Computer Algebra Recipes

7.2. ANALYTIC SOLITON SOLUTIONS 307
In 1971, Fred Tappert, of Bell Laboratories, found an exact two-soliton
analytic solution for the KdV equation, this equation now being entered.
> restart: with(plots):
> KdVE:=diff(psi(x,t),t)+psi(x,t)*diff(psi(x,t),x)
+diff(psi(x,t),x,x,x)=0;
μ μ μ
3
@
@
@
KdVE := Ã(x; t) + Ã(x; t) Ã(x; t) + Ã(x; t) =0
@t @x @x3 Tappert's two-soliton solution Ã(x; t) isgiven.
> psi(x,t):=72*(3+4*cosh(2*x-8*t)+cosh(4*x-64*t))
/(3*cosh(x-28*t)+cosh(3*x-36*t))^2;
72 (3 + 4 cosh(2 x ¡ 8 t)+cosh(4x ¡ 64 t))
Ã(x; t) :=
(3 cosh(x ¡ 28 t)+cosh(3x ¡ 36 t)) 2
The lhs of KdVE is extracted and simpli¯ed, Ã(x; t) having been automatically
substituted. A lengthy expression results, which is suppressed here in the text.
> check1:=simplify(lhs(KdVE));
The combine command, with the trig option, is applied to the numerator of
check1 , the result being zero, con¯rming that Ã(x; t) is a solution of KdVE .
> check2:=combine(numer(check1),trig);
check2 := 0
The two-soliton solution is now animated.
> animate(plot,[psi(x,t),x=-20..20],t=-1..1,frames=60,
numpoints=250,axes=frame,thickness=2);
In the animation, one initially has a taller, narrower solitary wave to the left of
a shorter, wider solitary wave. As the animation progresses, both pulses move
to the right. Having a larger velocity, the taller pulse overtakes the shorter
pulse and a collision occurs. During the collision, a nonlinear superposition
takes place, the resultant amplitude being less than the linear sum of the two
amplitudes. As time progresses, the taller, faster pulse passes through the
shorter one and emerges unchanged in shape, as does the shorter pulse. The
solitary waves are indeed solitons.
Next, we con¯rm that the sine{Gordon equation,
> SGE:=diff(U(x,t),x,x)-diff(U(x,t),t,t)-sin(U(x,t))=0;
μ μ
2
2
@ @
SGE := U (x; t) ¡ U (x; t) ¡ sin(U (x; t)) = 0
@x2 @t2 is satis¯ed by the following two-soliton kink{kink solution. [Jac90]
> U(x,t):=4*arctan(c*sinh(x/sqrt(1-c^2))
/cosh(c*t/sqrt(1-c^2)));
0 μ 1
x
B
c sinh p
U (x; t) := 4 arctan B 1 ¡ c2 C
@
μ C
ct A
cosh p
1 ¡ c2