Exact solution of the van der Waals model in the critical region

The celebrated van der Waals model describes simple fluids in the thermodynamic limit and predicts the existence of a critical point associated to the gas-liquid phase transition. However the behaviour of critical isotherms according to the equation of state, where a gasliquid phase transition occurs, significantly departs from experimental observations. The correct critical isotherms are heuristically re-established via the Maxwell equal areas rule. A long standing open problem in mean field theory is concerned with the analytic description of van der Waals isotherms for a finite size system that is consistent, in the thermodynamic limit, with the Maxwell prescription. Inspired by the theory of nonlinear conservation laws, we propose a novel mean field approach, based on statistical mechanics, that allows to calculate the van der Waals partition function for a system of large but finite number of particles N. Our partition function naturally extends to the whole space of thermodynamic variables, reproduces, in the thermodynamic limit N ! 1, the classical results outside the critical region and automatically encodes Maxwell’s prescription. We show that isothermal curves evolve in the space of thermodynamic variables like nonlinear breaking waves and the criticality is explained as the mechanism of formation of a classical hydrodynamic shock

Paraxial light in a Cole-Cole nonlocal medium: integrable regimes and singularities

Nonlocal nonlinear Schroedinger-type equation is derived as a model to describe paraxial light propagation in nonlinear media with different `degrees' of nonlocality. High frequency limit of this equation is studied under specific assumptions of Cole-Cole dispersion law and a slow dependence along propagating direction. Phase equations are integrable and they correspond to dispersionless limit of Veselov-Novikov hierarchy. Analysis of compatibility among intensity law (dependence of intensity on the refractive index) and high frequency limit of Poynting vector conservation law reveals the existence of singular wavefronts. It is shown that beams features depend critically on the orientation properties of quasiconformal mappings of the plane. Another class of wavefronts, whatever is intensity law, is provided by harmonic minimal surfaces. Illustrative example is given by helicoid surface. Compatibility with first and third degree nonlocal perturbations and explicit solutions are also discussed.Comment: 12 pages, 2 figures; eq. (36) corrected, minor change

Mechanism of wave breaking from a vacuum point in the defocusing nonlinear Schrödinger equation

We study the wave breaking mechanism for the weakly dispersive defocusing nonlinear Schrödinger equation with a constant phase dark initial datum that contains a vacuum point at the origin. We prove by means of the exact solution to the initial value problem that, in the dispersionless limit, the vacuum point is preserved by the dynamics until breaking occurs at a finite critical time. In particular, both Riemann invariants experience a simultaneous breaking at the origin. Although the initial vacuum point is no longer preserved in the presence of a finite dispersion, the critical behavior manifests itself through an abrupt transition occurring around the breaking time

High frequency integrable regimes in nonlocal nonlinear optics

We consider an integrable model which describes light beams propagating in nonlocal nonlinear media of Cole-Cole type. The model is derived as high frequency limit of both Maxwell equations and the nonlocal nonlinear Schroedinger equation. We demonstrate that for a general form of nonlinearity there exist selfguided light beams. In high frequency limit nonlocal perturbations can be seen as a class of phase deformation along one direction. We study in detail nonlocal perturbations described by the dispersionless Veselov-Novikov (dVN) hierarchy. The dVN hierarchy is analyzed by the reduction method based on symmetry constraints and by the quasiclassical Dbar-dressing method. Quasiclassical Dbar-dressing method reveals a connection between nonlocal nonlinear geometric optics and the theory of quasiconformal mappings of the plane.Comment: 45 pages, 4 figure

Exact solution of the van der Waals model in the critical region

The celebrated van der Waals model describes simple fluids in the thermodynamic limit and predicts the existence of a critical point associated to the gas-liquid phase transition. However the behaviour of critical isotherms according to the equation of state, where a gas-liquid phase transition occurs, significantly departs from experimental observations. The correct critical isotherms are heuristically re-established via the Maxwell equal areas rule. A long standing open problem in mean field theory is concerned with the analytic description of van der Waals isotherms for a finite size system that is consistent, in the thermodynamic limit, with the Maxwell prescription. Inspired by the theory of nonlinear conservation laws, we propose a novel mean field approach, based on statistical mechanics, that allows to calculate the van der Waals partition function for a system of large but finite number of particles $N$. Our partition function naturally extends to the whole space of thermodynamic variables, reproduces, in the thermodynamic limit $N\to \infty$, the classical results outside the critical region and automatically encodes Maxwell's prescription. We show that isothermal curves evolve in the space of thermodynamic variables like nonlinear breaking waves and the criticality is explained as the mechanism of formation of a classical hydrodynamic shock

On the models of nonlocal nonlinear optics

We show that under certain assumptions a general model of nonlocal nonlinear response in 1+1-dimension is equivalent to the model considered by Krolikowski and Bang for a Kerr-type medium. We derive the limit of weak nonlocality in high frequency regime and discuss the integrable cases.Comment: 6 page

Comprehensive analysis of large α\alpha yields observed in 6^{6}Li induced reactions

Background: Large $\alpha$ yields have been reported over the years in reactions with $^{6}$Li and $^{7}$Li projectiles. Previous theoretical analyses have shown that the elastic breakup (EBU) mechanism (i.e., projectile breakup leaving the target in its ground state) is able to account only for a small fraction of the total $\alpha$ inclusive breakup cross sections, pointing toward the dominance of non-elastic breakup (NEB) mechanisms. Purpose: We aim to provide a systematic study of the $\alpha$ inclusive cross sections observed in nuclear reactions induced by $^{6}$Li projectiles. In addition to estimating the total $\alpha$ singles cross sections, it is our goal to evaluate angular and energy distributions of these $\alpha$ particles and compare with experimental data, when available. Method: We compute separately the EBU and NEB components of the inclusive breakup cross sections. For the former, we use the continuum-discretized coupled-channels (CDCC) method, which treats this mechanism to all orders. For the NEB part, we employ the the model proposed in the eighties by Ichimura, Austern and Vincent [Phys. Rev. C32, 432 (1982)], within the DWBA approximation. Results: Overall, the sum of the computed EBU and NEB cross sections is found to reproduce very well the measured singles cross sections. In all cases analyzed, we find that the inclusive breakup cross section is largely dominated by the NEB component. Conclusions: The presented method provides a global and systematic description of inclusive breakup reactions induced by $^{6}$Li projectiles. It provides also a natural explanation of the previously observed underestimation of the measured $\alpha$ yields by CDCC calculations. The method used here can be extended to other weakly-bound projectiles, including halo nuclei.Comment: 11 pages, 16 figures, fig. 12 updated; some typos correcte

The puzzle of complete fusion suppression in weakly-bound nuclei: a Trojan Horse effect?

Experimental studies of nuclear collisions involving light weakly-bound nuclei show a systematic suppression of the complete fusion cross section by $\sim$30\% with respect to the expectation for tightly bound nuclei, at energies above the Coulomb barrier. Although it is widely accepted that the phenomenon is related to the weak binding of these nuclei, the origin of this suppression is not fully understood. In here, we present a novel approach that provides the complete fusion for weakly bound nuclei and relates its suppression to the competition between the different mechanisms contributing to the reaction cross section. The method is applied to the $^{6,7}$Li+$^{209}$Bi reactions, where we find that the suppression of complete fusion is mostly caused by the flux associated with non-elastic breakup modes, such as the partial capture of the projectile (incomplete fusion), whereas the elastic breakup mode is found to play a minor role. Finally, we demonstrate that the large $\alpha$ yields observed in these reactions can be naturally explained as a consequence of a {\it Trojan Horse} mechanism.Comment: 6 pages, 4 figures. Accepted for publication in Phys. Rev. Letter

On integrable conservation laws

We study normal forms of scalar integrable dispersive (non necessarily Hamiltonian) conservation laws via the Dubrovin-Zhang perturbative scheme. Our computations support the conjecture that such normal forms are parametrised by infinitely many arbitrary functions that can be identified with the coefficients of the quasilinear part of the equation. More in general, we conjecture that two scalar integrable evolutionary PDEs having the same quasilinear part are Miura equivalent. This conjecture is also consistent with the tensorial behaviour of these coefficients under general Miura transformations.Comment: 17 page

Puzzle of Complete Fusion Suppression in Weakly Bound Nuclei: A Trojan Horse Effect?

Experimental studies of nuclear collisions involving light weakly bound nuclei show a systematic suppression of the complete fusion cross section by ∼30% with respect to the expectation for tightly bound nuclei, at energies above the Coulomb barrier. Although it is widely accepted that the phenomenon is related to the weak binding of these nuclei, the origin of this suppression is not fully understood. Here, we present a novel approach that provides the complete fusion for weakly bound nuclei and relates its suppression to the competition between the different mechanisms contributing to the reaction cross section. The method is applied to the 6,7Li+209Bi reactions, where we find that the suppression of complete fusion is mostly caused by the flux associated with nonelastic breakup modes, such as the partial capture of the projectile (incomplete fusion), whereas the elastic breakup mode is found to play a minor role. Finally, we demonstrate that the large α yields observed in these reactions can be naturally explained as a consequence of a Trojan Horse mechanism.National Science Foundation under Contract No. NSF-PHY-1520972 with Ohio UniversityMinisterio de Ciencia,Innovación y Universidades (Projects No. FIS2014-53448-C2-1-P and No. FIS2017-88410-P)European Union’s Horizon 2020 (Grant Agreement No. 654002