Soliton equations: admitted solutions and invariances via Bäcklund transformations

Abstract

A couple of applications of Bäcklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on proving a new invariance admitted by a third order nonlinear evolution equation and, on the other one, on the construction of solutions. Indeed, via Bäcklund transformations, a {\it Bäcklund chart}, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third order nonlinear evolution equations of KdV type. On the basis of the Abelian wide Bäcklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the {\it Korteweg-deVries interacting soliton} (int.sol.KdV) equation is obtained and a related new explicit solution is constructed. Then, the corresponding non-Abelian {\it Bäcklund chart}, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered.

Figures (6)

• Figure 1: Induced invariance Bäcklund chart.
• Figure 2: The solution depicted in the case ${\sf d}=2$, $k_1=1+i$, $k_2=\overline{k_1}=1-i$, $B_1 = i-21+i2-i$, $B_2 =\overline{B_1}$, $-5\leq x \leq 5$ and $0\leq t \leq 2$, plot range $(-3.5, 3.5)$.
• Figure 3: The solution depicted in the case ${\sf d}=2$, $k_1=1+i$, $k_2=\overline{k_1}=1-i$, $B_1 = i-2i3i -1-1$, $B_2 =\overline{B_1}$, $-5\leq x \leq 5$ and $-1\leq t \leq 1$, plot range $(-5.5, 5.5)$.
• Figure 4: ${\sf d}=1$ with the input data $N=2$, $k_1=1$, $k_2=\sqrt{2}$, and $b_1=b_2=1$ in Theorem \ref{['class']}
• Figure 5: The solution depicted represents the case ${\sf d}=2$, $k_1=1$, $k_2=\sqrt{2}$, $B_1 = 1111$, $B_2 = \ 1-1-1\ 1$, when $-10 \leq x \leq 10$ and $-5\leq t\leq 5$, plot range $(-\sqrt{2}, \sqrt{2})$.