Related papers: On finite energy solutions of the KP-I equation
We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.
In this paper we obtain local in time existence and (suitable) uniqueness and continuous dependence for the KP-I equation for small data in the intersection of the energy space and a natural weighted $L^{2}$ space.
We prove that the solution map associated with the $1D$ half-wave cubic equation in the periodic setting cannot be uniformly continuous on bounded sets of the periodic Sobolev spaces $H^s$ with $s\in (1/4, 1/2)$
Of concern are lump solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. As in the classical Kadomtsev--Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak…
Of concern are traveling wave solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension breaking bifurcation. Moreover, the line solitary wave…
In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result…
The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and…
The Kadomtsev-Petviashvili (KP) nonlinear wave equation is the three dimensional generalization of the Korteveg-de Vries (KdV) equation. We show how to obtain the cylindrical KP (cKP) and cartesian KP in relativistic fluid dynamics. The…
In this article, we address the Cauchy problem for the KP-I equation \[\partial_t u + \partial_x^3 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0\] for functions periodic in $y$. We prove global well-posedness of this problem for any…
A family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the $xy$-plane are altered after…
The method of nonlinear self-adjointness is applied to the Kadomtsev-Petviashvili equation. The infinite set of conservation laws associated with the infinite algebra of Lie point symmetry of the KP equation is constructed.
In this paper, we study the three-dimensional gravity-capillary water wave problem involving an irrotational, perfect fluid with gravity and surface tension. We focus on steady waves propagating uniformly in one direction. Assuming constant…
In this paper, we consider the solution map of the initial value problem to the two-component Camassa-Holm equation on the line. We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s(\R)\times…
The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number $\beta>1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric…
We show the lack of uniform continuity of the flow map for the Camassa-Holm equation on the line, in the Sobolev spaces of index s > 3/2.
In this paper we study the $5$th Order Kadomstev-Petviashvili (KP) equations posed on the real line. In particular we adapt the energy estimate argument from Guo-Molinet (arXiv:2404.12364v1 [math.AP]) to conclude unconditional uniqueness of…
We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as $\partial_tu+\alpha\partial_x^3u+\partial^5_xu+\partial_x^{-1}\partial_y^2u+uu_x=0,$ while $\alpha\in \mathbb{R}$. We introduce an interpolated energy space $E_s$ to…
In this paper, we delve into the study of the generalized KP equation, which incorporates double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and…
We first review the known mathematical results concerning the KP type equations. Then we perform numerical simulations to analyze various qualitative properties of the equations : blow-up versus long time behavior, stability and instability…
The Kadomtsev-Petviashvili (KP) equation possesses a four-parameter family of one-dimensional periodic traveling waves. We study the spectral stability of the waves with small amplitude with respect to two-dimensional perturbations which…