Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inverse scattering transforms for the focusing and defocusing mKdV
  equations with nonzero boundary conditions   We explore systematically a rigorous theory of the inverse scattering
transforms with matrix Riemann-Hilbert problems for both focusing and
defocusing modified Korteweg-de Vries (mKdV) equations with non-zero boundary
conditions (NZBCs) at infinity. Using a suitable uniformization variable, the
direct and inverse scattering problems are proposed on a complex plane instead
of a two-sheeted Riemann surface. For the direct scattering problem, the
analyticities, symmetries and asymptotic behaviors of the Jost solutions and
scattering matrix, and discrete spectra are established. The inverse problems
are formulated and solved with the aid of the Riemann-Hilbert problems, and the
reconstruction formulae, trace formulae, and theta conditions are also found.
In particular, we present the general solutions for the focusing mKdV equation
with NZBCs and both simple and double poles, and for the defocusing mKdV
equation with NZBCs and simple poles. Finally, some representative
reflectionless potentials are in detail studied to illustrate distinct wave
structures for both focusing and defocusing mKdV equations with NZBCs.

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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn