Sine Gordon Equation and Other Equations with Trigonometric Nonlinearities

3.3.3. Sine Gordon Equation and Other Equations with Trigonometric Nonlinearities

1. 2 = a ∂t

+ b sin(λw).

∂x 2

Sine-Gordon equation . It arises in differential geometry and various areas of physics (superconduc- tivity, dislocations in crystals, waves in ferromagnetic materials, laser pulses in two-phase media, and others).

1 ◦ . Suppose w = ϕ(x, t) is a solution of the sine-Gordon equation. Then the functions

2 πn ❊

w ❊ 1 = ϕ(C 1 x, C 2 t), n = 0, 1, 2, ...;

sinh σ

w 2 = ϕ x cosh σ + t

a sinh σ, x √

+ t cosh σ ,

where C 1 , C 2 , and σ are arbitrary constants, are also solutions of the equation. The plus or minus signs in the first expression are chosen in any sequence.

2 w(x, t) = 2

4 ❋ bλ(kx + µt + θ 0 )

arctan exp

if bλ(µ − ak ) > 0,

bλ(µ 2

− ak 2 )

4 bλ(kx + µt + θ 0 )

w(x, t) = − +

arctan exp

if bλ(µ 2 − ak 2 ) < 0,

bλ(ak 2 − µ 2 )

where k, µ, and θ 0 are arbitrary constants. The first expression corresponds to a single-soliton solution.

3 ◦ . Functional separable solution:

4 w(x, t) = arctan

f (x)g(t) ,

where the functions f = f (x) and g = g(t) are determined by the first-order autonomous separable ordinary differential equations

aB + bλ)g 2 − aA,

where

A, B, and C are arbitrary constants. Note some exact solutions that follow from (1) and (2).

3.1. For

A = 0, B = k 2 > 0, and

C > 0, we have

4 µ sinh(kx + A 1 )

w(x, t) = arctan √ , µ 2 = ak 2 + bλ > 0, (3)

a cosh(µt + B 1 )

where k, A 1 , and B 1 are arbitrary constants. Formula (3) corresponds to the two-soliton solution of Perring–Skyrme (1962).

w(x, t) = 2 arctan 2 √ , µ = bλ − ak > 0,

4 µ sin(kx + A 1 )

a cosh(µt + B 1 )

where k, A 1 , and B 1 are arbitrary constants.

3.3. For A=k 2 > 0, B=k 2 γ 2 > 0, and

C = 0,

4 γ e µ(t+A 1 ) + ak 2 e − µ(t+A 1 )

w(x, t) = arctan

2 µ 2 ak γ + bλ > 0,

µ e kγ(x+B 1 ) + e − kγ(x+B 1 )

●✝❍ where k, A 1 , B 1 , and γ are arbitrary constants.

Reference : R. Steuerwald (1936), G. L. Lamb (1980), S. P. Novikov, S. V. Manakov, L. B. Pitaevskii, and V. E. Zakharov (1984).

4 ◦ . An N -soliton solution is given by (a = 1, b = −1, and λ = 1)

w(x, t) = arccos 1−2

where ●✝❍ µ i and C i are arbitrary constants.

Reference : R. K. Bullough and P. J. Caudrey (1980), S. P. Novikov, S. V. Manakov, L. B. Pitaevskii, and V. E. Zakharov (1984).

Item 3 ◦ .

6 ◦ . The sine-Gordon equation is integrated by the inverse scattering method; see the book by Novikov, Manakov, Pitaevskii, and Zakharov (1984). Belokolos (1995) obtained a general formula for the solution of the sine-Gordon equation with arbitrary initial and boundary conditions.

7 ◦ . The transformation

z = x − at, y = x + at

■✝❏ leads to an equation of the form 3.5.1.5:

∂ 1 zy w=− a −2

4 sin w.

References for equation 3.3.3.1: R. Steuerwald (1936), M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur (1973), V. E. Zakharov, L. A. Takhtajan, and L. D. Faddeev (1974), G. B. Whitham (1974), I. M. Krichever (1980), R. K. Bullough and P. J. Caudrey (1980), M. J. Ablowitz and H. Segur (1981), S. P. Novikov, S. V. Manakov, L. B. Pitaevskii, and V. E. Zakharov (1984), J. Weiss (1984), M. J. Ablowitz and P. A. Clarkson (1991).

2. 2 = a 2 + b sin(λw) + c sin 1

2 λw .

∂t ∂x

Double sine-Gordon equation . It arises in nonlinear optics (propagation of ultrashort pulses in a res- onance degenerate medium) and low temperature physics (propagation of spin waves in anisotropic spin liquids).

1 ◦ . Traveling-wave solutions:

λ 4 b 2 − c 2 ( kx + µt + θ 0 )

w(x, t) =

4 bλ(ak 2 − µ 2

−4 b 2 λ c 2 −4 b 2 ( kx + µt + θ 0 )

w(x, t) =

4 bλ(ak 2 − µ 2 )

Here, k, µ, and θ 0 are arbitrary constants. It is assumed that bλ(ak 2 − µ 2 ) > 0 in both formulas.

2 ◦ . Traveling-wave solutions:

w(x, t) = A + arctan B 1 e θ + C 1 +

arctan B 2 e θ + C 2 , θ = µt ❑ kx + θ 0 ,

1 , A, B B 2 , C 1 , C 2 , µ, and k are related by algebraic constraints with the parameters

where the parameters

a, b, c, and λ of the original equation; θ 0 is an arbitrary constant. Note some special cases of interest that arise in applications.

2.1. For a = 1, b = −1, c = − 1 2 , λ = 1:

w(x, t) = 4 arctan e

+ 4 arctan e ;

w(x, t) = 2π + 4 arctan e

− 4 arctan e ;

1 w(x, t) = δ − 2π + 4 arctan √ e + √

δ is any, k=µ+ 15 16 µ −1 .

2.4. For a = 1, b = 1, c = 1 2 , λ = 1:

w(x, t) = 2π − δ + 4 arctan √ e θ − √

δ is any, k=µ+

References : R. K. Bullough and P. J. Caudrey (1980), F. Calogero and A. Degasperis (1982).

3. 2 = a + b cos(λw).

∂t

∂x 2

The substitution w=u+

leads to an equation of the form 3.3.3.1:

= a − b sin(λu). ∂t 2 ∂x 2

4. ∂t 2

+ be βt sin k ( λw).

∂x 2

This is a special case of equation 3.4.1.7 with f (w) = b sin k ( λw). Therefore, for k = 1, the equation is reduced to a simpler equation of 3.3.3.1.

be 2 βx + sin k ( λw).

∂t ∂x This is a special case of equation 3.4.1.6 with f (w) = b sin k ( λw). Therefore, for k = 1, the equation is reduced to a simpler equation of 3.3.3.1.

2 2 + be βt cos ( ∂t λw). ∂x This is a special case of equation 3.4.1.7 with f (w) = b cos k ( λw). Therefore, for k = 1, the equation

is reduced to a simpler equation of 3.3.3.3. ∂ 2 w

+ be βx cos k ( λw).

∂t

∂x 2

This is a special case of equation 3.4.1.6 with f (w) = b cos k ( λw). Therefore, for k = 1, the equation reduced to a simpler equation of 3.3.3.3.

n 8. ∂w ∂t 2

+ k sin(λw).

∂x

∂x

This is a special case of equation 3.4.2.1 with f (w) = k sin(λw) and b = an. ∂ 2 w

This is a special case of equation 3.4.4.6 with f (w) = a cos n ( λw). ∂ 2 w

This is a special case of equation 3.4.4.6 with f (w) = a sin n ( λw).

3.3.4. Equations of the Form ∂ w 2 + a ∂w =∂

f (w) ∂w

∂t

∂t

∂x

∂x

The transformation τ = t + ln |w|, dz = aw −2 w x dt + (w + w t ) dx, u = 1/w

( dz = z t dt + z x dx), where the subscripts denote the corresponding partial derivatives, leads to the linear telegraph

References : C. Rogers and T. Ruggeri (1985), C. Rogers and W. F. Ames (1989).

1 ◦ . Solution for n ≠ −1:

w(x, t) = (x + C 2 ) 1 /(1+n) ( C 1 e − kt + C 2 )− b,

where C 1 and C 2 are arbitrary constants.

2 ◦ . Solution:

w(x, t) = (x + C) 2 /n u(t) − b,

C is an arbitrary constant, and the function u = u(t) is determined by the ordinary differential equation

This equation is easy to integrate for n = −2 and n = −1. For n = −3/2 and −3, its exact solutions are given in the handbook by Polyanin and Zaitsev (2003).

3 ◦ . Traveling-wave solution in implicit form:

Z w+b bu n − λ 2 dw = x + λt + C 2 ,

0 kλu + C 1

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

4 ◦ . Solution for n = −1:

2 at + C 1 e − kt + C 2

w(x, t) =

− b,

k(x + C 3 ) 2

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

5 ◦ . Generalized separable solution for n = 1:

w(x, t) = f (t)x 2 + g(t)x + h(t) − b,

where the functions f (t), g(t), and h(t) are determined by the system of ordinary differential equations

References : N. H. Ibragimov (1994), A. D. Polyanin and V. F. Zaitsev (2002).

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function

2 w 1 = wC 1 x+C 2 , t+C 3 − ln | C 1 |, λ

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Additive separable solution:

w(x, t) = ln( C 1 x+C 2 )+ C 3 e at + C 4 ,

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

1 2 w(x, t) = ln( λC 1 x + C 2 x+C 3 )+ u(t), λ

where C 1 , C 2 , and C 3 are arbitrary constants, and the function u = u(t) is determined by the ordinary differential equation

4 ◦ . Traveling-wave solution in implicit form:

Z be λw − λ 2 dw = x + λt + C 2 ,

aλw + C 1

where P✝◗ C 1 , C 2 , and λ are arbitrary constants.

References : N. H. Ibragimov (1994), A. D. Polyanin and V. F. Zaitsev (2002).