博欣内斯克方程

博欣内斯克方程是一个二元非线性偏微分方程：^[1]

${\frac {\partial ^{2}u}{\partial t^{2}}}-{\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u^{2}}{\partial ^{2}y^{2}}}+{\frac {\partial ^{4}u}{\partial x^{4}}}=0$

博欣内斯克方程有如下行波解：^[2]

$u(x,t)=(3/2*(1-c^{2}))*sech((1/2)*{\sqrt {(1-c^{2})*(x-c*t))^{2}}}$

$u(x,y,t)={\frac {4*k^{4}+\omega ^{2}-k^{2}}{2*l^{2}}}+6*k^{2}*{\frac {\cosh(k(x+x0)+l*(y+y0)-\omega *t)^{2}}{l^{2}}}$

$u(x,y,t):={\frac {(1/2)*(4*k^{4}*cosh(k*x+k*x0+l*y+l*y0-\omega *t)^{2}+\omega ^{2}*cosh(k*x+k*x0+l*y+l*y0-\omega *t)^{2}-k^{2}*cosh(k*x+k*x0+l*y+l*y0-\omega *t)^{2}-12*k^{4})}{/(cosh(k*x+k*x0+l*y+l*y0-\omega *t)^{2}*l^{2})}}$

参考文献

[1] [1]
Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations p339-376 Academy Press

[2] [2]
Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations p409 Academy Press

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