Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local Isometric immersions of pseudo-spherical surfaces and evolution
  equations   The class of differential equations describing pseudo-spherical surfaces,
first introduced by Chern and Tenenblat [3], is characterized by the property
that to each solution of a differential equation, within the class, there
corresponds a 2-dimensional Riemannian metric of curvature equal to $-1$. The
class of differential equations describing pseudo-spherical surfaces carries
close ties to the property of complete integrability, as manifested by the
existence of infinite hierarchies of conservation laws and associated linear
problems. As such, it contains many important known examples of integrable
equations, like the sine-Gordon, Liouville and KdV equations. It also gives
rise to many new families of integrable equations. The question we address in
this paper concerns the local isometric immersion of pseudo-spherical surfaces
in ${\bf E}^{3}$ from the perspective of the differential equations that give
rise to the metrics. Indeed, a classical theorem in the differential geometry
of surfaces states that any pseudo-spherical surface can be locally
isometrically immersed in ${\bf E}^{3}$. In the case of the sine-Gordon
equation, one can derive an expression for the second fundamental form of the
immersion that depends only on a jet of finite order of the solution of the
pde. A natural question is to know if this remarkable property extends to
equations other than the sine-Gordon equation within the class of differential
equations describing pseudo-spherical surfaces. In an earlier paper [11], we
have shown that this property fails to hold for all other second order
equations, except for those belonging to a very special class of evolution
equations. In the present paper, we consider a class of evolution equations for
$u(x,t)$ of order $k\geq 3$ describing pseudo-spherical surfaces. We show that
whenever an isometric immersion in ${\bf E}^3$ exists, depending on a jet of
finite order of $u$, then the coefficients of the second fundamental forms are
functions of the independent variables $x$ and $t$ only.

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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th