霍赫洛夫-沯波咯慈卡娅方程

**霍赫洛夫 - 沯波咯慈卡婭方程**(Khokhlov--Zabolotskaya equation) 是一個非線性偏微分方程：

$$
u_{xx}+[(\alpha *u+\beta )*u_{y}]_{y}=0
$$

## 解析解

霍赫洛夫 - 沯波咯慈卡婭方程有行波解：

　
: p[2] := 1.32+1.4934776966447732662\*(1.56+1.7969454312181156991\*x^1.2+1.2\*C[2]^1.2\*y^1.2)^1.2

　
: p[3] := 1.32+1.4934776966447732662\*(.2707963267948966192-1.4974545260150964159\*x^1.2-1.\*C[2]^1.2\*y^1.2)^1.2

　
: p[7] := 1.32+1.4934776966447732662((55.009468881881296225-14.965237496723309046\*I)\*sqrt(1.-.

　
: 66321499013806706114\*JacobiNS(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2-(.38969456396968710805\*I)\*JacobiNS(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2)\*sqrt(1.-.66321499013806706114\*JacobiNS(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2+(.38969456396968710805\*I)\*JacobiNS(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2)\*EllipticF((.84629952125971224961+.23023442302651244686\*I)\*JacobiNS(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3), .86217948717948717949-.50660293316059324046\*I)/sqrt(3000.\*JacobiNS(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^4-6725.\*JacobiNS(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2+5070.))^1.2

　
: p[8] := 1.32+1.4934776966447732662\*(-(34.214441730088728277\*I)\*sqrt(1.+2.1356058039711429821\*JacobiDN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2)\*sqrt(1.-2.2476058039711429821\*JacobiDN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2)\*EllipticF((1.4613712067681992557\*I)\*JacobiDN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3), 1.0258869993454412308\*I)/sqrt(3000.\*JacobiDN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^4+70.\*JacobiDN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2-625.))^1.2

　
: p[9] := 1.32+1.4934776966447732662\*((38.347855408516105018-11.263642905975212858\*I)\*sqrt(1.-1.3164251207729468599\*JacobiCN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2-(.84634523908200302082\*I)\*JacobiCN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2)\*sqrt(1.-1.3164251207729468599\*JacobiCN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2+(.84634523908200302082\*I)\*JacobiCN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2)\*EllipticF((1.2003002147146243158+.35255564762321759608\*I)\*JacobiCN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3), .84115756322748992840-.54079011994043611908\*I)/sqrt(5070.\*JacobiCN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^4-5450.\*JacobiCN(1.68+1.9520491881558575047\*x^1.2+1.2\*C[2]^1.2\*y^1.2, 1.3)^2+2070.))^1.2

p[10] := 1.32+1.4934776966447732662\*arctan(1/sqrt(1.2\*csc(1.56+1.7969454312181156991\*x^1.2+1.2\*C[2]^1.2\*y^1.2)^2-1.2))^1.2

　
: p[11] := 1.32+1.4934776966447732662\*arctan(1/sqrt(1.2\*sec(1.56+1.7969454312181156991\*x^1.2+1.2\*C[2]^1.2\*y^1.2)^2-1.2))^1.2

　
: p[12] := 1.32+1.4934776966447732662\*arctan(1/sqrt(1.2\*sech(1.56+1.7969454312181156991\*x^1.2+1.2\*C[2]^1.2\*y^1.2)^2-1.2))^1.2

　
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## 參考文獻

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