• Corpus ID: 119275130

On the tritronquée solutions of P$_I^2$

  title={On the tritronqu{\'e}e solutions of P\$\_I^2\$},
  author={Andrei A. Kapaev and Christian Klein and Tamara Grava},
  journal={arXiv: Mathematical Physics},
For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,...,6$, called {\em tritronqu\'ee}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal… 

On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations

It is argued that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P$$_I$$I) equation or its fourth-order analogue P$$-I^2$$I2.

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

. We provide a general solution to a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of

Truncated Solutions of Painlevé Equation PV

  • R. Costin
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2018
We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlev\'e equation with nonzero parameters, valid in half

Tronquée Solutions of the Third and Fourth Painlevé Equations

  • X. Xia
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2018
Recently in a paper by Lin, Dai and Tibboel, it was shown that the third and fourth Painleve equations have tronquee and tritronquee solutions. We obtain global information about these tronquee and

Spectral approach to Korteweg-de Vries equations on the compactified real line. (English)

  • Mathematics
  • 2022
Summary: We present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation

Numerical Approach to Painlevé Transcendents on Unbounded Domains

  • C. KleinN. Stoilov
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2018
A multidomain spectral approach for Painlev\'e transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic



Asymptotics of the instantons of Painleve I

The 0-instanton solution of Painlev\'e I is a sequence $(u_{n,0})$ of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models and

Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

AbstractWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation $$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$for

Weakly nonlinear solutions of equationP12

AbstractUsing the isomonodromic deformation method, we study the equation P12, $$\frac{1}{{10}}y^{(4)} + y''y + \frac{1}{2}(y')^2 + y^3 = x$$ , which is the first higher equation in the hierarchy of

On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

It is argued that the critical behavior near the point of “gradient catastrophe” of the solution of the Cauchy problem for the focusing nonlinear Schrödinger equation is approximately described by a particular solution to the Painlevé-I equation.

Квантования высших гамильтоновых аналогов уравнений Пенлеве I и II с двумя степенями свободы@@@“Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom

We construct a solution of an analog of the Schr\"{o}dinger equation for the Hamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy.

The KdV hierarchy: universality and a Painlevé transcendent

We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e\to 0$. For negative analytic initial data with a single negative hump, we prove that for

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

Resurgent transseries have recently been shown to be a very powerful construction for completely describing nonperturbative phenomena in both matrix models and topological or minimal strings. These

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

We consider unitary random matrix ensembles $$Z_{n,s,t}^{-1}e^{-n tr V_{s,t}(M)}dM$$ on the space of Hermitian n × n matrices M, where the confining potential Vs,t is such that the limiting mean

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian H1(z, t, q1, q2, p1, p2) corresponding to the second equation P12 in the Painlevé I hierarchy. This solution is