五阶KdV方程(Fifth order KdV equation)是一个非线性偏微分方程，简称fKdV方程：
u_{t}+\alphau^2u_{x}+\betau_{x}u_{xx}+\gammauu_{xxx}+\delta*u_{xxxxx}=0
解析解
:u(x, t) = 6_C3^2(-(6(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))\gamma^2/\alpha+(60(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))\delta-72\delta\gamma^2+720\delta^2\alpha-120\delta\beta^2)JacobiND(_C2+_C3x-(6(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))_C3^5t/\alpha, \sqrt(2))^2/(\beta(6\beta^2-120\delta\alpha+12\gamma^2+12\beta\gamma+6\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2*\gamma^2)))
:u(x, t) = 6_C3^2(-(6(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))\gamma^2/\alpha+(60(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))\delta-72\delta\gamma^2+720\delta^2\alpha-120\delta\beta^2)JacobiNS(_C2+_C3x-(6(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))_C3^5t/\alpha, I)^2/(\beta(6\beta^2-120\delta\alpha+12\gamma^2+12\beta\gamma+6\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2*\gamma^2)))
:u(x, t) = -3_C3^2(-(3/2)(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2))\gamma^2/\alpha+(15(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))\delta-18\delta\gamma^2+180\delta^2\alpha-30\delta\beta^2)JacobiCN(_C2+_C3x-(3/2)(-12\delta\alpha+\beta^2+2\beta\gamma-\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2))_C3^5t/\alpha, (1/2)\sqrt(2))^2/(\beta((3/2)\beta^2-30\delta\alpha+3\gamma^2+3\beta\gamma+(3/2)\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))
:u(x, t) = -6_C3^2(-(6(-12\delta\alpha+\beta^2+2\beta\gamma+\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))\gamma^2/\alpha+(60(-12\delta\alpha+\beta^2+2\beta\gamma+\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))\delta-72\delta\gamma^2+720\delta^2\alpha-120\delta\beta^2)JacobiDN(_C2+_C3x-(6(-12\delta\alpha+\beta^2+2\beta\gamma+\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2\gamma^2)))_C3^5t/\alpha, \sqrt(2))^2/(\beta(6\beta^2-120\delta\alpha+12\gamma^2+12\beta\gamma-6\sqrt(-40\delta\alpha\beta^2+\beta^4+4\beta^3\gamma+4\beta^2*\gamma^2)))
:u(x, t) = _C5-(3(-4_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\beta^2_C5^2-8_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\gamma^2_C5^2-(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\beta_C5^3\alpha-(3/2)\gamma(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^3\alpha+(1/4)\gamma(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^3\beta^2/\delta+(2/5)\gamma^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^3\beta/\delta-\beta^2_C5^4\alpha+4800_C3^6\beta\delta^2_C5-160_C3^4\beta^2_C5^2\delta-800\delta^2\alpha_C3^4_C5^2+16\gamma^2_C5^3\beta_C3^2-320\gamma^2_C3^4_C5^2\delta+7200\gamma_C3^6\delta^2_C5+10\gamma\beta^2_C5^3_C3^2-3\gamma_C5^4\beta\alpha-48000_C3^8\delta^3+2\beta^3_C5^3_C3^2+8\gamma^3_C5^3_C3^2-2\gamma^2_C5^4\alpha+10_C5^4\delta\alpha^2+20\gamma_C5^3\delta\alpha_C3^2-480\gamma_C3^4_C5^2\beta\delta-12\gamma_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^2\beta+40_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta\alpha_C5^2+120_C3^4(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta\gamma_C5+120_C3^4(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta\beta_C5-1200_C3^6(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta^2+40_C5^3\delta\alpha\beta_C3^2+(1/20)(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\beta^3_C5^3/\delta+(1/5)\gamma^3(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^3/\delta))JacobiSN(_C2+_C3x+(1/25)_C3(3_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5\beta\gamma+45_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta_C5\alpha+(1/2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2)))\alpha_C5^2\beta-150_C3^4(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta\beta-9_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\gamma^2_C5-(9/4(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2)))\gamma\alpha_C5^2+90_C3^4(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta\gamma+(1/20)(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^2\beta\gamma^2/\delta-(1/20)(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^2\beta^2\gamma/\delta+(3/10)\gamma^3(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^2/\delta+420_C3^4\delta_C5\beta\gamma+3600_C3^6\delta^2\gamma+300_C3^4\delta_C5\beta^2-6000_C3^6\delta^2\beta+2_C5^2\beta\gamma^2_C3^2-2_C5^2\beta^2\gamma_C3^2+_C5^3\beta\gamma\alpha-130_C5^2\delta\alpha\beta_C3^2+15_C5^3\delta\alpha^2+12\gamma^3_C5^2_C3^2-3\gamma^2_C5^3\alpha-360_C3^4\gamma^2\delta_C5)t/(\delta(-\alpha_C5+2\gamma_C3^2+(1/20)\gamma(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))/\delta)), (1/20)\sqrt(10)\sqrt((2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))/\delta)/_C3)^2/(60_C3^4(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta\beta-6_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\gamma^2_C5+60_C3^4(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta\gamma-3_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5\beta^2-(3/2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2)))\gamma\alpha_C5^2-(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\alpha_C5^2\beta+(1/20)(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\beta^3_C5^2/\delta-120_C3^4\delta_C5\beta^2+10_C5^2\beta^2\gamma_C3^2+16_C5^2\beta\gamma^2_C3^2-240_C3^4\gamma^2\delta_C5-3_C5^3\beta\gamma\alpha-\beta^2_C5^3\alpha+2400_C3^6\delta^2\beta+2400_C3^6\delta^2\gamma+10_C5^3\delta\alpha^2+8\gamma^3_C5^2_C3^2-2\gamma^2_C5^3\alpha+2\beta^3_C5^2_C3^2-360_C3^4\delta_C5\beta\gamma+20_C5^2\delta\alpha\beta_C3^2+(2/5)(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^2\beta\gamma^2/\delta+(1/4)(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5^2\beta^2\gamma/\delta-9_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))_C5\beta\gamma+30_C3^2(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))\delta_C5\alpha+(1/5)\gamma^3(2\gamma_C5+_C5\beta-40\delta_C3^2+\sqrt(4\gamma^2_C5^2+4\gamma_C5^2\beta+_C5^2\beta^2-40\delta\alpha_C5^2))*_C5^2/\delta)
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行波图
参考文献
*谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社
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李志斌编著 《非线性数学物理方程的行波解》 科学出版社
#王东明著 《消去法及其应用》 科学出版社 2002
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#Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
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#Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
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