Rational Solutions to the Boussinesq Equation

Rational solutions to the Boussinesq equation are constructed as a quotient of two polynomials in $x$ and $t$. For each positive integer $N$, the numerator is a polynomial of degree $N(N+1)-2$ in $x$ and $t$, while the denominator is a polynomial of degree $N(N+1)$ in $x$ and $t$. So we obtain a hierarchy of rational solutions depending on an integer $N$ called the order of the solution. We construct explicit expressions of these rational solutions for $N=1$ to $4$.