# (3+1)维Mel’nikov方程的Lump解研究Lump Solutions to the (3+1)-Dimensional Mel’nikov Equation

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In this article, lump solutions of the (3+1)-dimensional Mel’nikov are obtained via the Hirota bi-linear method and symbolic computation with Maple. A class of lump solutions rationally localized in all directions in the space is obtained. And we propose the conditions for the analyticity and ra-tional localization of the lump solutions. By selecting special value of parameters involved, the dynamic characteristics of the solutions are illustrated.

1. 引言

$\left\{\begin{array}{l}{\left({u}_{t}+6u{u}_{x}+{u}_{xxx}+8{|\varphi |}_{x}^{2}\right)}_{x}-{u}_{yy}+{u}_{zz}=0,\\ i{\varphi }_{y}=2{\varphi }_{xx}+2u\varphi ,\\ i{\varphi }_{z}={\varphi }_{xx}+u\varphi ,\end{array}$ (1)

2. Mel’nikov的Lump解

$\left\{\begin{array}{l}u=2{\left(\mathrm{ln}f\right)}_{xx},\\ \varphi =\frac{g}{f},\end{array}$ (2)

$\left\{\begin{array}{l}\left({D}_{x}{D}_{t}+{D}_{x}^{4}-{D}_{y}^{2}+{D}_{z}^{2}\right)f\cdot f+8\left(g\stackrel{¯}{g}-{f}^{2}\right)=0,\\ \left(i{D}_{y}-2{D}_{x}^{2}\right)g\cdot f=0,\\ \left(i{D}_{z}-{D}_{x}^{2}\right)g\cdot f=0.\end{array}$ (3)

${D}_{x}^{m}{D}_{t}^{n}\left(a,b\right)={{\left(\frac{\partial }{\partial x}-\frac{\partial }{\partial {x}^{\prime }}\right)}^{m}{\left(\frac{\partial }{\partial t}-\frac{\partial }{\partial {t}^{\prime }}\right)}^{n}a\left(x,t\right)b\left({x}^{\prime },{t}^{\prime }\right)|}_{{x}^{\prime }=x,{t}^{\prime }=t}$ (4)

$\left\{\begin{array}{l}f=1+{s}^{2}+{h}^{2},\\ {g}_{R}={b}_{0}+{b}_{1}s+{b}_{2}h+{b}_{3}{s}^{2}+{b}_{4}{h}^{2},\\ {g}_{I}={c}_{0}+{c}_{1}s+{c}_{2}h+{c}_{3}{s}^{2}+{c}_{4}{h}^{2},\end{array}$ (5)

$\left\{\begin{array}{l}s={a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}t+{a}_{5},\\ h={a}_{6}x+{a}_{7}y+{a}_{8}z+{a}_{9}t+{a}_{10},\end{array}$ (6)

$\begin{array}{l}\left\{{a}_{2}=2{a}_{3},{a}_{4}=\frac{\left({a}_{3}^{2}-{a}_{1}^{4}\right)\left[3{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}+16{a}_{1}^{4}\right]}{{a}_{1}{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}},{a}_{6}=0,{a}_{7}=2{a}_{1}^{2},{a}_{8}=\frac{1}{2}{a}_{7},{a}_{9}=\frac{2{a}_{1}{a}_{3}\left[3{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}-16{a}_{1}^{4}\right]}{{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}},\\ {b}_{0}=\frac{{b}_{3}\left({a}_{3}^{2}-3{a}_{1}^{4}\right)}{{a}_{1}^{4}+{a}_{3}^{2}},{b}_{1}=k{c}_{1},{b}_{2}=k{c}_{2},{b}_{4}={b}_{3},{c}_{0}=-k{b}_{0},{c}_{1}=-\frac{4{a}_{3}{b}_{3}{a}_{1}^{2}}{{a}_{1}^{4}+{a}_{3}^{2}},{c}_{2}=\frac{4{a}_{1}^{4}{b}_{3}}{{a}_{1}^{4}+{a}_{3}^{2}},{c}_{3}=-k{b}_{3},{c}_{4}={c}_{3}\right\}\end{array}$ (7)

${b}_{3}^{2}\left(1+{k}^{2}\right)=1$ (8)

$\left\{\begin{array}{l}u=\frac{4{a}_{1}^{2}\left(1-{s}^{2}+{h}^{2}\right)}{{\left(1+{s}^{2}+{h}^{2}\right)}^{2}},\\ \varphi =\frac{{g}_{R}+i{g}_{I}}{1+{s}^{2}+{h}^{2}},\end{array}$ (9)

$\begin{array}{l}{g}_{R}=\frac{{b}_{3}\left({a}_{3}^{2}-3{a}_{1}^{4}\right)}{{a}_{1}^{4}\text{+}{a}_{3}^{2}}-k\frac{4{a}_{3}{b}_{3}{a}_{1}^{2}}{{a}_{1}^{4}\text{+}{a}_{3}^{2}}s+k\frac{4{a}_{1}^{4}{b}_{3}}{{a}_{1}^{4}\text{+}{a}_{3}^{2}}h+{b}_{3}\left({s}^{2}+{h}^{2}\right),\\ {g}_{I}=-k\frac{{b}_{3}\left({a}_{3}^{2}-3{a}_{1}^{4}\right)}{{a}_{1}^{4}\text{+}{a}_{3}^{2}}-\frac{4{a}_{3}{b}_{3}{a}_{1}^{2}}{{a}_{1}^{4}\text{+}{a}_{3}^{2}}s+\frac{4{a}_{1}^{4}{b}_{3}}{{a}_{1}^{4}\text{+}{a}_{3}^{2}}h-k{b}_{3}\left({s}^{2}+{h}^{2}\right),\end{array}$ (10)

$\begin{array}{l}s={a}_{1}x+2{a}_{3}y+{a}_{3}z+\frac{\left({a}_{3}^{2}-{a}_{1}^{4}\right)\left[3{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}\text{+}16{a}_{1}^{4}\right]}{{a}_{1}{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}}t+{a}_{5},\\ h=2{a}_{1}^{2}y+{a}_{1}^{2}z+\frac{2{a}_{1}{a}_{3}\left[3{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}-16{a}_{1}^{4}\right]}{{\left({a}_{1}^{4}+{a}_{3}^{2}\right)}^{2}}t+{a}_{10}.\end{array}$ (11)

Figure 1. Plots of lump solution for u with ${a}_{1}=1,\text{\hspace{0.17em}}{a}_{3}=2,\text{\hspace{0.17em}}{b}_{3}=-0.8,\text{\hspace{0.17em}}{a}_{5}={a}_{10}=0,\text{\hspace{0.17em}}k=0.75$ when $x=0,\text{\hspace{0.17em}}z=0$

Figure 2. Plots of lump solution for ${|\varphi |}^{2}$ with ${a}_{1}=1,\text{\hspace{0.17em}}{a}_{3}=2,\text{\hspace{0.17em}}{b}_{3}=-0.8,\text{\hspace{0.17em}}{a}_{5}={a}_{10}=0,\text{\hspace{0.17em}}k=0.75$ when $x=0,\text{\hspace{0.17em}}z=0$

Figure 3. Plots of lump solution for u with ${a}_{1}=2,\text{\hspace{0.17em}}{a}_{3}=-2,\text{\hspace{0.17em}}{b}_{3}=0.8,\text{\hspace{0.17em}}{a}_{5}={a}_{10}=0,\text{\hspace{0.17em}}k=0.75$ when $y=0,\text{\hspace{0.17em}}t=0$

Figure 4. Plots of lump solution for ${|\varphi |}^{2}$ with ${a}_{1}=2,\text{\hspace{0.17em}}{a}_{3}=-2,\text{\hspace{0.17em}}{b}_{3}=0.8,\text{\hspace{0.17em}}{a}_{5}={a}_{10}=0,\text{\hspace{0.17em}}k=0.75$ when $y=0,\text{\hspace{0.17em}}t=0$

3. 结论

NOTES

*通讯作者。

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