15 problems

Let $u(t)$ be a solution of the cubic nonlinear Schrödinger equation on $\mathbb{T}^2$, and let $s>0$ with $s\neq 1$. For every $\epsilon>0$, let $C_\epsilon$ be a constant. Bourga…

Let $v_0$, $(x_0,t_0)$, $a_0=a(v_0)$, $a_0'=a'(v_0)$, $c_0=c(v_0)$, and $\kappa=-(a”'(v_0)t_0+b”'(v_0))$ be the quantities associated with a generic solution, where $kappa eq0$ and…

Let $v=v(x,t)$ be a smooth solution of the unperturbed equation … defined for all $x\in\boldsymbol{R}$ and $0\leq t<t_0$, and monotone in $x$ for every $t$. Let $H_f$ be the Hamilt…

Trivial-symmetry global attraction conjecture. The solution satisfies

Global attraction conjecture. For generic such equations, every finite-energy solution admits the asymptotics

Let $G$ be a Lie symmetry group and consider a $G$-invariant Hamiltonian nonlinear PDE … where $\Psi(\cdot,t)$ belongs to $L^2(\mathbb R^3)\otimes\mathbb R^d$,…

Let $G$ be a Lie symmetry group acting by a linear representation $T$ on a suitable Hilbert or Banach phase space $\mathcal E$, and consider a generic $G$-invariant autonomous Hami…

Let a tau-symmetric integrable Hamiltonian deformation of the Riemann hierarchy be considered modulo normal Miura-type transformations. Write $w_k=\partial_x^k w$. The canonical de…

Let a tau-symmetric integrable Hamiltonian deformation of the Riemann hierarchy be given by a hierarchy of scalar Hamiltonian evolutionary PDEs with Hamiltonian densities and Hamil…

Comoving local convergence conjecture. The convergence to the asymptotic solitons holds in the local seminorms in the comoving frame, namely in $L^2(|x-v_\pm t|<R)$ for every $R>0$…

Global attraction conjecture. The solution converges globally to the corresponding steady states:

Let $r_\pm(x,t;\epsilon)$ solve the relevant Hamiltonian perturbation of a hyperbolic system, with an $\epsilon$-independent Cauchy problem, no $\mathcal O(\epsilon)$ terms, and a…

At a Type II critical point, let $r_+$ and $r_-$ be the complex-conjugate Riemann invariants, with characteristic variables $x_+$ and $x_-$. For the unperturbed system, $r_+$ is ho…

Consider a Hamiltonian PDE with a small dispersion parameter $\epsilon$, and a generic solution developing a critical point. The leading term in the asymptotic expansion as…

Let $U=U(X;T)$ solve … Here $T\in\mathbb R$, and a solution is smooth on the entire real axis if it has no singularities for any $X\in\mathbb R$. Dubrovins' conjecture. For every r…