Stokes phenomena in discrete Painlevé I

Joshi, Nalini; Lustri, Christopher J.

doi:10.1098/rspa.2014.0874

Cited by 15 publications

(50 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we study the exponentially small oscillations present in traveling wave solutions to the seventh-order KdV equation (1). Using exponential asymptotics, we will show that these oscillations are switched on across special curves known as Stokes lines, and recover the condition that GSW solutions appear for ≤ 1 4 .…”

Section: Seventh-order Kdvmentioning

confidence: 88%

“…We first motivated this result by considering exponentially small terms in the seventh-order KdV equation (1), and subsequently a higher order KdV hierarchy (31). This extends the work of Refs.…”

Section: Discussionmentioning

confidence: 99%

“…where is the truncation remainder. We apply this expression to (1), and use (7) to eliminate series terms, leaving…”

Section: Exponential Asymptoticsmentioning

confidence: 99%

“…If < 1 4 , there are no imaginary solutions of (2) . Hence, the GSWs in the solution contain oscillatory terms associated with the singulant (1)…”

Section: Lattice Equations With Parametersmentioning

confidence: 99%

See 3 more Smart Citations

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Joshi

Lustri

2019

Stud Appl Math

Self Cite

View full text Add to dashboard Cite

Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infiniteorder differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite-difference discretization, in which a lattice generalized solitary wave solution is found. K E Y W O R D S asymptotic analysis, solitons and integrable systemsStud Appl Math. 2019;142:359-384.wileyonlinelibrary.com/journal/sapm

show abstract

Section: Seventh-order Kdvmentioning

confidence: 88%

Section: Discussionmentioning

confidence: 99%

“…where is the truncation remainder. We apply this expression to (1), and use (7) to eliminate series terms, leaving…”

Section: Exponential Asymptoticsmentioning

confidence: 99%

“…If < 1 4 , there are no imaginary solutions of (2) . Hence, the GSWs in the solution contain oscillatory terms associated with the singulant (1)…”

Section: Lattice Equations With Parametersmentioning

confidence: 99%

See 2 more Smart Citations

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Joshi

Lustri

2019

Stud Appl Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…This analysis differs in some technical details from the fourth-order equation (3.1) due to the difference terms. We therefore follow the method established in for differential-difference equations in [29], and subsequently utilised for difference equations in [25,26].…”

Section: 2mentioning

confidence: 99%

Standing lattice solitons in the discrete NLS equation with saturation

et al. 2019

View full text Add to dashboard Cite

We consider standing lattice solitons for discrete nonlinear Schrödinger equation with saturation (NLSS), where so-called transparent points were recently discovered. These transparent points are the values of the governing parameter (e.g., the lattice spacing) for which the Peierls-Nabarro barrier vanishes. In order to explain the existence of transparent points, we study a solitary wave solution in the continuous NLSS and analyse the singularities of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic singularities nearest to the real axis is proven and applied to two settings: (i) the fourth-order differential equation arising as the next-order continuum approximation of the discrete NLSS and (ii) the advance-delay version of the discrete NLSS.In the context of (i), the fourth-order differential equation generally does not have solitary wave solutions due to small oscillatory tails. Nevertheless, we show that solitary waves solutions exist for specific values of governing parameter that form an infinite sequence. We present an asymptotic formula for the distance between two subsequent elements of the sequence in terms of the small parameter of lattice spacing. To derive this formula, we used two different analytical techniques: the semi-classical limit of oscillatory integrals and the beyond-all-order asymptotic expansions. Both produced the same result that is in excellent agreement with our numerical data.In the context of (ii), we also derive an asymptotic formula for values of lattice spacing for which approximate standing lattice solitons can be constructed. The asymptotic formula is in excellent agreement with the numerical approximations of transparent points. However, we show that the asymptotic formulas for the cases (i) and (ii) are essentially different and that the transparent points do not generally imply existence of continuous standing lattice solitons in the advance-delay version of the discrete NLSS.

show abstract

On Stokes Phenomena for the Alternate Discrete PI Equation

Joshi

Takei

2017

Trends in Mathematics

View full text Add to dashboard Cite

Stokes phenomena in discrete Painlevé I

Cited by 15 publications

References 37 publications

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Standing lattice solitons in the discrete NLS equation with saturation

On Stokes Phenomena for the Alternate Discrete PI Equation

Contact Info

Product

Resources

About