DCDS
The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy
Fritz Gesztesy Helge Holden Johanna Michor Gerald Teschl
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 151-196 doi: 10.3934/dcds.2010.26.151
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.
keywords: initial value problem. complex-valued solutions Ablowitz-Ladik hierarchy
DCDS
The continuum limit of Follow-the-Leader models — a short proof
Helge Holden Nils Henrik Risebro
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 715-722 doi: 10.3934/dcds.2018031

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

keywords: Follow-the-Leader model Lighthill-Whitham-Richards model traffic flow continuum limit conservation laws
DCDS
Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics
Giuseppe Maria Coclite Helge Holden
Discrete & Continuous Dynamical Systems - A 2010, 27(1): 117-132 doi: 10.3934/dcds.2010.27.117
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 $\beta\to 0$.
keywords: point interaction. Schrödinger-Maxwell system
DCDS
Dissipative solutions for the Camassa-Holm equation
Helge Holden Xavier Raynaud
Discrete & Continuous Dynamical Systems - A 2009, 24(4): 1047-1112 doi: 10.3934/dcds.2009.24.1047
We show that the Camassa--Holm equation $u_t-$uxxt+3uux-$2u_xuxx-uuxxx=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.
keywords: dissipative solutions. Camassa--Holm equation
DCDS
A convergent numerical scheme for the Camassa--Holm equation based on multipeakons
Helge Holden Xavier Raynaud
Discrete & Continuous Dynamical Systems - A 2006, 14(3): 505-523 doi: 10.3934/dcds.2006.14.505
The Camassa--Holm equation $u_t$$-$uxxt+3u$u_x-2u_x$uxx-uuxxx=0 enjoys special solutions of the form $u(x,t)=$Σi=1n$p_i(t)e^{-|x-q_i(t)|}$, 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-uxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in Lloc(R, Hloc1(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.
keywords: numerical methods. peakons Camassa–Holm equation
NHM
Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow
Helge Holden Nils Henrik Risebro
Networks & Heterogeneous Media 2018, 13(3): 409-421 doi: 10.3934/nhm.2018018

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.

keywords: Follow-the-Leader model Lighthill-Whitham-Richards model traffic flow continuum limit
DCDS
Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics
Katrin Grunert Helge Holden Xavier Raynaud
Discrete & Continuous Dynamical Systems - A 2012, 32(12): 4209-4227 doi: 10.3934/dcds.2012.32.4209
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.
keywords: continuous semigroup. nonvanishing asymptotics conservative and global solutions Camassa--Holm equation
DCDS
Wellposedness for a parabolic-elliptic system
Giuseppe Maria Coclite Helge Holden Kenneth H. Karlsen
Discrete & Continuous Dynamical Systems - A 2005, 13(3): 659-682 doi: 10.3934/dcds.2005.13.659
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.
keywords: wellposedness Camassa-Holm equation. Parabolic-elliptic system
DCDS
Lipschitz metric for the Camassa--Holm equation on the line
Katrin Grunert Helge Holden Xavier Raynaud
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 2809-2827 doi: 10.3934/dcds.2013.33.2809
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).
keywords: conservative solutions. Camassa--Holm equation Lipschitz metric

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