Hi there, I'm learning about solitons and chanced upon this pdf talking about the inverse scattering method. However, I'm stuck trying to derive the coefficients using the LAX method (pg 5 of the attached pdf or from http://arxiv.org/pdf/0905.4746.pdf). Hope that someone can help shed some light on it...Thanks in advance! =)(adsbygoogle = window.adsbygoogle || []).push({});

[itex]L=-\partial_x^{\phantom{0}2}+u(x,t)[/itex]

[itex]A=\alpha_3\partial_x^{\phantom{0}3}+\alpha_2\partial_x^{\phantom{0}2}+ \alpha_1\partial_x+\alpha_0,[/itex]

where [itex]\alpha_j (j=0,1,2,3)[/itex] may depend on x and t.

Substituting these into the equation

[itex]L_t+LA-AL=0[/itex]

[itex]u_t+[-\partial_x^{\phantom{0}2} +u(x,t)][\alpha_3\partial_x^{\phantom{0}3}+\alpha_2\partial_x^{\phantom{0}2}+ \alpha_1\partial_x+ \alpha_0]-[\alpha_3\partial_x^{\phantom{0}3}+\alpha_2\partial_x^{\phantom{0}2}+ \alpha_1\partial_x+\alpha_0][-\partial_x^{\phantom{0}2} +u(x,t)]=0[/itex]

Focusing on the LHS,

[itex]u_t[/itex]

[itex]-\partial_x^{\phantom{0}2}(\alpha_3)-\partial_x^{\phantom{0}5}-\partial_x^{\phantom{0}2}(\alpha_2)-\partial_x^{\phantom{0}4}-\partial_x^{\phantom{0}2}(\alpha_1)-\partial_x^{\phantom{0}3}-\partial_x^{\phantom{0}2}(\alpha_0)[/itex]

[itex]+u\alpha_3\partial_x^{\phantom{0}4}+u\alpha_2\partial_x^{\phantom{0}2}+u\alpha_1\partial_x+u\alpha_0[/itex]

[itex]+\alpha_3\partial_x^{\phantom{0}5}+\alpha_2\partial_x^{\phantom{0}4}+ \alpha_1\partial_x^{\phantom{0}3}+\alpha_0\partial_x^{\phantom{0}2}[/itex]

[itex]-\alpha_3u_{xxx}-\alpha_2u_{xx}-\alpha_1u_{x}[/itex]

Rearranging in terms of [itex]\partial_x^{\phantom{0}j} (j=0,1,2,3)[/itex] terms,

[itex]u_t[/itex]

[itex]+(\alpha_3 -1)\partial_x^{\phantom{0}5}+(\alpha_2 -1)\partial_x^{\phantom{0}4}+(\alpha_1+u\alpha_3-1)\partial_x^{\phantom{0}3}+(\alpha_0+u \alpha_2)\partial_x^{\phantom{0}2}+(u \alpha_1)\partial_x[/itex]

[itex]-\alpha_3u_{xxx}-\alpha_2u_{xx}-\alpha_1u_{x}[/itex]

[itex]-\partial_x^{\phantom{0}2} (\alpha_3+\alpha_2+\alpha_1+\alpha_0)[/itex]

I'm not sure what to do with the last string of terms which involve partially differentiating the [itex]\alpha[/itex] terms by x and as such how to obtain the expression for the coefficients in eqn (4.6). Any help is much appreciated. Thanks! =)

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# Inverse Scattering Method (help please)

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