Periodic-background solutions of Kadomtsev-Petviashvili I equation

Abstract

A systematic method to obtain exact solutions of the Zakharov-Shabat spectral problems with Jacobi elliptic function potentials is constructed by using the Baker-Akhiezer functions which appear in the study quasi-periodic solutions. As a result, periodic-background lump (PB-lump for short) solutions of the Kadomtsev-Petviashvili I (KPI) equation are constructed. Like lump solutions, PB-lump solutions are localized in the xy-plane and propagate at constant velocities. Different from lump solutions, which are rational solutions and tend to a constant amplitude when x and y are large, PB-lump solutions are composed of polynomials, elliptic functions and other special functions, and tend to the periodic background when x and y are large. What’s more, the amplitude of the crest for the PB-lump solution changes periodically as it propagates on the xy-plane. The trajectories of the PB-lumps for the KPI equation are presented on the basis of the formal series expansion technique. As an illustration, both first- and second-order PB-lumps and their exact formulas are obtained.

Appendix A Special funcitons

Several special functions related elliptic integrals appeared in this paper. Because the denotations vary in different literatures, to avoid ambiguity, we write down the details in the following (Fig. 4).

Constants: k, $k'$ and $\bar{k}$.

$$\begin{aligned}&k\in (0,1)\text { is an arbitrary fixed constant},\\&k'=\sqrt{1-k^2}\text { is positive},\\&\bar{k}=\pm k'\text { can be either positive or negative}. \end{aligned}$$

Jacobi elliptic integrals: E(x, k), F(x, k), E(k) and K(k).

$$\begin{aligned}&E(x,k)=\int _0^x\sqrt{1-k^2\sin ^2\hat{x}}\mathrm d\hat{x},\quad E(k)=E(\tfrac{\pi }{2},k),\\&F(x,k)=\int _0^x\frac{1}{\sqrt{1-k^2\sin ^2\hat{x}}}\mathrm d\hat{x},\quad K(k)=F(\tfrac{\pi }{2},k). \end{aligned}$$

Jacobi elliptic functions: ${{\,\textrm{am}\,}}(x,k)$, ${{\,\textrm{sn}\,}}(x,k)$, ${{\,\textrm{cn}\,}}(x,k)$, ${{\,\textrm{dn}\,}}(x,k)$, Z(x, k) and $\mathcal E(x,k)$.

$$\begin{aligned}&\text {If } F(x,k)=y, \text { then } x={{\,\textrm{am}\,}}(y,k). \\&{{\,\textrm{sn}\,}}(x,k)=\sin ({{\,\textrm{am}\,}}(x,y)),\quad {{\,\textrm{cn}\,}}(x,k)=\cos ({{\,\textrm{am}\,}}(x,k)),\quad {{\,\textrm{dn}\,}}(x,k)=\sqrt{1-k^2{{\,\textrm{sn}\,}}^2(x,k)},\\&\mathcal E(x,k)=E({{\,\textrm{am}\,}}(x,k),k) ,\quad Z(x,k)=\mathcal E(x,k)-\tfrac{E(k)}{K(k)}x. \end{aligned}$$

Fig. 4

The crests and the moving windows (red stars; left, $u_7^+$; right, $u_7^-$; $k=0.9$, $t=120$)

Short forms: ${{\,\textrm{sn}\,}}x$, ${{\,\textrm{cn}\,}}x$, ${{\,\textrm{dn}\,}}x$, E, K and Z(x).

$$\begin{aligned}&{{\,\textrm{sn}\,}}x={{\,\textrm{sn}\,}}(x,k),\quad {{\,\textrm{cn}\,}}x={{\,\textrm{cn}\,}}(x,k),\quad {{\,\textrm{dn}\,}}x={{\,\textrm{dn}\,}}(x,k),\\&E=E(k),\quad K=K(k),\quad Z(x)=Z(x,k). \end{aligned}$$

Remark

Throughout this paper, all the Jacobi elliptic integrals and functions have the same modulus k. Therefore, we always suppress the dependence on k for notation simplicity.