New explicit and exact solutions for the Nizhnik-Novikov-Vesselov equation.
New solitons and periodic solutions for Thekadomtsev-Petviashvili equation.
Nonanalytic solutions of the KdV equation.
Nonclassical approximate symmetries of evolution equations with a small parameter.
Non-Euclidean geometry and differential equations
In this paper a geometrical link between partial differential equations (PDE) and special coordinate nets on two-dimensional smooth manifolds with the a priori given curvature is suggested. The notion of G-class (the Gauss class) of differential equations admitting such an interpretation is introduced. The perspective of this approach is the possibility of applying the instruments and methods of non-Euclidean geometry to the investigation of differential equations. The equations generated by the...
Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane
Nonlinear feedback stabilization of a two-dimensional Burgers equation
In this paper, we study the stabilization of a two-dimensional Burgers equation around a stationary solution by a nonlinear feedback boundary control. We are interested in Dirichlet and Neumann boundary controls. In the literature, it has already been shown that a linear control law, determined by stabilizing the linearized equation, locally stabilizes the two-dimensional Burgers equation. In this paper, we define a nonlinear control law which also provides a local exponential stabilization of...
Nonlinear models for laser-plasma interaction
In this paper, we present a nonlinear model for laser-plasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some well-posedness and ill-posedness result for some subsystems.
Nonlinear Scattering for Some Dispersive Equations Generalizing Benjamin-Bona-Mahony Equation.
Non-planar fronts in Boussinesq reactive flows
Nontrivial solitary waves of GKP equation in multi-dimensional spaces.
Note on coupled linear systems related to two soliton collision for the quartic gKdV equation.
Notes on symplectic non-squeezing of the KdV flow
We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle . The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our...
Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system
We prove almost optimal local well-posedness for the coupled Dirac–Klein–Gordon (DKG) system of equations in dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman–Machedon type. It has been known for some time that the Klein–Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.
Numerical comparisons of two long-wave limit models
The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...
Numerical comparisons of two long-wave limit models
The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...
Numerical investigation for solitary solutions of the Boussinesq equation.
Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method.
On a Caginalp phase-field system with a logarithmic nonlinearity
We consider a phase field system based on the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Dirichlet boundary conditions. In particular, we prove, in one and two space dimensions, the existence of a solution which is strictly separated from the singularities of the nonlinear term and that the problem possesses a finite-dimensional global attractor in terms of exponential attractors.
On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension.