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### Open Access Journals

DCDS

We show that the Camassa--Holm equation
$u_t-$u

_{xxt}+3uu_{x}-$2u_xu_{xx}-uu_{xxx}=0 possesses a global continuous semigroup of weak dissipative solutions for initial data $u|_{t=0}$ in $H^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. Stability in terms of $H^1$ and $L^\infty$ norm is discussed.
DCDS

The Camassa--Holm equation
$u_t$$-$u

_{xxt}+3u$u_x-2u_x$u_{xx}-uu_{xxx}=0 enjoys special solutions of the form $u(x,t)=$Σ_{i=1}^{n}$p_i(t)e^{-|x-q_i(t)|}$, denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data $u|_{t=0}=u_0$ in $H^1$(R) such that u-u_{xx}is a positive Radon measure, one can construct a sequence of multipeakons that converges in L_{loc}^{∞}(R, H_{loc}^{1}(R)) to the unique global solution of the Camassa--Holm equation. The approach also provides a convergent, energy preserving nondissipative numerical method which is illustrated on several examples.
DCDS

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy
with complex-valued initial data and prove unique solvability globally in
time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse
algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.

The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.

The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.

DCDS

We study a non-relativistic charged quantum particle moving
in a bounded open set $\Omega\subset\R^3$ with smooth boundary under the action of
a zero-range potential. In the electrostatic case the standing wave solutions take the form
$\psi(t,x)=u(x)e^{-i\omega t}$ where $u$ formally satisfies
$-\Delta u+\alpha\varphi u-\beta\delta_{x_0} u=\omega u$ and the electric potential $\varphi$ is given by
$-\Delta\varphi = u^2$. We introduce the definition of ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.

DCDS

We offer a simple and self-contained proof that the Follow-the-Leader model converges to the Lighthill-Whitham-Richards model for traffic flow.

DCDS

We show existence of global conservative solutions of the
Cauchy problem for the Camassa--Holm equation
$u_t-u_{txx}+\kappa
u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with
nonvanishing and distinct spatial asymptotics.

DCDS

We show existence of a unique, regular global solution of the parabolic-elliptic system
$u_t +f(t,x,u)_x+g(t,x,u)+P_x=(a(t,x) u_x)_x$ and
$-P_{x x}+P=h(t,x,u,u_x)+k(t,x,u)$ with initial data
$u|_{t=0} = u_0$. Here inf$_(t,x) a(t,x)>0$. Furthermore, we show that the solution is
stable with respect to variation in the initial data $u_0$ and the functions $f$, $g$ etc.
Explicit stability estimates are provided.
The regularized generalized Camassa--Holm equation is a special case of the model we discuss.

DCDS

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 for two solutions $u$ and $v$ of
the equation we have
$d_D(u(t),v(t))\le e^{Ct}
d_D(u_0,v_0)$. The relationship between this metric and the usual norms in $H^1$ and $L^\infty$ is clarified.
The method extends to the generalized hyperelastic-rod equation
$u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f''(u)(u_x)^2)_x=0$ (for $f$ without inflection points).

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