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1. General rogue waves in the three-wave resonant interaction systems

   https://doi.org/10.1093/imamat/hxab005  

   Yang, Bo; Yang, Jianke (March 2021, IMA Journal of Applied Mathematics)

   null (Ed.)

   Abstract General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction, respectively. It is shown that while the first family of solutions associated with a simple root exists for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, the ones generated by a $$2\times 2$$ block determinant in the double-root case, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained. 

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2. Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

   https://doi.org/10.1090/memo/1505  

   Bilman, Deniz; Miller, Peter (August 2024, Memoirs of the American Mathematical Society)

   We study fundamental rogue-wave solutions of the focusing nonlinear Schr\"odinger equation in the limit that the order of the rogue wave is large and the independent variables $(x,t)$ are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as $$x\to\pm\infty$$. The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions $$\mathcal{C}$$ (the ``channels''), $$\mathcal{S}$$ (the ``shelves''), and $$\mathcal{E}$$(the ``exterior domain'') contains the rescaled variables. On the region \mathcal{C}, the amplitude is small and the solution is highly oscillatory, while on the region \mathcal{S}, the solution is approximated by a modulated plane wave with a highly oscillatory correction term. The asymptotic behavior on these two domains is the same for all continuous orders. Assuming that the order belongs to the discrete sequence characteristic of rogue-wave solutions, the asymptotic behavior of the solution on the region $$\exterior$$ resembles that on \mathcal{S} but without the oscillatory correction term. Solutions of other continuous orders behave quite differently on $$\mathcal{E}$$. 

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3. Rogue wave patterns associated with Okamoto polynomial hierarchies

   https://doi.org/10.1111/sapm.12573  

   Yang, Bo; Yang, Jianke (March 2023, Studies in Applied Mathematics)

   Abstract We show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the τ functions of rogue wave solutions are determinants of Schur polynomials with index jumps of three, and an internal free parameter in these rogue waves gets large. We demonstrate these new rogue patterns in the Manakov system and the three‐wave resonant interaction system. For each system, we derive asymptotic predictions of its rogue patterns under a large internal parameter through Okamoto polynomial hierarchies. Unlike the previously reported rogue patterns associated with the Yablonskii–Vorob'ev hierarchy, a new feature in the present rogue patterns is that the mapping from the root structure of Okamoto‐hierarchy polynomials to the shape of the rogue pattern is linear only to the leading order, but becomes nonlinear to the next order. As a consequence, the current rogue patterns are often deformed, sometimes strongly deformed, from Okamoto‐hierarchy root structures, unless the underlying internal parameter is very large. Our analytical predictions of rogue patterns are compared to true solutions, and excellent agreement is observed, even when rogue patterns are strongly deformed from Okamoto‐hierarchy root structures. 

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4. A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions

   https://doi.org/10.1098/rspa.2020.0853  

   Feng, Bao-Feng; Ling, Liming; Zhu, Zuonong (March 2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Ling et al . 2016 Physica D 327 , 13–29 ( doi:10.1016/j.physd.2016.03.012 ); Feng et al . 2016 Phys. Rev. E 93 , 052227 ( doi:10.1103/PhysRevE.93.052227 )). 

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5. Multi‐breather and high‐order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

   https://doi.org/10.1111/sapm.12287  

   Feng, Bao‐Feng; Ling, Liming; Takahashi, Daisuke A. (October 2019, Studies in Applied Mathematics)

   Abstract We construct the multi‐breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and nontrivial phase‐modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multibreather solutions with different velocities in the limit, where the solution in the neighborhood of each breather tends to the simple one‐breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue wave and higher order rogue wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena, depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi‐ and higher order rogue wave solution. 

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