58 problems

For a general dispersive equation, let a generic solution mean a solution in the class under consideration, and let a multisolitary wave be a sum of solitary waves. Soliton resolut…

Dubrovin's universality conjecture. Near a generic gradient-catastrophe point, the dispersive solution should be modeled by this particular solution $\mathcal{D}(X;T)$.

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…

Berry's conjecture. The graphs of $\Re u(x,t)$, $\Im u(x,t)$, and $|u(x,t)|^2$ have fractal dimension $D=n+\frac{1}{2}$ at most all irrational times $t$.

Consider the free wave equation with initial data $u_0,u_1\in L^2$ and solution $u(t,\cdot)$. The preceding result gives almost linear growth for data whose Fourier transform satis…

Consider the $L^2$-supercritical generalized Korteweg–de Vries equation in the odd-power case, and the convergence around spatial zero asserted in Theorems … . Odd-power convergenc…

The Whitham equation is … for $(x,t)\in\mathbb{R}\times\mathbb{R}^+$, where $\zeta(x,t)\in\mathbb{R}$ is the surface elevation and $\sqrt{\mathcal{K}_{\mu}}(D)$ is the square root…

Klein–Saut conjecture. The following assertions hold:

Consider the dissipative modified Korteweg–de Vries equation on the real line … The scaling-critical index of the mKdV equation is $-\frac{1}{2}$, while that of the equation with o…

Endpoint logarithmic Strichartz conjecture. For $\lambda\gg1$,

The Klein–Gordon–Zakharov system is considered in spacetime dimension $\mathbb{R}^{1+d}$, with scattering behavior as described in Theorem … may be a phenomenon specific to the two…

Let $u$ solve the initial-value problem associated to the fractional KdV equation … with parameter $\alpha$ and initial data in the relevant Sobolev space. Global well-posedness an…

Classification of minimal-mass blowup solutions. Only the following two cases occur:

Consider the one-dimensional nonlinear Schrödinger equation … with standing-wave and traveling-wave solutions generated by a ground state $Q$. Soliton resolution conjecture. Genera…

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 periodically forced Dirac equation with mass $m=1$, and let $M^n$ denote its $n$-period propagator. For $\beta>0$, write…

Periodic fifth-order Airy Strichartz conjecture. The estimate holds for $a=0$.

One-dimensional dispersive flows with cubic defocusing nonlinearities and small initial data are considered. Global small-data scattering conjecture. Such flows should have global-…

Let $FIO(S,M^{\infty,q}_{1\otimes v_s})$ denote the class of continuous linear operators whose Wigner kernels have the localization and modulation-space structure specified above,…

Let $s$ denote the Sobolev regularity exponent, and let Theorems and be the local well-posedness results stated in the source for the two-dimensional and $n$-dimensional cubic prob…

Consider quasilinear dispersive flows in two and higher spatial dimensions with cubic nonlinearities and sufficiently small initial data. Two-dimensional and higher-dimensional qua…

Consider one-dimensional dispersive flows on the real line with cubic conservative nonlinearities and initial data of size $epsilon \ll 1$. Non-localized data long-time well-posedn…

Uniform two-thirds decay conjecture. For any $\zeta\in\mathcal{C}_c^{\infty}(\mathbb{R}^d)$,

Consider a one-dimensional dispersive flow which is locally well-posed in some Sobolev space $H^s$, has phase rotation symmetry, and is conservative. For initial data $u_0$ satisfy…

Berry–Klein's conjecture. At irrational times, the solution profiles of the periodic FLS equation have fractal dimension