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ExactN-Wave Solutions of Generalized Burgers Equations

1996
*
Studies in applied mathematics (Cambridge)
*

We sincerely thank Professor Sir James Lighthill

doi:10.1002/sapm1996974349
fatcat:txpeohbxi5fdnb73mh42ky5xpm
*and*Professor D. G. Crighton for their careful perusal*of*the original version*of*the paper, constructive criticism*and*many useful suggestions. ... For almost all these*equations*no*Hopf*-*Cole*-like*transformation*exists. ... Conclusions The non-planar Burgers*equation*(3.1), like most*generalized*Burgers*equations*, does not admit a*Hopf*-*Cole**transformation*,*and*therefore, must be treated directly. ...##
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Nonlinear dynamics: Challenges and perspectives

2005
*
Pramana (Bangalore)
*

In spite

doi:10.1007/bf02706208
fatcat:2g347braujdhdcigo27w6ukemu
*of*these developments, several challenges, some*of*them fundamental*and*others on the efficacy*of*these methods in developing cutting edge technologies, remain to be tackled. ... The study*of*nonlinear dynamics has been an active area*of*research since 1960s, after certain path-breaking discoveries, leading to the concepts*of*solitons, integrability, bifurcations, chaos*and*spatio-temporal ... Acknowledgement This work is sponsered by the Department*of*Science*and*Technology, Government*of*India. The author thanks Dr M Senthilvelan for assistance in the preparation*of*this article. ...##
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(2+1)-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions

2012
*
Physical Review Letters
*

The simple

doi:10.1103/physrevlett.109.170602
pmid:23215169
fatcat:l254bhepjjaujfodg67qesuiaa
*Hopf*-*Cole**transformation*, h(x, t)=(2ν/λ)lnZ(x, t)), maps KPZ stochastic growth onto the equilibrium statistical physics*of*directed polymers in a random medium (DPRM); i.e., a Schrodinger*equation*... Motivated by the*exact*1+1 results*of*Sasamoto & Spohn [11] , we conjecture that the*solution**of*the 2+1 KPZ*equation*for 3d wedge (i.e. ...##
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Geometrical and Dynamical Aspects of Nonlinear Higher-Order Riccati Systems
[article]

2015
*
arXiv
*
pre-print

Using these results we show the existence

arXiv:1507.00512v1
fatcat:65byflc3irgolcgcb57hjvr4wq
*of*a Lagrangian description for the related system,*and*the*Painlevé*II*equation*is analysed. ... We study a geometrical formulation*of*the nonlinear second-order Riccati*equation*(SORE) in terms*of*the projective vector field*equation*on S^1, whichn in turn is related to the stability algebra*of*Virasoro ... J FC*and*MFR acknowledge support from research projects MTM-2012-33575,*and*E24/1 (DGA). ...##
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Detecting discrete integrability: the singularity approach
[article]

2018
*
arXiv
*
pre-print

We describe the various types

arXiv:1809.00853v1
fatcat:cixwwk7bqfbihfdvshadrfxedu
*of*singularities that can arise for second order rational mappings*and*we discuss the historical*and*present-day, practical, role the singularity confinement property plays ... Acknowledgements RW*and*TM would like to acknowledge support from the Japan Society for the Promotion*of*Science (JSPS),*through*JSPS grants number 18K03355*and*18K13438, respectively. ... These have the form x n+1 x n x n−1 + ax n x n−1 + bx n−1 + c = 0, (17)*and*can be linearised*through*a*Cole*-*Hopf**transformation*x n = w n+1 /w n to the linear*equation*w n+2 + a n w n+1 + b n w n + c ...##
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On the origins of Riemann-Hilbert problems in mathematics
[article]

2020
*
arXiv
*
pre-print

The secondary purpose

arXiv:2003.14374v2
fatcat:keqqyloszjg75njjcnklx2aqpe
*of*this note is to invite a new*generation**of*mathematicians to the fascinating world*of*Riemann-Hilbert techniques*and*their modern appearances in nonlinear mathematical physics. ... We set out to achieve this goal with six examples, including a new proof*of*the integro-differential*Painlevé*-II formula*of*Amir, Corwin, Quastel that enters in the description*of*the KPZ crossover distribution ... Using the*Hopf*-*Cole**transformation*H(X, T ) := − ln Z(X, T ) the SHE is mapped to the celebrated Kardar-Parisi-Zhang (KPZ)*equation*H T = 1 2 H ZZ − 1 2 (H Z ) 2 +Ẇ (9.1) which has become the quintessential ...##
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Complete Integrability of Shock Clustering and Burgers Turbulence

2011
*
Archive for Rational Mechanics and Analysis
*

We consider scalar conservation laws with convex flux

doi:10.1007/s00205-011-0461-8
fatcat:ukf23xfafjf3bhrbmqtgckb7z4
*and*random initial data. The*Hopf*-Lax formula induces a deterministic evolution*of*the law*of*the initial data. ... Our results are rigorous for N-dimensional approximations*of*the Lax*equations*,*and*yield formulas for the limit N →∞. ... This work is part*of*a*general*program developed with Ravi Srinivasan*and*I am particularly indebted to him. Thanks also to Michele Benzi for informing me*of*[32] . ...##
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Extremal paths, the stochastic heat equation, and the three-dimensional Kardar-Parisi-Zhang universality class

2013
*
Physical Review E
*

common scaling phenomena exhibited by the pt-pt directed polymer in a random medium (DPRM), the stochastic heat

doi:10.1103/physreve.88.042118
pmid:24229127
fatcat:2bmf6ffxebh3vljmkr3zclr4zm
*equation*(SHE) with multiplicative noise in three dimensions,*and*kinetic roughening phenomena ... within this 3D KPZ class, revealing shared values for the limit distribution skewness*and*kurtosis, along with universal first*and*second moments. ... formal analysis has been based firmly on the SHE, rather than the KPZ*equation*itself, i.e., an ultraviolet regularization*of*KPZ via*Hopf*-*Cole**transformation*to the SHE [13] . ...##
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The Kardar-Parisi-Zhang equation - a statistical physics perspective
[article]

2016
*
arXiv
*
pre-print

its

arXiv:1601.00499v1
fatcat:gix7bazl4ngn5k6gnzm76x4caa
*generalization*to several components which is used to study equilibrium time correlations*of*anharmonic chains*and**of*the discrete nonlinear Schrödinger*equation*. ... The article covers the one-dimensional Kardar-Parisi-Zhang*equation*, weak drive limit, universality, directed polymers in a random medium, replica*solutions*, statistical mechanics*of*line ensembles,*and*... As noted already by*Hopf*[41]*and**Cole*[42] for the dissipative Burgers*equation*, (4.11) with zero noise, the*transformation*Z(x, t) = e λh(x,t) (5.1) "linearizes" the KPZ*equation*as ∂ t Z(x, t) = ...##
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Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
[article]

2022
*
arXiv
*
pre-print

In turn, such universal limits are described by the integro-differential

arXiv:2201.12941v1
fatcat:mcqthczxlrd5thhzrnsl3deusi
*Painlev*\'e II*equation*,*and*in particular they connect the random matrix models considered with the narrow wedge*solution*to the ... We consider one-cut regular polynomial potentials*and*a large class*of*multiplicative statistics. ... G. wishes to thank Ivan Corwin*and*Alexandre Krajenbrink for many helpful conversations*and*Alexei Borodin for comments on the earlier version*of*this manuscript. G. ...##
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Notes on Lie symmetry group methods for differential equations
[article]

2021
*
arXiv
*
pre-print

Fundamentals on Lie group methods

arXiv:1901.01543v8
fatcat:g6mgelak5jbbljk6rfax5zkacm
*and*applications to differential*equations*are surveyed. ... Many examples are included to elucidate their extensive applicability for analytically solving both ordinary*and*partial differential*equations*. ... Willy Hereman for testing the symmetry calculations*of*some Examples with his Maxima program symmgrp2020*and*spotting various misprints. ...##
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A Singular Mathematical Promenade
[article]

2017
*
arXiv
*
pre-print

We play with a bit

arXiv:1612.06373v4
fatcat:3x2nfhmw3jgo3lfwvyt7njnssq
*of*algebra, topology, geometry, complex analysis*and*computer science. ... Hopefully, some motivated undergraduates*and*some more advanced mathematicians will enjoy some*of*these panoramas. ... For instance, an*explicit*formula for the*generating*series ∑ a n t n would give the*exact*exponential growth rate*of*a n . Unfortunately, I was not able to compute this series . ...##
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A pedestrian's view on interacting particle systems, KPZ universality and random matrices

2010
*
Journal of Physics A: Mathematical and Theoretical
*

We then introduce the phenomenological Kardar-Parisi-Zhang (KPZ)

doi:10.1088/1751-8113/43/40/403001
fatcat:jrz5o5ikknc5dnlrkx4756ltie
*equation**and*explain the associated universality conjecture for surface fluctuations in growth models. ... These notes are based on lectures delivered by the authors at a Langeoog seminar*of*SFB/TR12 "Symmetries*and*universality in mesoscopic systems" to a mixed audience*of*mathematicians*and*theoretical physicists ... We are grateful to H Spohn, P L Ferrari*and*to an unknown referee for valuable remarks,*and*to M Prähofer*and*K A Takeuchi for providing figures. ...##
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Riemann surfaces for KPZ with periodic boundaries

2020
*
SciPost Physics
*

Known

doi:10.21468/scipostphys.8.1.008
fatcat:yv5fzttsercajh77gbppixdlhm
*exact*results for fluctuations*of*the KPZ height with periodic boundaries are expressed in terms*of*meromorphic functions on this Riemann surface, summed over all the sheets*of*a covering map to ... Connections to stationary large deviations, particle-hole excitations*and*KdV solitons are discussed. ... ∂ t h λ = 1 2 ∂ 2 x h λ −λ(∂ x h λ ) 2 +η with η a unit space-time Gaussian white noise,*and*which is related by the*Cole*-*Hopf**transform*Z λ (x, t) = e −2λh λ (x,t) to the stochastic heat*equation*with ...##
###
Riemann surfaces for KPZ with periodic boundaries
[article]

2019
*
arXiv
*
pre-print

Known

arXiv:1908.08907v2
fatcat:mnfdprkswrgkxbexnvrahhc6i4
*exact*results for fluctuations*of*the KPZ height with periodic boundaries are expressed in terms*of*meromorphic functions on this Riemann surface, summed over all the sheets*of*a covering map to ... Connections to stationary large deviations, particle-hole excitations*and*KdV solitons are discussed. ... ∂ t h λ = 1 2 ∂ 2 x h λ − λ(∂ x h λ ) 2 + η with η a unit space-time Gaussian white noise,*and*which is related by the*Cole*-*Hopf**transform*Z λ (x, t) = e −2λh λ (x,t) to the stochastic heat*equation*with ...