Dodd-Bullough-Mikhailov equation

Dodd-Bullough-Mikhailov equation is a nonlinear partial differential equation introduced by Roger Dodd, Robin Bullough, and Alexander Mikhailov^[1]。

$u_{xt}+\alpha*e^u+\gamma*e^{-2*u} = 0$

In 2005 Mathematician Wazwar combined the Tzitzeica equation with Dodd-Bullough-Mikhailov equation into Tzitz´eica-Dodd-Bullough-Mikhailov equation.^[2]

Dodd-Bullough-Mikhailov equation has traveling wave solutions.

Analytic solution

Like Tzitzeica equation,the analytic solution of Dodd-Bullough-Mikhailov equation also as the general form of ln(f(a(bx+ct+d))), where a,b,c,d are arbitrary constants, and f is a trigonometric function.

Perform transformation:

$v=e^u$ to transform Dodd-Bullough-Mikhailov equation into

$v*v_{xt}-v_{t}*v_{x}+\alpha*v^3+\gamma = 0$

obtain the traveling wave solutions of v(x,t): $v(x, t) = (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*cot(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2$ $v(x, t) = (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*coth(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^2$ $v(x, t) = (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*tan(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2$ $v(x, t) = (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*tanh(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^2$ $v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma^(1/3))*cot(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2$ $v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma^(1/3))*tan(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2$ $v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*coth(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2$ $v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*tanh(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2$

Then apply the inverse transform

$u(x,t)=ln(v(x,t))$

to get the traveling wave solutions of Dodd-Bullough-Mikhailov equation:

$u(x, t) =ln( (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*cot(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2)$ $u(x, t) =ln( (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*coth(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^)$ $u(x, t) =ln( (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*tan(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2)$ $u(x, t) =ln( (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*tanh(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^2)$ $u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma(1/3))*cot(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)$ $u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma^(1/3))*tan(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)$ $u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*coth(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)$ $u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*tanh(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)$

Traveling wave plot

The above solutions when plotted graphically provides a visual image of wave forms, they appear as complicated waves traveling from left to right.Some waves are periodic, some are solitary,called soliton.^[3]