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： i u t + c 0 u x x + u y y = c 1 | u | 2 u + c 2 u ϕ x , {\displaystyle iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi _{x},\,} ϕ x x + c 3 ϕ y y = ( | u | 2 ) x . {\displaystyle \phi _{xx}+c_{3}\phi _{yy}=(|u|^{2})_{x}.\,} u ( x , y , t ) = 0 , v ( x , y ) = C 6 + C 7 ∗ c o t ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) {\displaystyle 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 ∗ c o t h ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) {\displaystyle 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 ∗ t a n ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) {\displaystyle 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 ∗ t a n h ( C 2 + C 3 ∗ x + C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) {\displaystyle 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 ∗ a r c s i n h ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) + C 8 ∗ a r c s i n h ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) 2 {\displaystyle 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 ∗ c o s ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) + C 8 ∗ c o s ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) 2 {\displaystyle 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 ∗ c o s h ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) + C 8 ∗ c o s h ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) 2 {\displaystyle 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 ∗ c o t ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) + C 9 ∗ c o t ( C 2 + C 3 ∗ x − C 3 ∗ y / ( − C 1 ) + C 5 ∗ t ) 3 {\displaystyle 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 ∗ J a c o b i C N ( C 3 + C 4 ∗ x − C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) + C 10 ∗ J a c o b i C N ( C 3 + C 4 ∗ x − C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) 3 {\displaystyle 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 ∗ J a c o b i D N ( C 3 + C 4 ∗ x − C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) + C 10 ∗ J a c o b i D N ( C 3 + C 4 ∗ x − C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) 3 {\displaystyle {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 ∗ J a c o b i N D ( C 3 + C 4 ∗ x − C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) + C 10 ∗ J a c o b i N D ( C 3 + C 4 ∗ x − C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) 3 {\displaystyle 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 ∗ J a c o b i N S ( C 3 + C 4 ∗ x + C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) + C 10 ∗ J a c o b i N S ( C 3 + C 4 ∗ x + C 4 ∗ y / ( − C 1 ) + C 6 ∗ t , C 2 ) 3 {\displaystyle 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}} 扎哈羅夫-舒爾曼系統 Davey Stewardson 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