相关论文: On the Whitham Equations for the Defocusing Comple…
Extending our earlier work on Lax-type shocks of systems of conservation laws, we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general Lax- or undercompressive-type shock…
In this work, an exact solution to a new generalized nonlinear KdV partial differential equations has been investigated using homotopy analysis techniques. The mentioned partial differential equation has been solved using homotopy…
The evolutionary conditions for the dissipative continuous magnetohydrodynamic (MHD) shocks are studied. We modify Hada's approach in the stability analysis of the MHD shock waves. The matching conditions between perturbed shock structure…
A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi-Pasta-Ulam lattice with a general cubic interaction force. Explicit periodic traveling wave solutions in terms of Jacobi…
Recently, Galley [Phys. Rev. Lett. {\bf 110}, 174301 (2013)] proposed an initial value problem formulation of Hamilton's principle applied to non-conservative systems. Here, we explore this formulation for complex partial differential…
The Hamiltonian formulations for the perturbed Vlasov-Maxwell equations and the perturbed ideal magnetohydrodynamics (MHD) equations are expressed in terms of the perturbation derivative $\partial{\cal F}/\partial\epsilon \equiv [{\cal F},…
We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves…
Recent concept of integrable nonholonomic deformation found for the KdV equation is extended to the mKdV equation and generalized to the AKNS system. For the deformed mKdV equation we find a matrix Lax pair, a novel two-fold integrable…
In this article we derive rigorously a nonlinear, steady, bifurcation through spectral bifurcation (i.e., eigenvalues of the linearized equation crossing the imaginary axis) for a class of hyperbolic-parabolic model in a strip. This is…
"Generalized Hydrodynamics" (GHD) stands for a model that describes one-dimensional \textit{integrable} systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic…
The N-cnoidal solution of the Korteweg-de Vries (KdV) evolution equation is presented based on the prolongation structure theory of Wahlquist and Estabrook [J. Math. Phys. \textbf{16}, 1 (1975)]. The generalized KdV cnoidal wave solutions…
We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Levy (jump-diffusion) processes. These equations are fully non-linear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity…
The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian…
The fractional discrete nonlinear Schr\"odinger equation (fDNLS) is studied on a periodic lattice from the analytic and dynamic perspective by varying the mesh size $h>0$ and the nonlocal L\'evy index $\alpha \in (0,2]$. We show that the…
We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx, invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space-time reflection-symmetric first order…
Kinematic equations for the motion of slowly propagating, weakly curved fronts in bistable media are derived. The equations generalize earlier derivations where algebraic relations between the normal front velocity and its curvature are…
We investigate Hamiltonian aspects of the integro-differential kinetic equation for dense soliton gas which results as a thermodynamic limit of the Whitham equations. Under a delta-functional ansatz, the kinetic equation reduces to a…
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the $I$-method and spectral analysis following Pego and Weinstein, we are…
The KdV-Burgers equation is a canonical model describing the interplay between nonlinearity, viscosity and dispersion, and it admits viscous-dispersive shocks as traveling wave solutions. In this paper, we establish an $L^2$-contraction…
Dispersive PDEs are important both in applications (wave phenomena e.g. in hy- drodynamics, nonlinear optics, plasma physics, Bose-Einstein condensates,...) and a mathematically very challenging class of partial differential equations,…