相关论文: Discrete Reductive Perturbation Technique

Stochastic partial differential equations (SPDEs) represent a very active research field with numerous recent developments and breakthrough results. There are several well-established approaches and methods used to construct solutions for…

Physics-Informed Neural Networks (PINNs) offer a flexible framework for solving nonlinear partial differential equations (PDEs), yet conventional implementations often fail to preserve key physical invariants during long-term integration.…

The long time behavior of solutions to the defocusing 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 of Deift…

We apply the Linear Delta Expansion (LDE) to the Lindstedt-Poincare (``distorted time'') method to find improved approximate solutions to nonlinear problems. We find that our method works very well for a wide range of parameters in the case…

We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the…

We study the modulational stability of the nonlinear Schr\"odinger equation (NLS) using a time-dependent variational approach. Within this framework, we derive ordinary differential equations (ODEs) for the time evolution of the amplitude…

A new exponentially fitted version of the Discrete Variational Derivative method for the efficient solution of oscillatory complex Hamiltonian Partial Differential Equations is proposed. When applied to the nonlinear Schroedinger equation,…

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this letter we derive infinitely many conserved quantities…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate…

The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known equations such as the Korteweg--de Vries, coupled KdV, Harry…

An approximate perturbed direct homotopy reduction method is proposed and applied to two perturbed modified Korteweg-de Vries (mKdV) equations with fourth order dispersion and second order dissipation. The similarity reduction equations are…

In this paper we examine a deformation of the derivative nonlinear Schr\"odinger (DNLS) equation, the so-called Camassa-Holm DNLS (CH-DNLS) equation. We use two asymptotic multiscale expansion methods to reduce this model to both the…

A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be…

Under investigation in this paper is the fractional integrable and non-integrable discrete modified Korteweg-de Vries hierarchies. The linear dispersion relations, completeness relations, inverse scattering transform, and fractional soliton…

We consider the spectral problem of the Lax pair associated to periodic integrable partial differential equations. We assume this spectral problem to be a polynomial of degree $d$ in the spectral parameter $\lambda$. From this assumption,…

The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…

We analyze the pressure and density equations of state of unpolarized non-relativistic fermions at finite temperature in one spatial dimension. For attractively interacting regimes, we perform a third-order lattice perturbation theory…

The use of symplectic numerical schemes on Hamiltonian systems is widely known to lead to favorable long-time behaviour. While this phenomenon is thoroughly understood in the context of finite-dimensional Hamiltonian systems, much less is…

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit…