| Abstract: |
| Wronskian solutions are known for many nonlinear partial differential equations including the well studied KdV and Boussineque Equations. In the talk we present some properties of solutions obtained from Wronskian determinants $W(\phi_1,\phi_2,\ldots,\phi_N)$ with generating functions $\phi_j(x,t)=\cosh\gamma_j(x,t)$ or $\phi_j(x,t)=\sinh\gamma_j(x,t)$ for $\gamma_j(x,t)=p_jx+\sigma_j(t), \; p_j>0$, for real $x$ and $t$ and arbitrary functions $\sigma_j$.
By following the Hirota bi-linear method we study the properties of the multi-soliton functions $u=(\log W)_{xx}$, including characterization of the non-singular choices of $\sinh$ and $\cosh$. We also present an explicit decomposition $u=\sum_{j=1}^N k_j\psi_j^2$ for $k_j>0$ and $\psi_j$ the eigenfunctions of the eigenvalue operator in the Lax Pair for KdV and the Boussinessq equations.
We also discuss particular non-linear choices of the functions $\sigma_j$. |
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