Modified Kadomtsev-Petviashvili Equation Research Articles

Novel localized wave of modified Kadomtsev–Petviashvili equation

In this paper, we investigate the data-driven localized solutions of Kadomtsev–Petviashvili (KP) and modified KP equation. Through the two-dimensional Miura transformation, the solutions of modified KP equation can be converted into the solutions of KP equation, but the process is not invertible in mathematics. Based on the neural network, the localized waves of modified KP equation are obtained under an unsupervised training with the aid of two-dimensional Miura transformation and the initial and boundary conditions of solution of the KP equation. As the result of the different hyperparameters, three types of localized waves are found after the training, including the shape of kink, dark and kink-bell. The evolution and error dynamics of the predicted solutions are analyzed through the graphics.

On the line-soliton solutions of a coupled modified Kadomtsev–Petviashvili system in two-layer shallow water

In this paper, the line-soliton solutions of a coupled modified Kadomtsev–Petviashvili (MKP) equation are studied. Based on the tau-functions and Grassmannian Gr(N,M), we present and study the coupled modified Kadomtsev–Petviashvili equation and its coupled soliton solutions v and vˆ which can describe two layers shallow water. By giving examples and theoretical analysis, we give a general classification of the structures of coupled soliton solutions v and vˆ. At the same time, we describe the solutions v, vˆ by Le-diagram, chord diagram and analyze the influence of parameter ϵ on the solutions v, vˆ.

Exact travelling wave solutions for generalized (3+1) dimensional KP and modified KP equations

Exact travelling wave solutions for generalized (3+1) dimensional KP and modified KP equations

On the superpropagating nonlinear waveform in critical superthermal fluid

The superthermality indices of positron (electron) contributions to the new nonlinear wave profiles characterized by the model as supersoliton, cnoidal, shocklike, and super-periodic wave propagation have been obtained in the superthermal ionosphere plasma pair. These profiles that exhibit critical behaviors are described by the modified KP equation. Furthermore, the novel solutions have been exposed and the super-solitary and period solutions are derived and explained using the Jacobi elliptic function expansion method for the Modified Korteweg–De Vries (MKdV) equation. These solutions may be useful in the space of ionosphere fluids. Indeed, the MKdV model is potent for the dynamics of fluid flow. The super propagating waves obtained under the influence of critical densities represent a source of forcing or collapsing energy waves in space plasma fluids. To check the impacts of superthermality on the progress of nonlinear formations, all the given solutions are deliberated by the superthermal factor in ionosphere fluid observations and applications.

Some electrostatic structures of the mKP equation for a nonthermal plasma system by a unified solver technique

The reductive perturbation technique is used to obtain the Modified KP equation at critical densities distinguished by the MKPE in plasma ion pair with fast electron positron. The new structures reveal that super-solitary and period waveforms are derived via the mathematical analysis of Jacobi-elliptic function expansions (MJEFE) for MKPE. The electrostatic new structures such as super-soliton, cnoidals, shock-like, super-dispersive and superperiodic plasma waves which existed at critical densities have been introduced. The positrons (electron) nonthermality supports on nonlinear dispersive new structures have been discussed. Also, many of the obtained electrostatic solutions are important and may be applicable in ionosphere plasma observations.

Rational, semi-rational solution and self-consistent sources extension of the variable-coefficient extended modified Kadomtsev-Petviashvili equation

In this paper, we apply Hirota bilinear method and determinant technique to derive the Nth-order rational solution expressed compactly in terms of Matsuno determinants for the variable-coefficient extended modified Kadomtsev-Petviashvili (mKP) equation. As a special case, we obtain the M-lump solution expressed in terms of 2M × 2M determinants for the mKPI equation and investigate the dynamical behaviors of 1- and 2-lump solutions. Furthermore, we present the Wronskian and Grammian solution for the variable-coefficient extended mKP equation. Based on the Grammian solution, we construct the line soliton, the line breather and the semi-rational solution on constant and periodic backgrounds for the mKPI equation. Through the asymptotic analysis, we show that the semi-rational solutions describe the fission and fusion of lumps and line solitons. In addition, we construct the variable-coefficient extended mKP equation with self-consistent sources via the source generation procedure and derive its N-soliton solution in the compact form of Grammian and Wronskian.

Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation

Algebro-geometric solutions of the lattice potential modified Kadomtsev–Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup–Newell spectral problem is employed to generate a Lax triad for the lpmKP equation, as well as to define commutative integrable symplectic maps which generate discrete flows of eigenfunctions. These maps share the same integrals with the finite-dimensional Hamiltonian system associated to the Kaup–Newell spectral problem. We investigate asymptotic behaviors of the Baker–Akhiezer functions and obtain their expression in terms of Riemann theta function. Finally, algebro-geometric solutions for the lpmKP equation are reconstructed from these Baker–Akhiezer functions.

Single-traveling-wave and double-wave polynomial solutions of the generalized (3 + 1)-dimensional modified Kadomtsev–Petviashvili equations with variable coefficients

In this paper, we explore the generalized (3 + 1)-dimensional modified Kadomtsev–Petviashvili equations with variable coefficients, which are usually used in the fields of ferromagnetism, magneto-optics, plasma physics and fluid mechanics. For the sake of uncovering more physical phenomena related to this system, we consider the single-traveling-wave polynomial solutions in the light of the unified method and the double-wave polynomial solutions in line with the generalized unified method. Remarkably, the solitary-, soliton- as well as elliptic-type solutions are all discussed in these two kinds of solutions. Furthermore, the physical explanations of the solutions are given graphically and analytically for different choices of the free parameters (especially of the nonlinear coefficients of the equations that related to the physical insights). By discussing the wave propagation of each solution that we procured from the perspectives of amplitude, shape, symmetry or periodicity, we are capable of realizing the inherent characteristics of this equation commendably and discovering the correlative physical world more efficiently.

Modified Kadomtsev–Petviashvili Equation for Description of Nonlinear Perturbations in Plasma of Dusty Lunar Exosphere

Modified Kadomtsev–Petviashvili Equation for Description of Nonlinear Perturbations in Plasma of Dusty Lunar Exosphere

The Sylvester Equation and Kadomtsev–Petviashvili System

In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation LM are bold, please chekc if bold neceaasry, if not, please remove all bold of equation −MK = rsT together with an evolution equation set of r and s, master function S(i,j)=sTKjC(I + MC)−1Lir is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ-function by τ = |I + MC|. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed.

Interaction of dust ion acoustic solitons with cubic nonlinearity in a magnetized dusty plasma with () distributed electrons

AbstractDust ion acoustic waves (DIAWs) are analysed in multi‐component plasmas via reductive perturbation technique comprising inertial ions, (r, q) distributed electrons, and stationary dust by using fluid theory of plasmas. The modified Kadomtsev‐Petviashvili (MKP) equation is obtained for the critical condition at which the quadratic nonlinearity vanishes. Hirota's bilinear formalism is employed, for the first time, to study the two‐soliton solutions of MKP equation in the context of plasma physics. The effects of flatness at low energy (represented by ) and superthermality at high energy (represented by ) of electrons in phase space are investigated for the linear and nonlinear propagation of DIAWs. It is found that the amplitude of the nonlinear DIAW is highest for the flat‐topped distribution, whereas it is lowest for kappa distribution. Using the plasma parameters of Saturn's B‐ring. The estimates are given of the spatial scales over which the MKP solitons form for flat‐topped, kappa, and Maxwellian distributions, respectively. It is also discussed in detail how the presence of dust and non‐Maxwellian electron distributions affect the interaction of MKP solitons in Saturn's B‐ring. In addition, the interaction of a compressive and rarefactive soliton is studied giving results not achievable for KdV and KP equations.

Algebraic structures on the flows of dispersionless modified KP equation

Algebraic structures on the flows of dispersionless modified KP equation

Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps

We established a method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev's electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig's parametrizations of the totally positive part of $SL_3$ and $SO_5$, which is obtained via folding the Dynkin diagram of $A_3$ into one of $B_2$.

New nonlinear periodic, solitonic, dissipative waveforms for modified-Kadomstev-Petviashvili-equation in nonthermal positron plasma

The positron nonthermality contributions on nonlinear wave aspects of periodic solitary, shocklikes, rational, and explosive waves which exists for the critical features depicted by modified KP equation in earths space of regions D and F ionic plasma pairs with nonthermal fast positrons have been studied. Furthermore, the derived solutions have been disclosed that hyperbolic, periodic, trigonometric, rational and explosive forms may be usable in plasma of ionosphere. This technique is sound in the fluid flow models. To explore the nonthermal effects on the nonlinear properties in this physical mode, the modified KP equation has been examined mathematically with the method of Riccati-Bernoulli (RB) sub-ODE. The obtained forms may leading realizations in the ionosphere fluids investigations and observations.

Nonlinearity contributions on critical MKP equation

The mathematical new plasma wave solutions are specified in the compose of trigonometric, rational, hyperbolic, periodic and explosive kinds that are realistic for Modified-Kadomtsev-Petviashvili (MKP) equation. Also, numeral studies for the acquired solutions have been reveals that periodic, shock and explosive new forms may applicable in D-F Earth's ionosphere plasma. The used method is influential and robust in comparison applications in plasma fluids. To depict the propagating soliton profiles in a plasma medium, it is needful to solve MKP equation at a critical mass ratio. The Riccati–Bernoulli sub-ODE technique has been utilized to introduce some new important and applicable solutions. The number of these MKP solutions give a leading deed in applied ion acoustics in ionosphere.

Построение периодических решений модифицированного уравнения Кадомцева–Петвиашвили методом дибар-одевания

Построение периодических решений модифицированного уравнения Кадомцева–Петвиашвили методом дибар-одевания

Conservation laws and line soliton solutions of a family of modified KP equations

A family of modified Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.

Rational and semi-rational solutions of the modified Kadomtsev–Petviashvili equation and the $$\varvec{(2 + 1)}$$-dimensional Konopelchenko–Dubrovsky equation

General rational and semi-rational solutions of the modified Kadomtsev–Petviashvili (mKP) equation and the Konopelchenko–Dubrovsky equation are obtained based on the bilinear method and the KP hierarchy reduction technique. These solutions are expressed in terms of $$N \times N$$ determinants. The dynamics of the solutions, which exhibit various patterns, are thoroughly analyzed. It is shown that the rational solutions may describe the elastic interaction of a single-peak wave with either a double-peak (M-shape) wave or another single-peak wave for $$N=1$$ . Depending on the choice of parameters, the semi-rational solutions are found to depict the inelastic interaction between two (Y-shape) or three waves for $$N=1$$ . The second-order ( $$N=2$$ ) rational solutions exhibit the elastic interaction of three single-peak waves with either one double-peak wave or another single-peak wave. Inelastic interaction is displayed by proper choices of the parameters for semi-rational solutions. When $$N > 2$$ , similar local dynamical behaviors of the rational and semi-rational solutions have been observed.

Bifurcation analysis of KP and modified KP equations in an unmagnetized dust plasma with nonthermal distributed multi-temperatures ions

In this research, we consider an unmagnetized dusty plasma containing electrons, dust particle and lower (higher) temperature ions. The bifurcation analysis for Kadomtsev–Petviashvili (KP) and modified KP equations are discussed. The solitary and periodic wave solutions including several parameters of ions are investigated. Our novel results show that nonthermal parameter is directly proportional to the width of periodic wave. The width of periodic wave causes the change behavior of wave from compressive periodic to rarefactive. The obtained results have numerous applications in plasma physics.

Bäcklund transformations, rational solutions and soliton–cnoidal wave solutions of the modified Kadomtsev–Petviashvili equation

In this paper, the truncated Painlevé expansion is employed to derive the Bäcklund transformation and the nth Bäcklund transformation of the modified Kadomtsev–Petviashvili equation. The method for constructing the rational solutions with the aid of the Bäcklund transformation is presented. It is proved that the modified Kadomtsev–Petviashvili equation is CRE solvable. Then the soliton–cnoidal wave solutions are explicitly presented. These results are useful in explaining nonlinear wave phenomena in oceanography.