Riemann–Hilbert approach to the modified nonlinear Schrödinger equation with non-vanishing asymptotic boundary conditions

Yang, Yiling; Fan, Engui

doi:10.1016/j.physd.2020.132811

Cited by 33 publications

(13 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we briefly describe the direct scattering problem of the modified nonlinear Schrödinger equation (1.3) with finite density type initial data. These results are known, and interested readers can refer to the reference [21], which has a detailed proof process.…”

Section: Jost Functionsmentioning

confidence: 99%

“…Recently, Fan's team studied the soliton solutions of equation (1.3) under non-zero boundary conditions (NZBCs) by using the Riemann-Hilbert (RH) problem [21], and studied the long-time asymptotic behavior of the equation under the condition that the initial value belongs to Schwarz space [22] by using the nonlinear steepest descent method proposed by Defit and Zhou (DZ) in [23]. Our purpose in the present work is to study the asymptotic stability of N-soliton solution of the mNLS equation (1.3) under the condition of NZBCs by using the classical DZ method and the Dbar method, i.e., under the NZBCs q(x, t) = q ± e −4iα 2 t+2iαx , x → ±∞, (1.4) where q ± are independent of variables x and t and |q ± | = 1, we will verify that the soliton resolution conjecture holds for the initial value q(x, 0) = q 0 satisfying the conditions q 0 − q ± ∈ H 1,1 (R).…”

mentioning

confidence: 99%

“…Since the RH method is used to obtain the soliton solution of the mNLS equation (1.3) under the NZBCs (1.4) shown in [21], and the classical DZ method is also used to obtain the long-time asymptotic behavior of the initial value belonging to the Schwarz space [22]. Hereafter the main purpose of this work is to study the long-time asymptotic behavior of the solution under the NZBCs (1.4), and analyze the asymptotic stability of the soliton solution by Dbar steepest descent method.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schrödinger equation

Yang,

Tian,

2021

Preprint

View full text Add to dashboard Cite

The Cauchy problem of the modified nonlinear Schrödinger (mNLS) equation with the finite density type initial data is investigated via ∂ steepest descent method. In the soliton region of space-time x/t ∈ (5, 7), the long-time asymptotic behavior of the mNLS equation is derived for large times. Furthermore, for general initial data in a non-vanishing background, the soliton resolution conjecture for the mNLS equation is verified, which means that the asymptotic expansion of the solution can be characterized by finite number of soliton solutions as the time t tends to infinity, and a residual error O(t −3/4 ) is provided.

show abstract

Section: Jost Functionsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schrödinger equation

Yang,

Tian,

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The RH method is a very effective and convenient way to solve nonlinear integrable models, such as Korteweg–de Vries equation, derivative Schrödinger equation, Sasa–Satsuma equation, and so on 41–47 . Recently, the RH method cannot only solve the integrable equations with nonzero boundary conditions, 47,48 but also be used to solve the rogue waves 49,50 . For the focusing nCNLS equation, some scholars have obtained its multisoliton solutions by RH method, 51 and found that the equation has the novel explicit rogue waves by using the modified Darboux transform 52 .…”

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert problem and interactions of solitons in the ‐component nonlinear Schrödinger equations

Tian

Yang

2021

Stud Appl Math

View full text Add to dashboard Cite

In this work, we systematically investigate the general n$n$‐component nonlinear Schrödinger equations by using the Riemann–Hilbert method. Starting from the Lax pair associated with an false(n+1false)×false(n+1false)$(n + 1)\times (n + 1)$ matrix spectral problem, we construct a Riemann–Hilbert problem and obtain the compact N$N$‐soliton solutions formula expressed by determinants. Taking two‐components nonlinear Schrödinger equations as an example, this work detailedly analyzes its multisoliton solutions by means of parameters modulation. Many interesting new phenomena are displayed including elastic collision, soliton reflection, parallel propagation, time‐periodic propagation and (space, time)‐periodic propagation. In addition, the interactions between solitons from weak to strong are presented. Finally, some conjectures about the dynamical behaviors of N$N$‐soliton solutions are proposed. We hope that the present analysis would be useful for a better understanding of soliton interactions in related fields such as nonlinear optics, plasma physics, oceanography, and Bose–Einstein condensates.

show abstract

“…Based on the importance of nonlinear integrable evolution equations, these are the focus of scholars' research from the beginning to the end. In order to solve these equations, many novel and effective methods have been produced, such as Hirota bilinear method [16], Darboux and Bäcklund transformation [26,25] and inverse scattering transform(IST) [9,2,11,27,14,23,32,13,35,38,39,41,34].…”

mentioning

confidence: 99%

Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

Xun¹,

Tian²,

Zhang³

2022

DCDS-B

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. We successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering problem is analyzed, and the potential function is recovered. By introducing the special conditions of reflectionless case, the time-periodic soliton solutions formula of the equation is derived successfully. Take <inline-formula><tex-math id="M1">\begin{document}$ J = \overline{J} = 1,2,3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.</p>

show abstract

Riemann–Hilbert approach to the modified nonlinear Schrödinger equation with non-vanishing asymptotic boundary conditions

Cited by 33 publications

References 47 publications

On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schrödinger equation

On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schrödinger equation

Riemann–Hilbert problem and interactions of solitons in the ‐component nonlinear Schrödinger equations

Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

Contact Info

Product

Resources

About