Related papers: Lipschitz metric for the two-component Camassa--Ho…
We study stability of conservative solutions of the Cauchy problem for the periodic Camassa-Holm equation $u_t-u_{xxt}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$…
We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that…
We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa--Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of…
We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa-Holm system, $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}+\eta\rho\rho_x=0$ and $\rho_t+(u\rho)_x=0$, for initial data…
This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa-Holm system on the real line. We obtain the global weak solutions via a coordinate…
We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation $u_t+uu_x=\frac14(\int_{-\infty}^xu_x^2 dx-\int_{x}^\infty u_x^2 dx)$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with…
We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter-Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving…
We establish the existence of conservative solutions of the initial value problem of the two-component Hunter--Saxton system on the line. Furthermore we investigate the stability of these solutions by constructing a Lipschitz metric.
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…
In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the two-component modified Camassa-Holm (2-mCH) equation based on its Lax pair. Further via a series of deformations to the RH problem by using the…
This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \begin{alignat*}{4} &m_t+\left((u^2-u_x^2)m\right)_x=0,&\quad&m:= u-u_{xx},&\quad&t>0,&\;&-\infty<x<+\infty,\\ &u(x,0)=u_0(x),&&&&&&-\infty<x<+\infty,…
This paper analyzes the Lipschitz behavior of the feasible set in two parametric settings, associated with linear and convex systems in R^n. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified…
We show existence of a global weak dissipative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line with nonvanishing and distinct spatial asymptotics. The influence from the second component in the 2CH…
This paper is devoted to the Cauchy problem for the modified multi-component Camassa-Holm system in higher dimensions. On the one hand, we establish an almost complete local well-posedness results for the system in the framework of Besov…
We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial data \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber \\…
The paper aims at developing the Riemann-Hilbert (RH) approach for the modified Camassa-Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at…
We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions…
This paper establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient and the initial displacement in a damped biharmonic wave equation. The data consist of the boundary Cauchy data for the Laplacian of…
In this article we prove the existence and uniqueness of a (weak) solution $u$ in $L_p\left((0,T) , \Lambda_{\gamma+m}\right)$ to the Cauchy problem \begin{align} \notag &\frac{\partial u}{\partial t}(t,x)=\psi(t,i\nabla)u(t,x)+f(t,x),\quad…
We present some new persistence results for the non-periodic two-component Camassa-Holm (2CH) system in weighted $L_p$ spaces. Working with moderate weight functions that are commonly used in time-frequency analysis, the paper generalizes…