Quantization of the Kadomtsev–Petviashvili equation

We propose a quantization of the Kadomtsev–Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses m = 1, 2,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms \(\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}\) and \(\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}\) for all combinations of particles with masses m ₁, m ₂, and m ₁ + m ₂ for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass M=8 sector.     

Direct linearization of the nonisospectral Kadomtsev–Petviashvili equation

Direct linearization method is used to solve nonisospectral Kadomtsev–Petviashvili equation. By suitable choices of contours and measure, exact solutions including rational solutions, soliton-like solutions and periodic solutions are derived.     

Self-adjointness and conservation laws for Kadomtsev–Petviashvili–Burgers equation

In this work we study the Kadomtsev–Petviashvili–Burgers equation, which is a natural model for the propagation of the two-dimensional damped waves. We show that the equation is nonlinear self-adjoint and it will become strict self-adjoint or weak self-adjoint in some equivalent form. By using Ibragimov’s theorem on conservation laws we find some conservation laws for this equation.     

On the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources

A new procedure called ‘source generation’ is applied to construct non-isospectral soliton equations with self-consistent sources. As results, the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources (KPESCS) and its Gramm-type determinant solutions are obtained. Furthermore, the non-isospectral Pfaffianized-KP equation with self-consistent sources is constructed. This coupled system can not only be reduced to the non-isospectral Pfaffianized-KP equation, but also reduced to the non-isospectral KPESCS.     

The Cauchy problem for the dissipation-modified Kadomtsev–Petviashvili equation

Considered herein is the dissipation-modified Kadomtsev–Petviashvili equation in two space-dimensional case. It is established that the Cauchy problem associated to this equation is locally well-posed in anisotropic Sobolev spaces. It is also shown in some sense that this result is sharp. In addition, the global well-posedness for this equation under suitable conditions is proved.     

Solitons and Bäcklund transformation for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid dynamics

Under investigation in this paper is a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation, which describes the propagation of nonlinear waves in fluid dynamics. Bilinear form and Bäcklund transformation are derived by virtue of the Bell polynomials. Besides, the one- and two-soliton solutions are constructed via the Hirota method.     

Elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation

Theoretical and Mathematical Physics - We study elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation and derive the equations of motion of their poles. The...     

Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev–Petviashvili equation

Exact periodic kink-wave solution, periodic soliton and doubly periodic solutions for the potential Kadomtsev–Petviashvii (PKP) equation are obtained using homoclinic test technique and extended homoclinic test technique, respectively. It is investigated that periodic soliton is degenerated into doubly periodic wave varying with direction of wave propagation.     

Dynamics of the breather waves,rogue waves and solitary waves in an extend Kadomtsev–Petviashvili equation

Under investigation in this work is an extend Kadomtsev–Petviashvili (eKP) equation, which appears in the study of multi-component plasmas. By considering Bell’s polynomials, an effective and straightforward way is presented to succinctly derive its bilinear form and soliton solutions. Moreover, the homoclinic breather limit method is employed to construct the breather wave and rogue wave solutions of the equation. Finally, the dynamic behaviors of breather waves, rogue waves and solitary waves are discussed with graphic analysis. It is hoped that our results can be useful for explaining and enriching the dynamic behavior of these KP-type equations.     

Global well-posedness and scattering for small data for the three-dimensional Kadomtsev–Petviashvili II equation

We study global well-posedness for the Kadomtsev–Petviashvili II equation in three space dimensions with small initial data. The crucial points are new bilinear estimates and the definition of the function spaces. As by-product we obtain that all solutions to small initial data scatter as t→±∞.     

Logunov–Tavkhelidze equation in the relativistic quark model

Based on the relativistic Logunov–Tavkhelidze equation, we obtain the mass spectra and probabilities of radiative decays of heavy quarkonia in the framework of the constituent quark model of hadrons.     

Kadomtsev–Petviashvili hierarchies of types B and C

Theoretical and Mathematical Physics - This is a short review of the Kadomtsev–Petviashvili hierarchies of types $$mathrm{B}$$ and $$mathrm{C}$$ . The main objects are the $$L$$ operator,...     

On the Well-Posedness of the Dissipative Kadomtsev–Petviashvili Equation

Mathematical Notes - The well-posedness of the initial-value problem associated with the dissipative Kadomtsev–Petviashvili equation in the case of two-dimensional space is studied. It is...     

Quasideterminant solutions of the extended noncommutative Kadomtsev–Petviashvili hierarchy

We construct a nonauto Darboux transformation for the extended noncommutative Kadomtsev–Petviashvili (ncKP) hierarchy and consequently derive its quasi-Wronskian solution. We also obtain the quasi-Wronskian solution of the ncKP equation with self-consistent sources (ncKPESCS) as a by-product. Finally, we use the direct verification method to prove the quasi-Wronskian solution of the ncKPESCS.     

Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy

We construct bilinear identities for wave functions of an extended B-type Kadomtsev–Petviashvili (BKP) hierarchy containing two types of (2+1)-dimensional Sawada–Kotera equations with a self-consistent source. Introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the τ -function and bilinear identities for this extended BKP hierarchy. The bilinear identities generate all the Hirota bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, we obtain the Hirota bilinear equations for the two types of (2+1)-dimensional Sawada–Kotera equations in explicit form.