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 so...
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| Published in | Communications on pure and applied mathematics Vol. 76; no. 11; pp. 3233 - 3299 |
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| Main Authors | , , , , |
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01.11.2023
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| 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. |
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| AbstractList | 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. 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. |
| Author | McLaughlin, Ken T‐R Girotti, Manuela Jenkins, Robert Grava, Tamara Minakov, Alexander |
| Author_xml | – sequence: 1 givenname: Manuela surname: Girotti fullname: Girotti, Manuela organization: Saint Mary's University – sequence: 2 givenname: Tamara surname: Grava fullname: Grava, Tamara organization: SISSA INFN sezione di Trieste and University of Bristol – sequence: 3 givenname: Robert surname: Jenkins fullname: Jenkins, Robert organization: University of Central Florida – sequence: 4 givenname: Ken T‐R surname: McLaughlin fullname: McLaughlin, Ken T‐R organization: Tulane University – sequence: 5 givenname: Alexander surname: Minakov fullname: Minakov, Alexander organization: Univerzita Karlova |
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| Snippet | 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... |
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| Title | Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation |
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