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### Open Access Journals

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 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 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 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.
NHM

We show how to view the standard Follow-the-Leader (FtL) model as a numerical method to compute numerically the solution of the Lighthill-Whitham-Richards (LWR) model for traffic flow. As a result we offer a simple proof that FtL models converge to the LWR model for traffic flow when traffic becomes dense. The proof is based on techniques used in the analysis of numerical schemes for conservation laws, and the equivalence of weak entropy solutions of conservation laws in the Lagrangian and Eulerian formulation.

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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