ◦ . One- or two-soliton solutions of the original equation are generated by the following solutions of equation (2):

√ u = 1 + A exp kx ✕ kt 1+ k 2 ,

u=1+A 1 exp( k 1 x+m 1 t) + A 2 exp( k 2 x+m 2 t) + A 1 A 2 p 12 exp ( k 1 + k 2 ) x + (m 1 + m 2 ) t , where

1 , A, A A 2 , k, k 1 , and k 2 are arbitrary constants, and

References : R. Hirota (1973), M. J. Ablowitz and H. Segur (1981).

2 ◦ . Rational solutions are generated by the following solutions of equation (2):

Reference : M. J. Ablowitz and H. Segur (1981).

3 ◦ . Solution of equation (2): u = exp(2kx − 2mt) + (Cx − At) exp(kx − mt) − B,

2 2 C(2k 2 + 1) C (4 k + 3)

where k and C are arbitrary constants.

Reference : O. V. Kaptsov (1998).

u = sin(kx − mt) + Ax + Bt, u = sin(kx) + C sin(mt) + E cos(mt),

where k and C are arbitrary constants,

1− C 2 2 2 4 + k C 2 −4 k 2 m= k − k , A=

3 2 m 2 A(2k − 1)

Reference : O. V. Kaptsov (1998).

5 ◦ . Solution (

C is an arbitrary constant):

4 k −1

u = sin(kx) + C exp t k 4 k 2

Reference : O. V. Kaptsov (1998).

6 1 ◦ . The substitution w=

6 ( U − 1) leads to an equation of the form 10.2.1.2:

The substitution w = U − (a/b) leads to an equation of the form 10.2.1.2: