Antiderivative of a function arised in KdV equationI am computing the third non-trivial conservation law of KdV equation u x + 6 u u x + u x x x = 0 via the power series expansion method (Here we consider real-valued solutions only).I was given an equivalent form of the PDE: ( 2 u 3 + 5 u x 2 ) t + ( 36 u 3 u x + 6 u 2 u x x x + 10 u x u x x x x + 60 u x 3 + 60 u u x u x x ) To finish the job one needs to express ( 36 u 3 u x + 6 u 2 u x x x + 10 u x u x x x x + 60 u x 3 + 60 u u x u x x ) in a form ( ⋯ ) x It is clear that ( 9 u 4 ) x is an antiderivative of 36 u 3 u x , but what is the antiderivative of 6 u 2 u x x x + 10 u x u x x x x + 60 u x 3 + 60 u u x u x x in terms of derivatives of u?

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Antiderivative of a function arised in KdV equationI am computing the third non-trivial conservation law of KdV equation       u          x        +  6  u      u          x        +      u          x      x      x        =  0 via the power series expansion method (Here we consider real-valued solutions only).I was given an equivalent form of the PDE:            (      2              u        3            +      5              u                  x                2            )              t        +      (    36          u      3              u              x              +    6          u      2              u              x        x        x              +    10          u              x                    u              x        x        x        x              +    60          u              x            3        +    60    u          u              x                    u              x        x              )  To finish the job one needs to express      (    36          u      3              u              x              +    6          u      2              u              x        x        x              +    10          u              x                    u              x        x        x        x              +    60          u              x            3        +    60    u          u              x                    u              x        x              )   in a form   (  ⋯      )          x      It is clear that   (  9      u    4        )          x       is an antiderivative of   36      u    3        u          x      , but what is the antiderivative of  6      u    2        u          x      x      x        +  10      u          x            u          x      x      x      x        +  60      u          x        3    +  60  u      u          x            u          x      x       in terms of derivatives of u?

Waldruhylm · Accepted Answer

Step 1I am fairly certain that this is an impossible task.The       u          x            u          x      x      x      x       term is okay because it can be produced by the following two terms:  (      u    x        u          x      x      x            )    x    =      u          x      x            u          x      x      x        +      u    x        u          x      x      x      x           and     (      u          x      x        2        )    x    =  2      u          x      x            u          x      x      x        .Now consider   (  u      u    x    2        )    x    =      u    x    3    +  2  u      u    x        u          x      x           and     (      u    2        u          x      x            )    x    =  2  u      u    x        u          x      x        +      u    2        u          x      x      x        .Step 2To produce the coefficients in your equation, using above two terms will leave out one term from       u    2        u          x      x      x        ,    u      u    x        u          x      x        ,        u    x    3    ,, any of which don&#039;t have closed form antiderivatives.To see this, we can use a qualitative argument. All of above terms have three copies of u, and totally 3 derivatives w.r.t. x. To find the antiderivative, we need three copies of u and total 2 derivatives to distribute among three u&#039;s, the only possible choices are   u      u    x    2   and       u    2        u          x      x      . Thus to produce all three, the coefficients must satisfy certain relations.The closest is:   u  (  u      u          x      x        −      u    x    2        /    2      )    x    =      u    2        u          x      x      x        .

I am computing the third non-trivial conservation law of KdV equation u_{x}+6u u_{x}+u_{x x x}=0

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