Related papers: (G'/G)-Expansion Method and Weierstrass Elliptic F…
A Gaussian beam method is presented for the analysis of the energy of the high frequency solution to the mixed problem of the scalar wave equation in an open and convex subset, with initial conditions compactly supported in this set, and…
We devise a lucid and straightforward way for explicitly constructing quasi-periodic wave solutions (also called multi-periodic wave solutions) of supersymmetric equations in superspace $\mathbb{R}_\Lambda^{2,1}$ over two-dimensional…
In the present paper, a hierarchy of the mKdV equation is integrated by the methods of algebraic geometry. The mKdV hierarchy in question arises on coadjoint orbits in the loop algebra of $\mathfrak{sl}(2)$, and employs a family of…
New exact solutions for the heat equation with a polynomial non-linearity and for the Fisher equation are found. An extended class of non-linear heat equations admitting solitary wave solutions is found. The generalization of the Fisher…
Different dynamics, described by kinetic equation and clipping method is shown as well as a role of approximate resonances in wave turbulence theory. Applications of clipping method are sketched for gravity-capillary and drift waves. Brief…
In this paper, by using bifurcation method, we successfully find the K(2,2)equation with osmosis dispersion possess two new types of travelling wave solu tions called kink-like wave solutions and antikink-like wave solutions. They are…
We propose a new regularization method for constructing a shock wave type solution with nonsmooth front (interaction of shock waves) for quasilinear equations in the one-dimensional case.
Elliptic equation $(y')^2=a_0+a_2y^2+a_4y^4$ is the foundation of the elliptic function expansion method of finding exact solutions to nonlinear differential equation. In some references, some new form solutions to the elliptic equation…
We study the behavior of solitary-wave solutions of some generalized nonlinear Schr\"odinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing…
The generalized Bretherton equation is studied. The classification of the meromorphic traveling wave solutions for this equation is presented. All possible exact solutions of the generalized Brethenton equation are given.
For both cubic and quintic nonlinearities of the one-dimensional complex Ginzburg-Landau evolution equation, we prove by a theorem of Eremenko the finiteness of the number of traveling waves whose squared modulus has only poles in the…
Motivated by the study of matter waves in Bose-Einstein condensates and coupled nonlinear optical systems, we study a system of two coupled nonlinear Schrodinger equations with inhomogeneous parameters, including a linear coupling. For that…
An exact Craik-Criminale (CC) solution to the incompressible Navier-Stokes (NS) equations describes the instability of an elliptical columnar flow interacting with a single Kelvin wave. These CC solutions are extended to allow…
The long-time asymptotics is analyzed for finite energy solutions of the 1D Schr\"odinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to…
In this study we establish the existence and uniqueness of the solution of a coupled system of general elliptic equations with anisotropic diffusion , non-uniform advection and variably influencing reaction terms on Lipschitz continuous…
In this paper, the linear spectral problem, which associated with the (n+1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation, with the Jacobi elliptic function as the external potential is investigated based on the Lam\'{e}…
We present a complete study of the ergodic theorem for the difficult problem of non-linear wave propagations through cylindrical measures /path integrals and the famous Ruelle-Amrein-Geogerscu-Enss (R.A.G.E.) theorem on the caracterization…
We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order.…
In this paper, we study the nonlinear dispersive waves including the rarefaction and dispersive shock waves in the discrete modified KdV equation through the numerical simulations of the dispersive Riemann problems. In particular, we…
The nonlinear fractional Boussinesq equations are known as the fractional differential equation class that has an important place in mathematical physics. In this study, a method called (G'G^2)-extension method which works well and reveals…