{
  "statement": "Prove H2: Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
  "starting_statement": null,
  "target_statement": "Let \\(\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)\\), and let \\(u\\in C(\\mathbb R;H^1(\\mathbb T;\\mathbb R))\\) satisfy\n\\[\n(1-\\partial_x^2)u_t+\\partial_x(u+u^2/2)=0\n\\]\nin distributions. Suppose \\(u=C\\) on \\(\\omega\\times(0,T_0)\\), where \\(C\\in\\mathbb R\\), \\(T_0>0\\), and \\(\\omega\\subset\\mathbb T\\) is nonempty and open.\nPut \\(z=u-C\\), \\(\\beta=1+C\\), and\n\\[\nw=(1-\\partial_x^2)^{-1}(\\beta z+z^2/2),\n\\]\nwhere the inverse is the periodic Fourier multiplier \\((1+n^2)^{-1}\\), \\(n\\in\\mathbb Z\\).\n\nFix \\(x_0\\in\\omega\\) and use the interval coordinate \\(x\\in[-\\pi,\\pi]\\) on \\(\\mathbb T\\) with both endpoints representing \\(x_0\\). For \\(\\alpha>2\\), define\n\\[\nH_\\beta(s)=\\beta s^2/2+s^3/6,\\quad\nN_\\alpha(t)=\\int_{-\\pi}^{\\pi}e^{\\alpha x}H_\\beta(z(x,t))\\,dx,\\quad\nW_\\alpha(x,t)=e^{\\alpha x/2}w(x,t).\n\\]\nThen \\(N_\\alpha\\in C^1(\\mathbb R)\\), and for every \\(t\\in\\mathbb R\\),\n\\[\nN_\\alpha'(t)=-\\frac\\alpha2\\int_{-\\pi}^{\\pi}\n\\left[(\\partial_xW_\\alpha)^2+(\\alpha^2/4-1)W_\\alpha^2\\right]dx.\n\\]",
  "case_id": "NE-1-H0-to-H2",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}