Abstract We obtain Fredholm type formulas for partial degenerations of theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of ‘many-soliton’ solutions on an elliptic (cnoidal) background standing wave for the Korteweg–de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay–Moses formula for N -solitons. In particular, we represent such a solution as a sum of the following two terms: a ‘shifted’ elliptic (cnoidal) background wave and a Kay–Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus N + 1 finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as N bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.
more »
« less
Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation
We analyse the case of a dense modified Korteweg–de Vries (mKdV) soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann–Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally, we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.
more » « less- NSF-PAR ID:
- 10442003
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- ISSN:
- 0010-3640
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
Abstract The mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction.
-
more » « less
ABSTRACT A narrow linear object extending ∼60 kpc from the centre of a galaxy at redshift z ∼ 1 has recently been discovered and interpreted as shocked gas filament forming stars. The host galaxy presents an irregular morphology, implying recent merger events. Supposing that each of the progenitor galaxies has a central supermassive black hole (SMBH) and the SMBHs are accumulated at the centre of the merger remnant, a fraction of them can be ejected from the galaxy with a high velocity due to interactions between SMBHs. When such a runaway SMBH (RSMBH) passes through the circumgalactic medium (CGM), converging flows are induced along the RSMBH path, and star formation could eventually be ignited. We show that the CGM temperature prior to the RSMBH perturbation should be below the peak temperature in the cooling function to trigger filament formation. While the gas is temporarily heated due to compression, the cooling efficiency increases, and gas accumulation becomes allowed along the path. When the CGM density is sufficiently high, the gas can cool down and develop a dense filament by z = 1. The mass and velocity of the RSMBH determine the scale of filament formation. Hydrodynamical simulations validate the analytical expectations. Therefore, we conclude that the perturbation by RSMBHs is a viable channel to form the observed linear object. Using the analytical model validated by simulations, we show that the CGM around the linear object to be warm ($T \lesssim 2 \times 10^5$ K) and dense ($n \gtrsim 2 \times 10^{-5} (T/2 \times 10^5 \, K)^{-1} \, {\rm cm^{-3}}$).
-
more » « less
The aqueous proton is a common and long-studied species in chemistry, yet there is currently intense interest devoted to understanding its hydration structure and transport dynamics. Typically described in terms of two limiting structures observed in gas-phase clusters, the Zundel H5O2+and Eigen H9O4+ions, the aqueous structure is less clear due to the heterogeneity of hydrogen bonding environments and room-temperature structural fluctuations in water. The linear infrared (IR) spectrum, which reports on structural configurations, is challenging to interpret because it appears as a continuum of absorption, and the underlying vibrational modes are strongly anharmonically coupled to each other. Recent two-dimensional IR (2D IR) experiments presented strong evidence for asymmetric Zundel-like motifs in solution, but true structure–spectrum correlations are missing and complicated by the anharmonicity of the system. In this study, we employ high-level vibrational self-consistent field/virtual state configuration interaction calculations to demonstrate that the 2D IR spectrum reports on a broad distribution of geometric configurations of the aqueous proton. We find that the diagonal 2D IR spectrum around 1200 cm−1is dominated by the proton stretch vibrations of Zundel-like and intermediate geometries, broadened by the heterogeneity of aqueous configurations. There is a wide distribution of multidimensional potential shapes for the proton stretching vibration with varying degrees of potential asymmetry and confinement. Finally, we find specific cross peak patterns due to aqueous Zundel-like species. These studies provide clarity on highly debated spectral assignments and stringent spectroscopic benchmarks for future simulations.
-
Context. We started a multi-scale analysis of star formation in G202.3+2.5, an intertwined filamentary sub-region of the Monoceros OB1 molecular complex, in order to provide observational constraints on current theories and models that attempt to explain star formation globally. In the first paper (Paper I), we examined the distributions of dense cores and protostars and found enhanced star formation activity in the junction region of the filaments. Aims. In this second paper, we aim to unveil the connections between the core and filament evolutions, and between the filament dynamics and the global evolution of the cloud. Methods. We characterise the gas dynamics and energy balance in different parts of G202.3+2.5 using infrared observations from the Herschel and WISE telescopes and molecular tracers observed with the IRAM 30-m and TRAO 14-m telescopes. The velocity field of the cloud is examined and velocity-coherent structures are identified, characterised, and put in perspective with the cloud environment. Results. Two main velocity components are revealed, well separated in radial velocities in the north and merged around the location of intense N 2 H + emission in the centre of G202.3+2.5 where Paper I found the peak of star formation activity. We show that the relative position of the two components along the sightline, and the velocity gradient of the N 2 H + emission imply that the components have been undergoing collision for ~10 5 yr, although it remains unclear whether the gas moves mainly along or across the filament axes. The dense gas where N 2 H + is detected is interpreted as the compressed region between the two filaments, which corresponds to a high mass inflow rate of ~1 × 10 −3 M ⊙ yr −1 and possibly leads to a significant increase in its star formation efficiency. We identify a protostellar source in the junction region that possibly powers two crossed intermittent outflows. We show that the H II region around the nearby cluster NCG 2264 is still expanding and its role in the collision is examined. However, we cannot rule out the idea that the collision arises mostly from the global collapse of the cloud. Conclusions. The (sub-)filament-scale observables examined in this paper reveal a collision between G202.3+2.5 sub-structures and its probable role in feeding the cores in the junction region. To shed more light on this link between core and filament evolutions, one must characterise the cloud morphology, its fragmentation, and magnetic field, all at high resolution. We consider the role of the environment in this paper, but a larger-scale study of this region is now necessary to investigate the scenario of a global cloud collapse.more » « less
