We study a nonlinear Schrödinger system with three-wave interaction: { - Δ u_1 = f_1(u_1) + α u_2u_3 in ℝ^N, - Δ u_2 = f_2(u_2) + α u_3u_1 in ℝ^N, - Δ u_3 = f_3(u_3) + α u_1u_2 in ℝ^N, u⃗=(u_1,u_2,u_3)∈ (H_rad^1(ℝ^N))^3, . where 3≤ N≤ 5 , α∈ℝ and each nonlinearity f_i(ξ ) satisfies the Berestycki-Lions conditions. Let S_i denote the set of all least energy solutions of the scalar equation -Δ u = f_i(u) in H_rad^1(ℝ^N) . A solution of the systems is called vector if all its components are nontrivial. We establish the existence of two distinct families of vector solutions {u⃗_α} with different asymptotic behaviors as α→ 0 . One family satisfies dist(u⃗_α,S_1× S_2× S_3) → 0 , while another satisfies dist(u⃗_α,S_1× S_2×{0}) → 0 . By contrast, we prove that no family of vector solutions satisfies dist(u⃗_α,S_1×{0}×{0}) → 0 . Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak.
Lie symmetries, conservation laws, reductions, and new exact solutions of the extended form of the (2+1) -dimensional generalized Zakharov–Kuznetsov (ZK) equation are studied. This generalized ZK model includes three arbitrary exponents m, n and k in the nonlinear terms. The Lie algebra for the generalized ZK model is four dimensional. Conservation laws are derived using the multiplier method and five cases arise under different restrictions on the parameters m, n and k of the model. Lie symmetries associated with the conserved vectors are determined for each case. Furthermore, several reductions and new exact solutions of ZK equation are established using the generalized double reduction theory. These solutions advance the analysis of nonlinear dynamics in physical systems including plasmas, shallow water flows, and other dispersive media.
In this paper, we study the 3D dissipative fluid–dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the Navier–Stokes/Poisson–Nernst–Planck system. We prove that if the partial derivatives of two velocity components (∂ _1 u_1, ∂ _2 u_2) satisfy ∫ _0^T‖ (∂ _1 u_1, ∂ _2 u_2)(· , t) ‖ ^8/5-4α_Ḃ_∞ ,∞^-α/1 + ln (e + ‖ (∂ _1 u_1, ∂ _2 u_2)(· , t) ‖ _Ḃ_∞ , ∞^-α ) dt< ∞ for 0< α < 1, then the local solution can be smoothly extended past the time t = T . Particularly, a regularity criterion is further established for the critical case α =0 . These results represent further improvements of previous studies by Zhao et al. (2016; 2019; 2025) and Wu (2019). Moreover, we extend the results by Zhang (2008) and Dong et al. (2010; 2011) for the incompressible Navier–Stokes equations.
We investigate the phenomenon of Landau damping within the context of the Vlasov-Riesz system. Employing a nonlinear analysis technique developed by Ionescu, Pausader, Wang, and Widmayer, our primary objective is to extend the understanding of Landau damping to kinetic models beyond the traditional Vlasov-Poisson system. Our analysis is grounded in the use of the Gevrey- γ norm, and we demonstrate that the critical Gevrey index γ =-4/3α +5/3 is dependent on the parameter α from the Riesz interaction, defined as U=( -Δ) ^-α(ρ -1) . This study not only broadens the applicative scope of Landau damping but also deepens the theoretical understanding of its mechanisms in more complex kinetic frameworks.
In this work, we provide an analytical proof of the existence of attracting invariant tori supporting quasiperiodic dynamics in a family of three-dimensional dissipative systems modeling self-oscillatory electronic devices. This family was originally proposed by Kuznetsov, Kuznetsov, and Stankevich, who observed numerically such quasiperiodic dynamics and evidences for the existence of invariant tori. Our results establish the existence of normally hyperbolic attracting tori using recent advances in averaging theory. We also show that the dynamics on these tori are always either periodic or quasiperiodic, and that quasiperiodic motion occurs for a set of parameter values with positive Lebesgue measure. These findings not only confirm and extend the prior numerical observations, but also demonstrate that quasiperiodicity arises with positive probability when parameters are randomly chosen. Moreover, our study highlights the effectiveness of averaging theory as a powerful analytical tool for detecting attractors in applied dynamical systems.
