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Register a new userSemi-rational solutions of the third-type Davey-Stewartson equation
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General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simplest) rogue waves are line rogue waves. It is also shown that the multi-rogue waves describe interactions of several fundamental rogue waves, which would generate interesting curvy wave patterns. The higher order rogue waves originate from a localized lump and retreat back to it. Several types of hybrid solutions composed of rogue waves, breathers and solitons have also been illustrated. Specifically, these semi-rational solutions have a new phenomenon: lumps form on dark solitons and gradual separation from the dark solitons is observed.
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In this paper, the partially party-time ($PT$) symmetric nonlocal Davey-Stewartson (DS) equations with respect to $x$ is called $x$-nonlocal DS equations, while a fully $PT$ symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely breather, rational and semi-rational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the $x$-nonlocal DS equations, the usual ($2+1$)-dimensional breathers are periodic in $x$ direction and localized in $y$ direction. Nonsingular rational solutions are lumps, and semi-rational solutions are composed of lumps, breathers and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both $x$ and $y$ directions with parallels in profile, but localized in time. Nonsingular rational solutions are ($2+1$)-dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semi-rational solutions describe interactions of line rogue waves and periodic line waves.
The integrable Davey-Stewartson system is a linear combination of the two elementary flows that commute: $mathrm{i} q_{t_1} + q_{xx} + 2qpartial_y^{-1}partial_x (|q|^2) =0$ and $mathrm{i} q_{t_2} + q_{yy} + 2qpartial_x^{-1}partial_y (|q|^2) =0$. In the literature, each elementary Davey-Stewartson flow is often called the Fokas system because it was studied by Fokas in the early 1990s. In fact, the integrability of the Davey-Stewartson system dates back to the work of Ablowitz and Haberman in 1975; the elementary Davey-Stewartson flows, as well as another integrable $(2+1)$-dimensional nonlinear Schrodinger equation $mathrm{i} q_{t} + q_{xy} + 2 qpartial_y^{-1}partial_x (|q|^2) =0$ proposed by Calogero and Degasperis in 1976, appeared explicitly in Zakharovs article published in 1980. By applying a linear change of the independent variables, an elementary Davey-Stewartson flow can be identified with a $(2+1)$-dimensional generalization of the integrable long wave-short wave interaction model, called the Yajima-Oikawa system: $mathrm{i} q_{t} + q_{xx} + u q=0$, $u_t + c u_y = 2(|q|^2)_x$. In this paper, we propose a new integrable semi-discretization (discretization of one of the two spatial variables, say $x$) of the Davey-Stewartson system by constructing its Lax-pair representation; the two elementary flows in the semi-discrete case indeed commute. By applying a linear change of the continuous independent variables to an elementary flow, we also obtain an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system.
This is a continuation of our previous paper arXiv:1904.07924, which is devoted to the construction of integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system; in this series of papers, we refer to a discretization of one of the two spatial variables as a semi-discretization. In this paper, we construct an integrable semi-discrete Davey-Stewartson system, which is essentially different from the semi-discrete Davey-Stewartson system proposed in the previous paper arXiv:1904.07924. We first obtain integrable semi-discretizations of the two elementary flows that compose the Davey-Stewartson system by constructing their Lax-pair representations and show that these two elementary flows commute as in the continuous case. Then, we consider a linear combination of the two elementary flows to obtain a new integrable semi-discretization of the Davey-Stewartson system. Using a linear transformation of the continuous independent variables, one of the two elementary Davey-Stewartson flows can be identified with an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system proposed in https://link.aps.org/doi/10.1103/PhysRevE.91.062902 .
We consider a focusing Davey-Stewartson system and construct the solution of the Cauchy problem in the possible presence of exceptional points (and/or curves).
In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit $lambda_{j}$ $rightarrow$ $lambda_{1}$ of the Lax pair eigenvalues used in the $n$-fold Darboux transformation that generates the order-$n$ periodic solution from a constant seed solution. Further, this special kind of breather solution of order $n$ can be used to generate the order-$n$ rational solution by taking the limit $lambda_{1}$ $rightarrow$ $lambda_{0}$, where $lambda_{0}$ is a special eigenvalue associated to the eigenfunction $phi$ of the Lax pair of the mKdV equation. This eigenvalue $lambda_0$, for which $phi(lambda_0)=0$, corresponds to the limit of infinite period of the periodic solution. %This second limit of double eigenvalue degeneration might be realized approximately in optical fibers, in which an injected %initial ideal pulse is created by a comb system and a programmable optical filter according to the profile of the analytical %form of the b-positon at a certain spatial position $x_{0}$. Therefore, we suggest a new way to observe the higher-order %rational solutions in optical fibers, namely, to measure the wave patterns at the central region of the higher order b-positon %generated by ideal initial pulses when the eigenvalue $lambda_{1}$ is approaching $lambda_{0}$. Our analytical and numerical results show the effective mechanism of generation of higher-order rational solutions of the mKdV equation from the double eigenvalue degeneration process of multi-periodic solutions.
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