卡东穆塞夫-彼得韦亚斯维利方程

卡东穆塞夫-彼得韦亚斯维利方程(Kadomtsev-Petviashvili equation)，简称KP方程，是1970年苏联物理学家波里斯·卡东穆塞夫 和弗拉基米尔-彼得韦亚斯维利创立以模拟非线性波动的非线性偏微分方程： ∂ x ( ∂ t u + u ∂ x u + ϵ 2 ∂ x x x u ) + λ ∂ y y u = 0 {\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0} 其中 λ = ± 1 {\displaystyle \lambda =\pm 1} . 卡东穆塞夫-彼得韦亚斯维利方程有解析解 u ( x , y , t ) = C 5 + 12. ∗ C 2 ∗ tanh ⁡ ( C 1 + C 2 ∗ x + C 3 ∗ y − ( .50000000000000000000 ∗ ( 8. ∗ C 2 4 + C 3 2 ) ) ∗ t / C 2 ) {\displaystyle u(x,y,t)=C5+12.*_{C}2*\tanh(_{C}1+_{C}2*x+_{C}3*y-(.50000000000000000000*(8.*_{C}2^{4}+_{C}3^{2}))*t/_{C}2)} 代人参数： C5 = 1, _C1 = 0, _C2 = 1, _C3 = 3 得： u = 1 + 12. ∗ t a n h ( x + 3 ∗ y − 8.5000000000000000000 ∗ t ) {\displaystyle u=1+12.*tanh(x+3*y-8.5000000000000000000*t)} Kadomtsev-Petviahivili equation 3D plot Kadomtsev-Petviashivili pde animation 利用sech函数展开法可得卡东穆塞夫-彼得韦亚斯维利方程的sech函数解和tanh函数解。 u := a ∗ s e c h ( a ∗ x + b ∗ y + c ∗ z − ( a 4 + 3 ∗ b 2 + 3 ∗ c 2 ) / a ) ∗ t {\displaystyle u:=a*sech(a*x+b*y+c*z-(a^{4}+3*b^{2}+3*c^{2})/a)*t} 参数：a = -2 .. 2, b = -2 .. 2, c = 0 Kadomtsev Petviashivili pde sech solution 3d plot 卡东穆塞夫-彼得韦亚斯维利方程sech函数亮孤立子 u := 2 ∗ a 2 ∗ t a n h ( a ∗ x + b ∗ y + ( 8 ∗ a 4 − 3 ∗ b 2 ) / a ) 2 ∗ t {\displaystyle u:=2*a^{2}*tanh(a*x+b*y+(8*a^{4}-3*b^{2})/a)^{2}*t} 。 参数：a = 2, b = -2； KP方程暗tanh函数暗孤立子 通过朗斯基行列式展开法可得卡东塞穆夫-彼得韦亚斯维利方程多个雅可比橢圓函數解。 u 4 := ( − 4 ∗ m 2 ∗ k [ 1 ] 2 ∗ g ) ( 1 − m 2 ∗ s n ( ξ [ 1 ] , k ) ∗ s i n ( ξ [ 2 ] ) + d n ( ξ [ 1 ] , k ) ∗ c o s ( ξ [ 2 ] ) ∗ c n ( ξ [ 1 ] , k ) ) 2 ) {\displaystyle u4:={\frac {(-4*m^{2}*k[1]^{2}*g)}{({\sqrt {1-m^{2}}}*sn(\xi [1],k)*sin(\xi [2])+dn(\xi [1],k)*cos(\xi [2])*cn(\xi [1],k))^{2})}}} 其中： g = ( m 2 − 1 ) ∗ s n ( ξ [ 1 ] , k ) 2 + ( 2 − 2 ∗ m 2 ) ∗ s n ( ξ [ 1 ] , k ) 4 + c o s ( ξ [ 2 ] ) 2 ; − 2 ∗ s n ( ξ [ 1 ] , k ) 2 ∗ c o s ( ξ [ 2 ] ) 2 + m 2 ∗ s n ( ξ [ 1 ] , k ) 4 ∗ c o s ( ξ [ 2 ] ) 2 {\displaystyle g=(m^{2}-1)*sn(\xi [1],k)^{2}+(2-2*m^{2})*sn(\xi [1],k)^{4}+cos(\xi [2])^{2};-2*sn(\xi [1],k)^{2}*cos(\xi [2])^{2}+m^{2}*sn(\xi [1],k)^{4}*cos(\xi [2])^{2}} ξ [ 1 ] = k [ 1 ] ∗ x + λ [ 1 ] ∗ y + ( 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t + γ [ 1 ] {\displaystyle \xi [1]=k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1]} ξ [ 2 ] = 1 − m 2 ∗ ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y + ( − 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t ) − γ [ 2 ] {\displaystyle \xi [2]={\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2]} 代入后得： f 4 := − 4 ∗ m 2 ∗ k [ 1 ] 2 ∗ ( ( m 2 − 1 ) ∗ J a c o b i S N ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y + ( 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 {\displaystyle f4:=-4*m^{2}*k[1]^{2}*((m^{2}-1)*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}} − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t + γ [ 1 ] , k ) 2 + ( 2 − 2 ∗ m 2 ) ∗ J a c o b i S N ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y {\displaystyle -3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{2}+(2-2*m^{2})*JacobiSN(k[1]*x+\lambda [1]*y} + ( 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t + γ [ 1 ] , k ) 4 + {\displaystyle +(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{4}+} c o s ( ( 1 − m 2 ) ∗ ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y + ( − 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t ) {\displaystyle cos({\sqrt {(1-m^{2})}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)} − γ [ 2 ] ) 2 ) / ( s q r t ( 1 − m 2 ) ∗ J a c o b i S N ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y + ( 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − {\displaystyle -\gamma [2])^{2})/(sqrt(1-m^{2})*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-} 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t + γ [ 1 ] , k ) ∗ s i n ( ( 1 − m 2 ) ∗ ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y + {\displaystyle 3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*sin({\sqrt {(}}1-m^{2})*(k[1]*x+\lambda [1]*y+} ( − 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t ) − γ [ 2 ] ) + J a c o b i D N ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y + {\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])+JacobiDN(k[1]*x+\lambda [1]*y+} ( 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t + γ [ 1 ] , k ) ∗ c o s ( 1 − m 2 ∗ ( k [ 1 ] ∗ x + λ [ 1 ] ∗ y + {\displaystyle (4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*cos({\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+} ( − 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t ) − γ [ 2 ] ) ∗ J a c o b i C N ( k [ 1 ] ∗ x + {\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])*JacobiCN(k[1]*x+} λ [ 1 ] ∗ y + ( 4 ∗ m 2 ∗ k [ 1 ] 3 + 16 ∗ k [ 1 ] 3 − 3 ∗ σ 2 ∗ λ [ 1 ] 2 / k [ 1 ] ) ∗ t + γ [ 1 ] , k ) ) 2 {\displaystyle \lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k))^{2}} Kadomtsev Petviashivili pde elliptic function solution 3d plot Kodomtsev,B.B and Petviashivili V.I. 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