The logarithmic derivative of the marginal distributions randomly fluctuating interfaces in one dimension on a large scale evolve according to Kadomtsev-Petviashvili (KP) equation. This is derived algebraically from Fredholm determinant obtained [MQR17, arXiv:1701.00018] for KPZ fixed point as limit transition probabilities TASEP, special solvable model universality class. Tracy-Widom appear se...

We study the problem of gravity surface waves for an ideal fluid model in (2+1)-dimensional case. apply a systematic procedure to derive Boussinesq equations given relation between orders four expansion parameters, amplitude parameter $\alpha$, long-wavelength $\beta$, transverse wavelength $\gamma$, and bottom variation $\delta$. derived only possible extensions Korteweg-de Vries equation, fif...

Three-dimensional solitary waves or lump solitons are known to be solutions to the Kadomtsev–Petviashvili I equation, which models small-amplitude shallow-water waves when the Bond number is greater than 1 3 . Recently, Pego and Quintero presented a proof of the existence of such waves for the Benney–Luke equation with surface tension. Here we establish an explicit connection between the lump s...

In this paper, we review various properties of the supersymmetric Two Boson (sTB) system. We discuss the equation and its nonstandard Lax representation. We construct the local conserved charges as well as the Hamiltoniam structures of the system. We show how this system leads to various other known supersymmetric integrable models under appropriate field redefinition. We discuss the sTB and th...

We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev–Petviashvili (KP) equations: (ut + 6uux + uxxx)x ± 3uyy ± 3uzz = 0. We also show that both of the 3+1 dimensional KP equations do not possess the three-soliton solution. This suggests that none of the 3+1 dimensional KP equations should be integrable, and partially explains why they do not pass the Pa...

We have recently solved the inverse scattering problem for oneparameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev Petviashvili (dKP) equation. We showed, in particu...

The paper addresses a new “inverse” problem for reconstructing the amplitudes of 2D surface waves from observation of the wave patterns (formed by wave crests). These patterns will depend on the amplitudes because of nonlinear effects. We show that the inverse problem can be solved when the waves are modelled by an equation that supports soliton solutions. Specifically, the explicit solution to...

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristi...

Purely dispersive partial differential equations such as the Korteweg–de Vries equation, the nonlinear Schrödinger equation, and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in...

We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its ZakharovShabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the n-th Gel’fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above mentioned ...