dorsal/arxiv
View SchemaNonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili equations
| Authors | Nalini Joshi, Johannes A. Petersen, Luke M. Schubert |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9905010 |
| URL | https://arxiv.org/abs/solv-int/9905010 |
Abstract
We study characteristic Cauchy problems for the Korteweg-deVries (KdV) equation $u_t=uu_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation $u_{yy}=\bigl(u_{xxx}+uu_x+u_t\bigr)_x$ with holomorphic initial data possessing nonnegative Taylor coefficients around the origin. For the KdV equation with initial value $u(0,x)=u_0(x)$, we show that there is no solution holomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\mathbb C}^2$ unless $u_0(x)=a_0+a_1x$. This also furnishes a nonexistence result for a class of $y$-independent solutions of the KP equation. We extend this to $y$-dependent cases by considering initial values given at $y=0$, $u(t,x,0)=u_0(x,t)$, $u_y(t,x,0)=u_1(x,t)$, where the Taylor coefficients of $u_0$ and $u_1$ around $t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic solution around the origin in ${\mathbb C}^3$ unless $u_0$ and $u_1$ are polynomials of degree 2 or lower.
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"abstract": "We study characteristic Cauchy problems for the Korteweg-deVries (KdV)\nequation $u_t=uu_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation\n$u_{yy}=\\bigl(u_{xxx}+uu_x+u_t\\bigr)_x$ with holomorphic initial data\npossessing nonnegative Taylor coefficients around the origin. For the KdV\nequation with initial value $u(0,x)=u_0(x)$, we show that there is no solution\nholomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\\mathbb C}^2$ unless\n$u_0(x)=a_0+a_1x$. This also furnishes a nonexistence result for a class of\n$y$-independent solutions of the KP equation. We extend this to $y$-dependent\ncases by considering initial values given at $y=0$, $u(t,x,0)=u_0(x,t)$,\n$u_y(t,x,0)=u_1(x,t)$, where the Taylor coefficients of $u_0$ and $u_1$ around\n$t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic\nsolution around the origin in ${\\mathbb C}^3$ unless $u_0$ and $u_1$ are\npolynomials of degree 2 or lower.",
"arxiv_id": "solv-int/9905010",
"authors": [
"Nalini Joshi",
"Johannes A. Petersen",
"Luke M. Schubert"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili equations",
"url": "https://arxiv.org/abs/solv-int/9905010"
},
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