KP solitons and total positivity for the Grassmannian

@article{Kodama2011KPSA,
  title={KP solitons and total positivity for the Grassmannian},
  author={Yuji Kodama and Lauren K. Williams},
  journal={Inventiones mathematicae},
  year={2011},
  volume={198},
  pages={637-699}
}
Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian $$Gr_{k,n}$$Grk,n. More recently, several authors (Biondini and Chakravarty, J Math Phys 47:033514, 2006; Biondini and Kodama, J. Phys A Math Gen 36:10519–10536… 
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References

SHOWING 1-10 OF 35 REFERENCES
KP solitons, total positivity, and cluster algebras
TLDR
This paper explains how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation and uses this framework to give an explicit construction of certain soliton contour graphs.
KP solitons in shallow water
The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev–Petviashvili (KP) equation. The KP equation describes weakly dispersive and small
Soliton Solutions of the KP Equation and Application to Shallow Water Waves
The main purpose of this paper is to give a survey of recent developments on a classification of soliton solutions of the Kadomtsev–Petviashvili equation. The paper is self‐contained, and we give
Classification of the line-soliton solutions of KPII
In the previous papers (notably, Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169–90, Biondini G and Chakravarty S 2006 J. Math. Phys. 47 033514), a large variety of line-soliton solutions of the
The Deodhar decomposition of the Grassmannian and the regularity of KP solitons
Young diagrams and N-soliton solutions of the KP equation
We consider N-soliton solutions of the KP equation, An N-soliton solution is a solution u(x, y, t) which has the same set of N line soliton solutions in both asymptotics y → ∞ and y → −∞. The
On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy
We describe the interaction pattern in the x–y plane for a family of soliton solutions of the Kadomtsev–Petviashvili (KP) equation, The solutions considered also satisfy the finite Toda lattice
The direct method in soliton theory
The bilinear, or Hirota's direct, method was invented in the early 1970s as an elementary means of constructing soliton solutions that avoided the use of the heavy machinery of the inverse scattering
Soliton solutions of the Kadomtsev-Petviashvili II equation
We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to the
KP line solitons and Tamari lattices
The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a
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1
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4
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