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ExactN-Wave Solutions of Generalized Burgers Equations

P. L. Sachdev, K. T. Joseph, B. Mayil vaganan
1996 Studies in applied mathematics (Cambridge)  
We sincerely thank Professor Sir James Lighthill 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.  ... 
doi:10.1002/sapm1996974349 fatcat:txpeohbxi5fdnb73mh42ky5xpm

Nonlinear dynamics: Challenges and perspectives

M. Lakshmanan
2005 Pramana (Bangalore)  
In spite 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.  ... 
doi:10.1007/bf02706208 fatcat:2g347braujdhdcigo27w6ukemu

(2+1)-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions

Timothy Halpin-Healy
2012 Physical Review Letters  
The simple 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.  ... 
doi:10.1103/physrevlett.109.170602 pmid:23215169 fatcat:l254bhepjjaujfodg67qesuiaa

Geometrical and Dynamical Aspects of Nonlinear Higher-Order Riccati Systems [article]

José F. Cariñena, Partha Guha, Manuel F. Rañada
2015 arXiv   pre-print
Using these results we show the existence 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).  ... 
arXiv:1507.00512v1 fatcat:65byflc3irgolcgcb57hjvr4wq

Detecting discrete integrability: the singularity approach [article]

Basil Grammaticos, Alfred Ramani, Ralph Willox, Takafumi Mase
2018 arXiv   pre-print
We describe the various types 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  ... 
arXiv:1809.00853v1 fatcat:cixwwk7bqfbihfdvshadrfxedu

On the origins of Riemann-Hilbert problems in mathematics [article]

Thomas Bothner
2020 arXiv   pre-print
The secondary purpose 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  ... 
arXiv:2003.14374v2 fatcat:keqqyloszjg75njjcnklx2aqpe

Complete Integrability of Shock Clustering and Burgers Turbulence

Govind Menon
2011 Archive for Rational Mechanics and Analysis  
We consider scalar conservation laws with convex flux 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] .  ... 
doi:10.1007/s00205-011-0461-8 fatcat:ukf23xfafjf3bhrbmqtgckb7z4

Extremal paths, the stochastic heat equation, and the three-dimensional Kardar-Parisi-Zhang universality class

Timothy Halpin-Healy
2013 Physical Review E  
common scaling phenomena exhibited by the pt-pt directed polymer in a random medium (DPRM), the stochastic heat 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] .  ... 
doi:10.1103/physreve.88.042118 pmid:24229127 fatcat:2bmf6ffxebh3vljmkr3zclr4zm

The Kardar-Parisi-Zhang equation - a statistical physics perspective [article]

Herbert Spohn
2016 arXiv   pre-print
its 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) =  ... 
arXiv:1601.00499v1 fatcat:gix7bazl4ngn5k6gnzm76x4caa

Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation [article]

Promit Ghosal, Guilherme L. F. Silva
2022 arXiv   pre-print
In turn, such universal limits are described by the integro-differential 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.  ... 
arXiv:2201.12941v1 fatcat:mcqthczxlrd5thhzrnsl3deusi

Notes on Lie symmetry group methods for differential equations [article]

F. Güngör
2021 arXiv   pre-print
Fundamentals on Lie group methods 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.  ... 
arXiv:1901.01543v8 fatcat:g6mgelak5jbbljk6rfax5zkacm

A Singular Mathematical Promenade [article]

Etienne Ghys
2017 arXiv   pre-print
We play with a bit 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 .  ... 
arXiv:1612.06373v4 fatcat:3x2nfhmw3jgo3lfwvyt7njnssq

A pedestrian's view on interacting particle systems, KPZ universality and random matrices

Thomas Kriecherbauer, Joachim Krug
2010 Journal of Physics A: Mathematical and Theoretical  
We then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) 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.  ... 
doi:10.1088/1751-8113/43/40/403001 fatcat:jrz5o5ikknc5dnlrkx4756ltie

Riemann surfaces for KPZ with periodic boundaries

Sylvain Prolhac
2020 SciPost Physics  
Known 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  ... 
doi:10.21468/scipostphys.8.1.008 fatcat:yv5fzttsercajh77gbppixdlhm

Riemann surfaces for KPZ with periodic boundaries [article]

Sylvain Prolhac
2019 arXiv   pre-print
Known 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  ... 
arXiv:1908.08907v2 fatcat:mnfdprkswrgkxbexnvrahhc6i4