Modified Kadomtsev Petviashvili 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.

Studies on electromagnetic waves for ferromagnetic materials

With the developing technology, magneto-optical and ferromagnetic materials are gaining importance and are used especially in magneto-optics, ferromagnetism, fluid mechanics, etc. These processes are modeled via Kadomtsev–Petviashvili-type models. In this work, a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system and special cases are considered that simulates electromagnetic, water, and powder-acoustic/ion-acoustic/dust-ion-acoustic waves. As to the novelty of this paper, the travelling wave, soliton solutions of the considered systems are hold by using Bernoulli method which is the well-known ansatz-based method and the analytical method. As far as we know, the obtained solutions are seen for the first time in this study and are important for the development of the use of magneto-optical and ferromagnetic materials in industry and applied sciences, fiber optic communication fields.

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.

Soliton interactions and Mach reflection in gas bubbles–liquid mixtures

In this study, we assume that blood is assumed to be a viscoelastic and incompressible homogeneous media in which several uniform sized oxygen bubbles are uniformly distributed. Based on this, we establish a (3 + 1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation to describe the long nonlinear pressure waves in the gas bubbles–liquid mixtures. Using bell polynomials, a new bilinear form of the mKP equation is constructed, and then the one- and two-soliton solutions of the equation are obtained by the Hirota method. Via the one-soliton solutions, parametric conditions of the existence of shock wave, elevation and depression solitons, and the Mach reflection characters in the mixtures are discussed. Soliton interactions have been discussed on the basis of the two-soliton solutions. We find that the (i) parallel elastic interactions can exist between the shock and elevation solitons; (ii) oblique elastic interactions can exist between the (a) shock and depression solitons and (b) the elevation and depression solitons; and (iii) oblique inelastic interactions can exist between the two depression solitons.

Archipelagos, islands, necklaces, and other exotic structures in external force-driven chaotic dusty plasmas

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn’s magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system.

Formation and interaction of multi-dimensional electrostatic ion-acoustic solitons in two-electron temperature plasmas

Ion-acoustic waves are investigated in an unmagnetized collisionless plasma comprising dynamical ions and inertialess cold and hot (C–H) electrons expressed by Maxwellian, kappa, and (r, q) distributions. The reductive perturbation theory is applied for deriving a modified Kadomtsev–Petviashvili (mKP) equation to examine the characteristics of ion-acoustic structures (solitary waves, IASWs). The Hirota bilinear formalism is used to investigate the propagation of a single soliton and the interaction of two solitons with special reference to space plasmas. It is found that ratio of number density of cold to hot electrons and non-Maxwellian nature of cold electrons not only affect the propagation characteristics of single mKP solitons but also alter the interaction time of the IASWs. It is found that the swiftness of the interaction of solitons for flat-topped distribution surpasses both kappa and Maxwellian distributions. It is also found that the bipolar and tripolar structures depend on the ratio of the propagation vectors. The ranges of electric field amplitude for the IASWs are calculated for C–H electron distributions corresponding to Saturn's B-ring and the region just beyond terrestrial magnetopause, and are shown to agree with Cassini wideband receiver observational data and wideband plasma wave instrument's waveform data.

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.

Integrability, similarity reductions and solutions for a (3+1)-dimensional modified Kadomtsev–Petviashvili system with variable coefficients

In this study, a (3+1)-dimensional modified Kadomtsev–Petviashvili system with variable-coefficients (vcmKP) arising in many fields of science like plasma physics, optics, and fluid mechanics is studied. The consistent Riccati expansion solvability technique is applied to the vcmKP system to obtain necessary integrability conditions between the variable coefficients. Moreover, many novel different types of travelling wave solutions are given. Additionally, the vcmKP system is reduced to a nonlinear third order ordinary differential equation and many other new solutions are obtained. Finally, some plots are given to illustrate how the arbitrary function choices affect the propagation of the wave solutions.

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.

The discrete Burgers equation

In this paper I give a short review on the continuous, semi-discrete and discrete Burgers equations, which are featured as integrable equations that are linearisable. The review focuses more on connections of these three kinds of equations and connections with (2+1)-dimensional systems. The continuous Burgers hierarchy is a reduction of the derivative nonlinear Schrödinger hierarchy. The later is a result of the squared eigenfunction symmetry constraint of the modified Kadomtsev–Petviashvili system. The semi-discrete Burgers hierarchy can be considered as Bäcklund transformations of the continuous Burgers hierarchy. They are also reductions of the relativistic Toda hierarchy, which is a consequence of the squared eigenfunction symmetry constraint of the semi-discrete modified Kadomtsev–Petviashvili system. The fully discrete Burgers equation is the Bianchi identity of the Bäcklund transformation for the continuous Burgers hierarchy. It is 3-point lattice equation, three-dimensionally consistent, and linearisable.

Formation of electrostatic solitary and periodic waves in dusty plasmas in the light of Voyager 1 and 2 spacecraft and Freja satellite observations

Motivated by the observations of Voyager 1 and 2 spacecraft and Freja satellite observations in Saturn’s magnetosphere, the formation of dust-acoustic (DA) localized and periodic waves in a complex plasma having superthermal electrons and ions are reported. In this regard, a modified Kadomtsev–Petviashvili (mKP) equation is derived by employing the weak turbulence theory for studying the characteristics of the nonlinear dust-acoustic waves (DAWs) in the model under consideration. The localized and periodic wave solutions to the mKP equation are derived using ansatz method in terms of Jacobi elliptic functions (JEFs). It is reported that the phase velocity of the DAWs in the Saturn’s magnetosphere is lower for kappa distributed ions and electrons by comparison with regions of space plasmas where the electrons and ions follow the Maxwellian distribution. The conditions for the existence of both localized and periodic waves are also presented. Estimates are also given of the spatial scales over which the dust-acoustic solitary/periodic structures form in Saturn’s magnetosphere.

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.

Certain electromagnetic waves in a ferromagnetic film

Electromagnetic-wave/ferromagnetic-material investigations with magneto-optics are carried out in magneto-optic recording, computer data-storage, waveguides, etc. In this paper, we investigate a variable-coefficient modified Kadomtsev–Petviashvili system for certain electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film with the potential application in magneto-optic recording. With respect to the electromagnetic waves in the ferromagnetic film, symbolic computation helps us work out (1) two sets of the variable-coefficient-dependent auto-Bäcklund transformations via the generalized Laurent series, along with some solitonic features, (2) one set of the variable-coefficient-dependent bilinear forms and (3) two sets of the variable-coefficient-dependent N-solitonic solutions with N being a positive integer. We note that the electromagnetic field can excite the magnetization of a ferromagnetic medium in the form of solitons with the applications in magneto-optic recording. Relevant constraints on the variable coefficients are presented. We hope that the results could help the studies on magneto-optic recording, computer data-storage and waveguides.

Squared eigenfunction symmetry of the DmKP hierarchy and its constraint

AbstractIn this paper, squared eigenfunction symmetry of the differential‐difference modified Kadomtsev–Petviashvili (DmKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the DmKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R‐Toda) hierarchy. An invertible transformation is given to connect the positive and negative R‐Toda hierarchies. The positive R‐Toda hierarchy is reduced to the differential‐difference Burgers hierarchy. We also consider another DmKP hierarchy and show that its squared eigenfunction symmetry constraint gives rise to the Volterra hierarchy. In addition, we revisit the Ragnisco–Tu hierarchy which is a squared eigenfunction symmetry constraint of the differential‐difference Kadomtsev–Petviashvili (DKP) system. It was thought the Ragnisco–Tu hierarchy did not exist one‐field reduction, but here we find a one‐field reduction to reduce the hierarchy to the Volterra hierarchy. Besides, the differential‐difference Burgers hierarchy is also investigated in the Appendix. A multidimensionally consistent three‐point discrete Burgers equation is given.

The reduction of the q-Wronskian solutions of the q-deformed modified KP hierarchy to its constrained case

By using the eigenfunction symmetry constraints of the [Formula: see text]-deformed modified Kadomtsev–Petviashvili ([Formula: see text]-mKP) hierarchy, we show a necessary and sufficient condition to reduce [Formula: see text]-Wronskian solutions of the [Formula: see text]-mKP hierarchy to the [Formula: see text]-cmKP ([Formula: see text]-deformed constrained modified Kadomtsev–Petviashvili) hierarchy defined by setting the Lax operator as [Formula: see text]. Then an illustrative example is given.

Propagation of dust-ion-acoustic solitary waves for damped modified Kadomtsev–Petviashvili–Burgers equation in dusty plasma with a q-nonextensive nonthermal electron velocity distribution

Propagation of dust-ion-acoustic solitary waves for damped modified Kadomtsev–Petviashvili–Burgers equation in dusty plasma with a q-nonextensive nonthermal electron velocity distribution

Obtaining multisoliton solutions of the (2+1)-dimensional Camassa–Holm system using Darboux transformations

We construct and study Darboux transformations for the $$(2{+}1)$$ -dimensional Camassa–Holm system. We apply a reciprocal transformation that relates the $$(2{+}1)$$ -dimensional Camassa–Holm system and the linear system associated with the modified Kadomtsev–Petviashvili hierarchy. Using three Darboux transformation operators, we obtain three types of solutions for the $$(2{+}1)$$ -dimensional Camassa–Holm system, of which one is a multisoliton solution. In addition, we briefly discuss rational solutions.

Dust acoustic rogue waves of fractional-order model in dusty plasma

In this paper, the fractional-order model is used to study dust acoustic rogue waves in dusty plasma. Firstly, based on control equations, the multi-scale analysis and reduced perturbation method are used to derive the (3+1)-dimensional modified Kadomtsev–Petviashvili (MKP) equation. Secondly, using the semi-inverse method and the fractional variation principle, the (3+1)-dimensional time-fractional modified Kadomtsev–Petviashvili (TF-MKP) equation is derived. Then, the Riemann–Liouville fractional derivative is used to study the symmetric property and conservation laws of the (3+1)-dimensional TF-MKP equation. Finally, the exact solution of the (3+1)-dimensional TF-MKP equation is obtained by using fractional order transformations and the definition and properties of Bell polynomials. Based on the obtained solution, we analyze and discuss dust acoustic rogue waves in dusty plasma.