19 problems

Let $u$ solve the Korteweg–de Vries (KdV) equation with almost periodic initial data. Deift conjecture. The corresponding solution should evolve almost periodically in time. This c…

Absolute convergence conjecture. The integral

Let $F(X;T)$ satisfy the fourth-order ordinary differential equation … At $T=0$, the equation is conjectured to have a smooth real solution on the real line with asymptotic behavio…

Let $N \ge 1$ be an integer and let $\varepsilon > 0$. Suppose $u_0 \in L^2(\mathbb{T})$ is frequency-localized as … Let $u(x,t)$ be the corresponding solution to the periodic line…

Let a pulse evolve according to either the KdV equation or the mKdV equation, and consider changing the overall sign of the solution. The mKdV equation is symmetric under this sign…

Consider a Gaussian initial pulse for the KdV equation, and let the predicted soliton amplitudes be obtained from relations between the pulse parameter and soliton amplitudes using…

Let $\eta=\{\eta_k\}_{k\geq1}$ and $\eta^{(1)}=\{\eta_k^{(1)}\}_{k\geq1}$ be the frequency collections associated with the comb domains describing the KdV flows. Comb-domain generi…

Let $\mathsf{E}$ be the spectral set, let $\mathcal{R}(\mathsf{E})$ denote the reflectionless potentials with spectrum $\mathsf{E}$, and let … be the divisor map. Let…

Let $m\in\mathbb{N}$ and $\tilde{c}>0$. For sufficiently small $\varepsilon>0$, consider the FPU system and the speed $c=1-\varepsilon^2\tilde{c}$. A generalized moving pulse solut…

Let $u(x)$) and $y(x)$ be functions satisfying equation … possesses four step-like solutions for any $H_0>-4$ and $H_1>0$. Two of them have the asymptotic behaviour … and another t…

Let $f$) be a periodic function on $\mathbb{T}^2$, let $\hat{f}$ be its Fourier transform, and for a sufficiently large integer $N$ and $p\geq 2$ define $A_{p,N}$ as the best const…

Let … should also hold far beyond $H^{-3/4}(\mathbb{R})$ and could be achieved by a suitable extension of the inverse scattering transform method for $\mathbb{L}_{q}$. This would e…

Let the Cauchy problem for the KdV equation be posed with measure initial data. The Miura transform is given by $\mathcal{M}(u)=\partial_x u+u^2$ and maps suitable $L^2$ solutions…

Let $u(x,t)$ be the solution considered in Theorem 1.3, with $t>0$. Meromorphic–Gevrey representation conjecture. The function $u(x,t)$ could be represented, for any $t>0$, as a me…

A well-posed KdV problem is an initial-value problem for the KdV equation whose solution exists and depends appropriately on its initial data. The Hirota $\tau$-representation is t…

Let $L_{+}^{2}$ denote the positive-half-line subspace, let $\chi_{+}$ be the projection onto it, and let $\mathcal{F}$ be the Fourier transform. For $t\geq 0$ and $c\in\mathbb{R}$…

Finiteness conjecture. There are no Jacobi structures for the KdV equation other than $Q_0,Q_1,Q_2,Q_3$.

Let $n$ be a positive integer, let $k=[(n+1)/2]$, and consider the pole asymptotics $x_j(t)\sim a_jt^{1/3}$ as $t\rightarrow +0$ for the KdV solution with initial value…

Pole-free solution conjecture. This equation has a real pole-free solution $U(X,T)$.