戴维-斯图尔森方程组

戴维-斯图尔森方程组(Davey–Stewartson equation)是Davey、Stewartson在1974年用以模拟水面波包传播的非线性偏微分方程^[1]：

iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi _{x},\,

\phi _{xx}+c_{3}\phi _{yy}=(|u|^{2})_{x}.\,

行波解

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*cot(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*coth(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*tan(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*tanh(_{C}2+_{C}3*x+_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*arcsinh(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)+_{C}8*arcsinh(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)^{2}

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*cos(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)+_{C}8*cos(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)^{2}

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*cosh(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)+_{C}8*cosh(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)^{2}

u(x,y,t)=0,v(x,y)=_{C}6+_{C}7*cot(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)+_{C}9*cot(_{C}2+_{C}3*x-_{C}3*y/{\sqrt {(}}-_{C}1)+_{C}5*t)^{3}

u(x,y,t)=0,v(x,y)=_{C}7+_{C}8*JacobiCN(_{C}3+_{C}4*x-_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)+_{C}10*JacobiCN(_{C}3+_{C}4*x-_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)^{3}

{u(x,y,t)=0,v(x,y)=_{C}7+_{C}8*JacobiDN(_{C}3+_{C}4*x-_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)+_{C}10*JacobiDN(_{C}3+_{C}4*x-_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)^{3}}

u(x,y,t)=0,v(x,y)=_{C}7+_{C}8*JacobiND(_{C}3+_{C}4*x-_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)+_{C}10*JacobiND(_{C}3+_{C}4*x-_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)^{3}

u(x,y,t)=0,v(x,y)=_{C}7+_{C}8*JacobiNS(_{C}3+_{C}4*x+_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)+_{C}10*JacobiNS(_{C}3+_{C}4*x+_{C}4*y/{\sqrt {(}}-_{C}1)+_{C}6*t,_{C}2)^{3}

戴维-斯图尔森方程组行波图

Boiti, M.; Martina, L.; Pempinelli, F., Multidimensional localized solitons, Chaos Solitons Fractals, 1995, 5 (12): 2377–2417, Bibcode:1995CSF.....5.2377B, MR 1368226, doi:10.1016/0960-0779(94)E0106-Y
Davey, A.; Stewartson, K., On three dimensional packets of surface waves, Proc. R. Soc. A, 1974, 338 (1613): 101–110, Bibcode:1974RSPSA.338..101D, doi:10.1098/rspa.1974.0076
Sattinger, David H.; Tracy, C. A.; Venakides, S. (编), Inverse Scattering and Applications, Contemporary Mathematics 122, Providence, RI: American Mathematical Society, 1991, ISBN 0-8218-5129-2, MR 1135850