相关论文: Higher Order Modulation Equations for a Boussinesq…

The Hamiltonian formulation of the water wave problem due to Zakharov, and the reduced Zakharov equation derived therefrom, have great utility in understanding and modelling water waves. Here we set out to review the cubic Zakharov equation…

We introduce two exponential-type integrators for the "good" Bousinessq equation. They are of orders one and two, respectively, and they require lower regularity of the solution compared to the classical exponential integrators. More…

We find Baikov-Gazizov-Ibragimov approximate point symmetries of the second-order Boussinesq ODE, and we find the higher-order approximate symmetries corresponding to the unstable point symmetries (the point symmetries that disappear fron…

The Balitsky-Kovchegov (BK) evolution equation is an equation derived from perturbative Quantum Chromodynamics that allows one to evolve with collision energy the scattering amplitude of a pair of quark and antiquark off a hadron target,…

We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schr\"odinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of…

The Comments are devoted to the paper 'Derivation of lump solutions to a variety of Boussinesq equations with distinct dimensions' (Int J Numer Methods Heat Fluid Flow. 2022;32:3072{3082), in which three new generalizations of the classical…

We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real…

The soliton resolution for the focusing modified Korteweg-de vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation…

In this article, we derive a viscous Boussinesq system for surface water waves from Navier-Stokes equations. We use neither the irrotationality assumption, nor the Zakharov-Craig-Sulem formulation. During the derivation, we find the bottom…

The high-energy evolution in perturbative QCD suffers from a severe lack-of-convergence problem, due to higher order corrections enhanced by double and single transverse logarithms. We resum double logarithms to all orders within the…

A classification of the time evolution of the two-soliton solutions of the Boussinesq equation is given, based on the number of extrema of the wave. For solitons moving in the same directions, three different scenarios are found, while it…

We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known, that for such regimes, a generalization of the KdV…

We derive exact equations governing the large-scale dynamics of hard rods, including diffusive effects that go beyond ballistic transport. Diffusive corrections are the first-order terms in the hydrodynamic gradient expansion and we obtain…

In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated…

We study asymptotic reductions and solitary waves of a weakly nonlocal defocusing nonlinear Schr\"odinger (NLS) model. The hydrodynamic form of the latter is analyzed by means of multiscale expansion methods. To the leading-order of…

In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et. al. [Stud. Appl. Math., 53 (1974) 249--315] and one by Hirota and…

A long wave multi-dimensional approximation of shallow water waves is the bi-directional Benney-Luke equation. It yields the well-known Kadomtsev-Petviashvili equation in a quasi one-directional limit. A direct perturbation method is…

The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the…

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic…

We consider here two discrete versions of the modified KdV equation. In one case, some solitary wave solutions, B\"acklund transformations and integrals of motion are known. In the other one, only solitary wave solutions were given, and we…