More Like this

1. Riemann-Hilbert problems and N-soliton solutions for a coupled mKdV system

   Ma, Wen-Xiu (October 2018, Journal of geometry and physics)

   A 3×3 matrix spectral problem is introduced and its associated AKNS integrable hierarchy with four components is generated. From this spectral problem, a kind of Riemann–Hilbert problems is formulated for a system of coupled mKdV equations in the resulting AKNS integrable hierarchy. N-soliton solutions to the coupled mKdV system are presented through a specific Riemann–Hilbert problem with an identity jump matrix. 

   more » « less
2. Generalized matrix exponential solutions to the AKNS hierarchy

   Zhang, Jian-bing; Gu, Canyuan; Ma, Wen-Xiu (February 2018, Advances in mathematical physics)

   Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure. 

   more » « less
3. Reverse Space‐Time Nonlocal Sine‐Gordon/Sinh‐Gordon Equations with Nonzero Boundary Conditions

   https://doi.org/10.1111/sapm.12222  

   Ablowitz, Mark J.; Feng, Bao‐Feng; Luo, Xu‐Dan; Musslimani, Ziad H. (July 2018, Studies in Applied Mathematics)

   Abstract Nonlocal reverse space‐time Sine/Sinh‐Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well‐known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1‐soliton and 2‐soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered. 

   more » « less
4. On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems*

   https://doi.org/10.1088/1361-6544/ac20a5  

   Kuijlaars, Arno; Tovbis, Alexander (September 2021, Nonlinearity)

   Abstract We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states u ( z ) in the spectral theory of soliton gases for the one dimensional integrable focusing nonlinear Schrödinger equation (fNLS) and for the Korteweg–de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimising (positive) measure for a certain energy functional is absolutely continuous and its density u ( z ) ⩾ 0 solves the required integral equation. In a similar fashion we show that v ( z ), the temporal analog of u ( z ), is the difference of densities of two absolutely continuous measures. Together, the integral equations for u , v represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis (2020 Phys. Rev. E 101 052207). It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems. 

   more » « less
5. Integrable nonlocal derivative nonlinear Schrödinger equations

   https://doi.org/10.1088/1361-6420/ac5f75  

   Ablowitz, Mark J; Luo, Xu-Dan; Musslimani, Ziad H; Zhu, Yi (April 2022, Inverse Problems)

   Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation. 

   more » « less