and the true solution ${u_\epsilon}(x,t)$ of the Cauchy problem for the BO equation with initial data $u_0$. Some comments about our continuing work can be found in the conclusion, §\[sec:conclusion\]. But before we embark on our study of the BO equation, we pause to consider some familiar analogues of Theorem \[MainTheorem\].

Elementary examples
-------------------

The key role played in the zero-dispersion limit of the BO Cauchy problem by the multivalued solution of the equation with the same initial data is reminiscent of two basic example problems from the theory of linear and nonlinear waves.

### The zero-viscosity limit of the viscous Burgers equation

The Burgers equation with viscosity $\epsilon>0$ is the nonlinear wave equation $$\frac{\partial w_\epsilon}{\partial t} + 2w_\epsilon
\frac{\partial w_\epsilon}{\partial x} -
\epsilon \frac{\partial^2w_\epsilon}{\partial x^2}=0,
\quad x\in\mathbb{R},\quad t>0$$ and we take fixed initial data $w_\epsilon(x,0)=u_0(x)$. As is well-known, this Cauchy problem is solved by the Cole-Hopf transformation, leading to the exact solution formula $$w_\epsilon(x,t)=\frac{1}{2t}\frac{\displaystyle\int_\mathbb{R}
e^{R(\xi;x,t)/\epsilon}(x-\xi)\,d\xi}
{\displaystyle\int_\mathbb{R}
e^{R(\xi;x,t)/\epsilon}\,d\xi},\quad t>0,$$ where the exponent function is defined as $$R(\xi;x,t):=-\int_0^\xi u_0(\eta)\,d\eta -\frac{(x-\xi)^2}{4t}.$$ One examines the asymptotic behavior in the limit $\epsilon\downarrow
0$ by using Laplace’s method to analyze the integrals (see [@Miller], §3.6). The dominant contributions to the integrals come from neighborhoods of 