Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 3233-3299 |
| Number of pages | 67 |
| Journal | Communications on Pure and Applied Mathematics |
| Volume | 76 |
| Issue number | 11 |
| Early online date | 23 Jun 2023 |
| DOIs | |
| Publication status | Published - 1 Nov 2023 |
Bibliographical note
Funding Information:This manuscript was partially developed upon work supported by the National Science Foundation under Grant No. DMS-1928930 while T.G., M.G. and K.M. participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall '21 semester “Universality and Integrability in Random Matrix Theory and Interacting Particle Systems”. T.G., M.G., R.J., K.M. would also like to thank the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, for support (EPSRC grant No. EP/R014604/1) and hospitality during the programme “Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves” (HYD2) in the Summer and Fall '22 semesters, where part of the work on this paper was undertaken. T.G. acknowledges the funding from the European Union's H2020 research and innovation programme under the Marie Skłodowska–Curie grant No. 778010 IPaDEGAN, the research project “Mathematical Methods in Non Linear Physics” (MMNLP), Gruppo 4-Fisica Teorica of INFN and the support of GNFM-INDAM group. K.M. was supported in part by the National Science Foundation under grant DMS-1733967. M.G. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) grant No. RGPIN-2022-04106 and the partial support of the Simons Foundation Fellowship while visiting the Isaac Newton Institute. R.J. acknowledges the support of the Simons Foundation under grant 853620. The authors thank Iryna Egorova for interesting and useful discussions regarding decay rate of the potentials to the elliptic background for, and Percy Deift for many insightful suggestions.
Funding Information:
K.M. was supported in part by the National Science Foundation under grant DMS‐1733967. M.G. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) grant No. RGPIN‐2022‐04106 and the partial support of the Simons Foundation Fellowship while visiting the Isaac Newton Institute. R.J. acknowledges the support of the Simons Foundation under grant 853620.
Funding Information:
This manuscript was partially developed upon work supported by the National Science Foundation under Grant No. DMS‐1928930 while T.G., M.G. and K.M. participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall '21 semester “Universality and Integrability in Random Matrix Theory and Interacting Particle Systems”. T.G., M.G., R.J., K.M. would also like to thank the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, for support (EPSRC grant No. EP/R014604/1) and hospitality during the programme “Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves” (HYD2) in the Summer and Fall '22 semesters, where part of the work on this paper was undertaken.
Funding Information:
T.G. acknowledges the funding from the European Union's H2020 research and innovation programme under the Marie Skłodowska–Curie grant No. 778010 IPaDEGAN, the research project “Mathematical Methods in Non Linear Physics” (MMNLP), Gruppo 4‐Fisica Teorica of INFN and the support of GNFM‐INDAM group.
Publisher Copyright:
© 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
Keywords
- Soliton gas, generalised hydrodynamic