A modified Korteweg-de Vries equation soliton gas under the nonzero background
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914861016940544 |
|---|---|
| author | Zhang, Xiaoen Ling, Liming |
| author_facet | Zhang, Xiaoen Ling, Liming |
| contents | In this paper, we consider a soliton gas of the focusing modified Korteweg-de Vries generated from the $N$-soliton solutions under the nonzero background. The spectral soliton density is chosen on the pure imaginary axis, excluding the branch cut $Σ_{c}=\left[-i, i\right]$. In the limit $N\to\infty$, we establish the Riemann-Hilbert problem of the soliton gas. Using the Deift-Zhou nonlinear steepest-descent method, this soliton gas under the nonzero background will decay to a constant background as $x\to+\infty$, while its asymptotics as $x\to-\infty$ can be expressed with a Riemann-Theta function, attached to a Riemann surface with genus-two. We also analyze the large $t$ asymptotics over the entire spatial domain, which is divided into three distinct asymptotic regions depending on the ratio $ξ=\frac{x}{t}$. Using the similar method, we provide the leading-order asymptotic behaviors for these three regions and exhibit the dynamics of large $t$ asymptotics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05384 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A modified Korteweg-de Vries equation soliton gas under the nonzero background Zhang, Xiaoen Ling, Liming Exactly Solvable and Integrable Systems 35Q55, 35Q51, 35Q15, 37K40, 37K15, 37K10 In this paper, we consider a soliton gas of the focusing modified Korteweg-de Vries generated from the $N$-soliton solutions under the nonzero background. The spectral soliton density is chosen on the pure imaginary axis, excluding the branch cut $Σ_{c}=\left[-i, i\right]$. In the limit $N\to\infty$, we establish the Riemann-Hilbert problem of the soliton gas. Using the Deift-Zhou nonlinear steepest-descent method, this soliton gas under the nonzero background will decay to a constant background as $x\to+\infty$, while its asymptotics as $x\to-\infty$ can be expressed with a Riemann-Theta function, attached to a Riemann surface with genus-two. We also analyze the large $t$ asymptotics over the entire spatial domain, which is divided into three distinct asymptotic regions depending on the ratio $ξ=\frac{x}{t}$. Using the similar method, we provide the leading-order asymptotic behaviors for these three regions and exhibit the dynamics of large $t$ asymptotics. |
| title | A modified Korteweg-de Vries equation soliton gas under the nonzero background |
| topic | Exactly Solvable and Integrable Systems 35Q55, 35Q51, 35Q15, 37K40, 37K15, 37K10 |
| url | https://arxiv.org/abs/2407.05384 |