8 problems

Let $u$ solve the fifth-order modified KdV type equation on the periodic domain with parameters satisfying the hypotheses of Theorem 1.1, and let $s$ denote the Sobolev regularity…

Consider the evolution of a localized Gaussian initial profile in the higher-time-derivative modified KdV equation, whose evolution tends to decay into $N$-soliton solutions with i…

Consider the rotated version of the higher-time-derivative modified KdV equation with parameter $\sigma$, and let $N$ denote the number of solitons in a solution. For…

The elastica model describes DNA shapes through solutions of the focusing modified Korteweg–de Vries (MKdV) equation on hyperelliptic curves, with reality conditions restricting wh…

Consider the modified KdV equation for $v(t,x)$ with parameter ${\kappa}>0$ and sufficiently smooth Cauchy data. The $x$-evolution means evolution of the data according to this equ…

Let $V$ be a self-similar solution of the modified Korteweg–de Vries equation defined on $\mathbb{R}$, with mean parameter $c$, and suppose that $V\in\mathcal{S}'(\mathbb{R})$. In…

Higher-order mKdV breather energy conjecture. For every $n\in\mathbb{N}$, the $(2n+1)$-st order energy of $B$ is

The higher-order mKdV Lyapunov functional conjecture. The functional $\mathcal{H}_{2n+1}$ generates the associated nonlinear ODE