Related papers: Cauchy problem and multi-soliton solutions for a t…
The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a…
In the present paper, we mainly study the integrable semi-discretization of a multi-component short pulse equation. Firstly, we briefly review the bilinear equations for a multi-component short pulse equation proposed by Matsuno (J. Math.…
We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to…
The failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system. In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations which is a…
Explicit multi-soliton and multi-peakon solutions of the Dullin-Gottwald-Holm equation are constructed via Darboux transformation and direct computation, respectively. To this end we first map the Dullin-Gottwald-Holm equation to a negative…
An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder…
In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using the method of approximation of smooth solutions, a regularization technique and the special structure of the system, we prove the…
We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of…
The matrix 2x2 spectral differential equation of the second order is considered on x in ($-\infty,+\infty$). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second…
The global existence of the solution for the second-type derivative nonlinear Schr\"odinger (DNLSII) equation with solitons is presented for the first time on the line with weighted Sobolev initial data in $H^2( \mathbb{R}) \cap…
The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$ of…
Nonlinear Schr\"odinger equation, short pulse equation and complex short pulse equation have important application in nonlinear optics. They can be derived from the Maxwell equation. In this paper, we investigate a coupled…
This paper studies the properties of solutions for a double nonlinear time-dependent parabolic equation with variable density, not in divergence form with a source or absorption. The problem is formulated as a partial differential equation…
A system of two operator equations is considered - one of pseudomonotone type and the other of strongly monotone type - both being strongly coupled. Conditions are given that allow to reduce the solvability of this system to a single…
A direct method is developed for constructing the bright $N$-soliton solution of a multi-component modified nonlinear Schr\"odinger equation. Specifically, the two different expressions of the solution are obtained both of which are…
Solutions for all Adler-Bobenko-Suris equations excluding Q4 and several lattice Boussinesq-type equations are reconsidered by employing the Cauchy matrix approach. Through introducing a ``fake'' nonautonomous plane wave factor, we derive…
Analytic smooth solutions of a general, strongly parabolic semi-linear Cauchy problem of $2m$-th order in $\mathbb{R}^N\times (0,T)$ with analytic coefficients (in space and time variables) and analytic initial data (in space variables) are…
We consider a hyperbolic system of three conservation laws in one space variable. The system is a model for fluid flow allowing phase transitions; in this case the state variables are the specific volume, the velocity and the mass density…
This paper establishes the global existence of smooth solutions to the 2D isentropic and irrotational Euler equations for Chaplygin gases with a general class of short pulse initial data, which, in particular, resolves in this special case,…
In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial…