Dromion solutions of PT-symmetric $(x,y)$-nonlocal Davey-Stewartson I equation

Abstract

A $\mathcal{PT}$-symmetric nonlocal Davey-Stewartson I equation is considered, in which $\overline{u}(x,y,t)$ in the classical equation is replaced by $\overline{u}(-x,-y,t)$. Using the nonlinear constraint from $2+1$ dimensions to $1+1$ dimensions and Darboux transformation in 1+1 dimensions, $2m\times 2n$ dromion solutions are obtained. It is proved that under certain conditions, the derived solutions are always globally defined and decay exponentially at spacial infinity. Moreover, each asymptotic solution as $t\to \pm \infty$ has exactly $4mn$ peaks. The local behavior of each peak is also given.

Introduction

The $\mathcal{PT}$-symmetric equation was first introduced by Bender and Boettcher in 1998 [1]. It has applications in quantum mechanics, Bose-Einstein condensation and nonlinear optics. In 2013, Ablowitz and Musslimani introduced the ($\mathcal{PT}$-symmetric) nonlocal nonlinear Schrödinger equation [2], and derived its exact solutions by inverse scattering. Quite a lot of work were done after that for this equation and the others [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].

In [15], Fokas studied high dimensional equations including the nonlocal Davey-Stewartson (DS) equations. The classical DS equations are equations describing two dimensional water wave. The DSI equation is

$$\begin{array}{c}i{u}_t=u_{xx}+u_{yy}+2u^2\overline{u}+2(v_1+v_2)u,\hfill \\ v_{1,y}-v_{1,x}={\left(u\overline{u}\right)}_x,v_{2,y}+v_{2,x}=-{\left(u\overline{u}\right)}_x,\hfill \end{array}$$

where $\overline{u}$ is the complex conjugation of $u$. In the equation, $u$ is a complex unknown, $v_1,v_2$ are real unknowns. Usually we only consider the behavior of $u$, which is physically meaningful. It was known that Eq.  (1) has dromion solutions [16], [17].

There are various types of nonlocal DS equations, say $(x,y)$-nonlocal equation where $\overline{u}(x,y,t)$ is replaced by $\overline{u}(-x,-y,t)$, or $x$-nonlocal equation where $\overline{u}(x,y,t)$ is replaced by $\overline{u}(-x,y,t)$ etc. with certain change of signs in the classical DS equation. In this paper, we only consider the $(x,y)$-nonlocal equation.

For a function $\phi (x,y,t)$, denote ${\phi }^{*}(x,y,t)=\overline{\phi }(-x,-y,t)$. Likewise, for a matrix function $M$, denote $M^{*}(x,y,t)={\overline{M}}^T(-x,-y,t)$ where $M^T$ is the transpose of $M$.

The nonlocal DSI equation that we will consider is

$$\begin{array}{c}-i{u}_t=u_{xx}+u_{yy}-2u^2{u}^{*}-2(v_1+v_2)u,\hfill \\ v_{1,x}-v_{1,y}=v_{2,x}+v_{2,y}=-{\left(u{u}^{*}\right)}_x,\hfill \end{array}$$

where $v_1$ and $v_2$ satisfy $v_1^{*}=v_1$, $v_2^{*}=v_2$.

Eq.  (2) can be derived from Eq.  (1) by formal substitution [18]

$$(x,y,t,u,\overline{u},v_1,v_2)\mapsto (ix,iy,t,u,u^{*},v_1,v_2).$$

Eq.  (2) is $\mathcal{PT}$-symmetric in the sense that if $(u(x,y,t),v_1(x,y,t),v_2(x,y,t))$ is a solution of Eq.  (2), then so is $(\overline{u}(-x,-y,-t),v_1(-x,-y,-t),v_2(-x,-y,-t))$. This leads to a conserved density $u{u}^{*}$, which is invariant under $(x,y)\to (-x,-y)$ together with complex conjugation, and satisfies the conservation law

$$i{\left(u{u}^{*}\right)}_t={(u{\left(u^{*}\right)}_x-u_x{u}^{*})}_x+{(u{\left(u^{*}\right)}_y-u_y{u}^{*})}_y.$$

Due to the substitution (3), the nonlocal equation (2) has the similar algebraic structure as the classical equation (1). Hence they have similar Lax pairs, bilinear forms, Darboux transformations, nonlinear constraints etc. However, the analytic properties of Eqs.  (2) and (1) are quite different.

There are many researches on various versions of nonlocal DS equations [15], [18], [19], [20], [21], [22], [23], [24]. Especially, for this $(x,y)$-nonlocal DSI equation, quite a lot of non-localized exact solutions, including line solitons, line breathers, rogue waves, lumps etc. are obtained by bilinear method or binary Darboux transformation [25], [26], [27], [28]. However, it is still interesting to obtain localized solutions.

The nonlinear constraint from 2+1 dimensions to 1+1 dimensions was presented by [29], [30] for the KP equation. Based on this idea, $m\times n$ dromion solutions of the classical equation (1) were constructed by using nonlinear constraint and Darboux transformation in [31]. The derived solutions were globally defined. However, like the other nonlocal equations, usually the exact solutions of Eq.  (2) may have singularities. It is essential to find certain conditions to guarantee the globality of the derived solutions. Moreover, unlike Eq.  (1), although similar Darboux transformations of degree $m+n$ generating dromion solutions of Eq.  (1) still generate exact solutions of Eq.  (2), these exact solutions do not have properties as good as those of the classical DSI equation. In order to get dromion solutions, we need to use Darboux transformation of degree $2m+2n$. In the construction, the spectral parameters should be chosen symmetrically as ${\lambda }_j$, $-{\lambda }_j$$(j=1,\cdots ,m+n)$ and the solutions of the Lax pair should also be chosen in a special way. The main task of this paper is to show that these derived solutions are really dromions by analyzing their globality and the asymptotic behaviors at both spacial and temporal infinity.

In Section 2, the nonlinear constraint and Darboux transformation are presented, which are parallel to those for the classical DSI equation. In Section 3, a special form of Darboux transformation is constructed by choosing the spectral parameters and the solutions of the Lax pair properly. The derived solutions look like $2m\times 2n$ dromion solutions if they are globally defined. Theorem 2 in Section 4 shows that under certain conditions on the parameters, the derived solutions have no singularities. Theorem 3 in Section 5 demonstrates that the derived solution tends to zero at spacial infinity and gives the estimate of the decay rate. In 6 Asymptotic behavior of the solution at temporal infinity, Theorem 4 and Theorem 5 show that the asymptotic solution as time tends to infinity has exactly $2m\times 2n$ peaks. The local behavior of each peak is also given in Theorem 5.