Group Classification of a Generalized Lane-Emden System

Table 4

Lie symmetries for 𝑛=3, for various functions 𝐻(𝑣) and 𝐺(𝑢).

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𝐻(𝑣) arbitrary, 𝐺(𝑢)=𝑐, 𝑐 a constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥⁻²𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+8𝑣)𝜕𝑢

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢) arbitrary

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥⁻²𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+8𝑢)𝜕𝑣

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥⁻²𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+8𝑢)𝜕𝑣, 𝑋₄𝜕=𝛽ln𝑥𝜕𝑣−2𝑥⁻²𝜕𝜕𝑢,

𝑋₅=𝛽𝑥²𝜕𝜕𝜕𝑣−8𝜕𝑢,

𝑋₆𝜕=96𝑥𝜕𝑥−24𝑑𝑥²𝜕𝜕𝑢+(𝛽𝑑𝑥⁴𝜕+192𝑣)𝜕𝑣,

𝑋₇=(192𝑢+24𝑑𝑥⁴)𝜕𝜕𝑢+(192𝑣−𝛽𝑑𝑥²+24𝛼𝑥²)𝜕𝜕𝑣

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑢^−𝑝, 𝛼,𝑝, and 𝛽 constants (𝛽,𝑝≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥⁻²𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+8𝑢)𝜕𝑣,

𝑋₄𝜕=4𝑥𝜕𝜕𝑥+8𝑢𝜕𝑢−(𝛼𝑝𝑥²𝜕−8𝑣+8𝑝𝑣)𝜕𝑣

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑒^−𝑘𝑢, 𝛼,𝑘, and 𝛽 constants (𝛽,𝑘≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥⁻²𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+8𝑢)𝜕𝑣,

𝑋₄𝜕=8𝜕𝑢−(𝛼𝑘𝑥²𝜕+8𝑘𝑣)𝜕𝑣

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽ln𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥⁻²𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+8𝑢)𝜕𝑣,

𝑋₄𝜕=4𝑥𝜕𝜕𝑥+8𝑢𝜕𝑢+(8𝑣−𝛽𝑥²)𝜕𝜕𝑣,

𝑋₅=𝑥⁻¹𝜕𝜕𝑥−2𝑥⁻²𝑢𝜕𝜕𝑢−(2𝑥⁻²𝜕𝑣−𝛽ln𝑥)𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝑐, 𝑐 constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥⁻²𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+8𝑣)𝜕𝑢,𝑋₄𝜕=𝑏ln𝑥𝜕𝑢−2𝑥⁻²𝜕𝜕𝑣,

𝑋₅=𝑏𝑥²𝜕𝜕𝜕𝑢−8𝜕𝑣,

𝑋₆𝜕=96𝑥𝜕𝑥+(𝑏𝑐𝑥⁴𝜕+192𝑢)𝜕𝑢−24𝑐𝑥²𝜕𝜕𝑣,

𝑋₇=(24𝑎𝑥²+192𝑢−𝑏𝑐𝑥⁴)𝜕𝜕𝑢+(24𝑐𝑥²𝜕+192𝑣)𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁𝜕=𝐹(𝑥)𝜕𝑣,𝑋₂𝜕=𝑊(𝑥)𝜕𝑢,𝑋₃𝜕=𝑢𝜕𝜕𝑢+𝑣𝜕𝑣,

𝑋₄𝜕=𝑏𝑣𝜕𝜕𝑢+𝛽𝑢𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝, 𝛽 and 𝑝 constants (𝛽,𝑝≠0)

𝑋₁𝜕=(𝑏𝑥+𝑏𝑝𝑥)𝜕𝜕𝑥+4𝑏𝑢𝜕𝜕𝑢+(2𝑎+2𝑏𝑣−2𝑎𝑝−2𝑏𝑝𝑣)𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0) 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝛽 and 𝑘 constants (𝛽,𝑘≠0)

𝑋₁𝜕=𝑏𝑘𝑥𝜕𝜕𝑥+4𝑏𝜕𝜕𝑢−(2𝑎𝑘+2𝑏𝑘𝑣)𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣^−𝑚, 𝑎,𝑚, and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝑐, 𝑐 a constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥⁻²𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+8𝑣)𝜕𝑢,

𝑋₄𝜕=4𝑥𝜕𝑥+(8𝑢−𝑎𝑚𝑥²𝜕−8𝑚𝑢)𝜕𝜕𝑢+8𝑣𝜕𝑣

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𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁𝜕=(𝛽𝑥+𝛽𝑚𝑥)𝜕𝜕𝑥+(2𝛼−2𝛼𝑚+2𝛽𝑢−2𝛽𝑚𝑢)𝜕𝜕𝑢+4𝛽𝑣𝜕𝑣

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𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚 and  𝑏 constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝,𝛽 and 𝑝 constants (𝛽,𝑝≠0)

𝑋₁𝜕=(𝑝𝑚𝑥−𝑥)𝜕𝜕𝑥+(2𝑚𝑢−2𝑢)𝜕𝜕𝑢+(2𝑝𝑣−2𝑣)𝜕𝑣

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𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚 and 𝑏 constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢,𝛽 and 𝑘 constants (𝛽,𝑘≠0)

𝑋₁𝜕=𝑘𝑚𝑥𝜕𝜕𝑥+(2𝑚−2)𝜕𝜕𝑢+2𝑘𝑣𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑒^−𝑚𝑣, 𝑎,𝑚, and 𝑏 constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝑐, 𝑐 constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥⁻²𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+8𝑣)𝜕𝑢,𝑋₄=(𝑎𝑚𝑥²𝜕+8𝑚𝑢)𝜕𝜕𝑢−8𝜕𝑣

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𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚 and 𝑏 constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants(𝛽≠0)

𝑋₁𝜕=𝛽𝑚𝑥𝜕𝜕𝑥−(2𝛼𝑚+2𝛽𝑚𝑢)𝜕𝜕𝑢+4𝛽𝜕𝑣

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𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚 and 𝑏 constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝,𝛽 and 𝑝 constants (𝛽,𝑝≠0)

𝑋₁𝜕=𝑚𝑝𝑥𝜕𝜕𝑥+2𝑚𝑢𝜕𝜕𝑢+(2𝑝−2)𝜕𝑣

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𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚 and 𝑏 constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝛽 and 𝑘 constants(𝛽,𝑘≠0)

𝑋₁𝜕=𝑚𝑘𝑥𝜕𝜕𝑥+2𝑚𝜕𝜕𝑢+2𝑘𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏ln𝑣, 𝑎,𝑚, and 𝑏 constants(𝑏≠0), 𝐺(𝑢)=𝑐, 𝑐 constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥⁻²𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+8𝑣)𝜕𝑢,𝑋₄𝜕=4𝑥𝜕𝑥+(8𝑢−𝑏𝑥²)𝜕𝜕𝜕𝑢+8𝑣,𝜕𝑣

𝑋₅=𝑥⁻¹𝜕𝜕𝑥−(2𝑥⁻²𝜕𝑢−𝑏ln𝑥)𝜕𝑢−2𝑥⁻²𝑣𝜕𝜕𝑣

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