Special Session 94: Dynamics and Variational Methods of Quasi-Hamiltonian Systems

Invariant Tori in Hamiltonian and Conformally Symplectic Systems: A Posteriori KAM Theory and Computation

Renato C Calleja IIMAS-UNAM Mexico Co-Author(s):    Pedro Porras, Alex Haro, Arturo Vieiro

Abstract: This talk presents recent and ongoing work on invariant tori in Hamiltonian systems with quasi-periodic time dependence and in conformally symplectic systems. In joint work with Pedro Porras and Alex Haro, we developed an a posteriori KAM theorem for Lagrangian invariant tori in Hamiltonian flows with quasi-periodic forcing. The proof is based on the parameterization method and a Newton-like scheme using adapted symplectic frames and an intrinsic torsion matrix, leading to efficient quadratically convergent algorithms under standard Diophantine and nondegeneracy conditions. I will also discuss current work with Alex Haro and Arturo Vieiro on secondary tori near elliptic points and on resonance capture in near-symplectic dynamics via Birkhoff normal forms. These problems illustrate how the same functional analytic and geometric ideas extend from conservative to weakly dissipative settings. The parameterization method provides a common framework throughout, connecting rigorous existence theory, effective numerical computation, and the study of breakdown and bifurcation phenomena for quasi-periodic invariant objects.

On selection problems for second-order Hamilton-Jacobi equations

Qinbo Chen Nanjing University Peoples Rep of China Co-Author(s):

Abstract: This talk concerns the simultaneous effects of the discounted approximation and the potential perturbation in second-order degenerate elliptic Hamilton-Jacobi equations. Our analysis relies on generalized (stochastic) Mather measures selected via the nonlinear adjoint method, which enables us to derive a new selection mechanism for second-order Hamilton-Jacobi equations.

Long time stability of Hamiltonian derivative nonlinear Schr\odinger equations

Shengqing Hu Shenzhen SMU-BIT University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we study an abstract Birkhoff normal form theorem for some unbounded infinite dimensional Hamiltonian systems. Based on this result, we obtain that the solution to derivative nonlinear Schr\odinger equations with typical small enough initial value remains small in Sobolev norm over a subexponential long time interval.

Cartan-Schouten Connections: Geometric Reduction and a Connection-Dependent Variational Principle

Qiao Huang Southeast University Peoples Rep of China Co-Author(s):

Abstract: We study the family of Cartan-Schouten connections on Lie groups, parameterized by $\lambda\in[0,1]$, whose geodesics through the identity are one-parameter subgroups. We compute their curvature, torsion, parallel transport, and geodesics, and develop Euler-Poincar\'e and Lie-Poisson reduction for mechanical systems via these connections, unifying the ``minus'' and ``plus'' cases. These inspire us to introduce a connection-dependent variational principle where the Lagrangian is expressed in terms of the parallel-transported velocity, leading to an integro-differential Euler-Lagrange equation that explicitly involves torsion and curvature memory terms.

Comparison principle of general Hamilton-Jacobi equations and applications

Gengyu Liu Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Zhang Jianlu, Tu Son

Abstract: In this talk, we investigate contact Hamilton-Jacobi equations $H(x, du, u) = c$ and $H(x, du, u) = c + \Delta u$ under the condition that the contact direction is not strictly positive definite. Correspondingly, we describe the structure of $\mathfrak{C}$ containing all the $c \in\mathbb{R}$ that makes the equations solvable and establish general comparison principles. As applications, we employ these comparison principles to obtain quantitative homogenization results.

Mean Field Games of Controls with Boundary Conditions \& Invariance Constraints

Kyle Rosengartner Baylor University USA Co-Author(s):    Jameson Graber

Abstract: In a mean field game of controls, a large population of identical players seek to minimize a cost that depends on the joint distribution of the states of the players and their controls. We first consider the classes of mean field games of controls in which the value function and the distribution of player states satisfy either Dirichlet or Neumann boundary conditions. We prove that such systems are well-posed either with sufficient smallness conditions or in the case of monotone couplings. Next, we consider mean field games of controls under invariance constraints imposed on the state space. We prove the existence and uniqueness of weak solutions to our mean field game system, and then we prove higher regularity of solutions under some additional assumptions.

On the vanishing viscosity limit of Hamilton-Jacobi equations

Zibo Wang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: In this talk, we try to develop a unified framework for the vanishing viscosity limit of Hamilton-Jacobi equations. Our approach aims to establish the convergence of viscous approximations within the class of Tonelli Hamiltonians. Moreover, we provide quantitative estimates on the rate of convergence. This is a joint work with Jianlu Zhang.

Uniqueness results of ground states for mixed Laplacian and fractional Laplacian.

Jiwen Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will report some recent uniqueness results for the radial solutions to linear and nonlinear Schr\{o}dinger equations involving mixed Laplacian and fractional Laplacian. We prove that the linear equation admits at most one bounded radial solution, provided that the potential is radial, nondecreasing and H\{o}lder continuous. Moreover, the nonlinear Schr\{o}dinger equation possesses a unique ground state solution and is nondegenerate when $s$ is close to $0$ or $1$.

Periodic, Quasi-Periodic and Almost Periodic Solutions of the Hamilton-Jacobi Equation

Kai Zhao Tongji university Peoples Rep of China Co-Author(s):    Kai Zhao

Abstract: It is well known that there exist no non-trivial periodic solutions for classical Hamilton-Jacobi equations. Combining the weak KAM method for contact Hamiltonian systems with the viscosity solution theory of the Hamilton-Jacobi equation, we have recently investigated the problem of time periodic solutions for contact Hamilton-Jacobi equations. Under certain conditions, quasi-periodic solutions and almost periodic solutions can also emerge. Furthermore, we have explored the chaotic nature of the large time behavior of the solutions.