Integrability of One Bilinear Equation: Singularity Analysis and Dimension

The integrability of a four-dimensional sixth-order bilinear equation associated with the exceptional affine Lie algebra D(1)4 is studied by means of the singularity analysis. This equation is shown to pass the Painlevé test in three distinct cases of its coefficients, exactly when the equation is effectively a three-dimensional one, equivalent to the BKP equation.

Solitary wave solutions along with Painleve analysis for the Ablowitz–Kaup–Newell–Segur water waves equation

This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method.

Some new dispersive dromions and integrability analysis for the Davey–Stewartson (DS-II) model in fluid dynamics

This paper focuses on the Davey–Stewartson (DS)-II equation, and the extended modified auxiliary equation mapping (EMAEM) architectonic is used to develop a new set of solutions such as kink, singular kink, rational, combined soliton-like solutions, bell-type solutions, trigonometric and hyperbolic solutions. Furthermore, this study reveals that the used technique is efficient for solving other nonlinear evolution equations (NLEEs). The Painleve test ([Formula: see text]-test) will also be used to examine the integrability of the DS-II equation. Finally, graphical simulations are designed to show the exact behavior of solutions as well as the efficacy of the suggested strategy.

Singularity analysis and analytic solutions for the Benney–Gjevik equations

We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

Painlevé analysis for various nonlinear Schrödinger dynamical equations

In this paper, our objective is to analyze integrability of three famous nonlinear models, namely unstable nonlinear Schrödinger equation (UNLSE), modified UNLSE (MUNLSE) as well as (2+1)-dimensional cubic NLSE (CNLSE) by utilizing Painlevé test ([Formula: see text]-test). The non-appearance of some sort of singularities such as moveable branch points indicates a sound probability of complete integrability of the concerned NLSE. In case an NLSE passes the [Formula: see text]-test, the studied model can be solved by implementing inverse scattering transformation (IST).

Painlevé analysis of a nonlinear Schrödinger equation discussing dynamics of solitons in optical fiber

In this paper, we investigated a new form of nonlinear Schrödinger equation (NLSE), namely the Biswas–Arshed model (BAM) for the analysis of complete integrability with the help of Painlevé test ([Formula: see text]-test). By applying this test, we analyze the singularity structure of the solutions of BAM, knowing the fact that the absence of specific sort of singularities like moveable branch points is a patent signal for the complete integrability of the discussed model. Passing the [Formula: see text]-test is a powerful indicator that the studied model is resolvable by means of inverse scattering transformation (IST).

Conserved quantities along with Painlevé analysis and optical solitons for the nonlinear dynamics of Heisenberg ferromagnetic spin chains model

In this paper, we will investigate a famous model of nonlinear sciences namely [Formula: see text]-dimensional nonlinear spin dynamics of Heisenberg ferromagnetic spin chains (HFSC) model for the evaluation of optical travelling waves by implementing unified method (UM). We will extract dark as well as bright solitary waves along with elliptic waves. We will draw conserved quantities of our governing model by utilizing dilation symmetry. At the end, the integrability of nonlinear spin dynamics of HFSC model with the help of Painlevé test will also be studied in this paper.

Painlevé integrability and multisoliton solutions of a generalized KdV system

The integrability of a generalized KdV model, which has abundant physical applications in many fields, is investigated by employing Painlevé test. Eventually, we discover a new generalized P-type KdV model in sense of WTCKruskal method. Subsequently, Hereman’s simplified bilinear method is used to examine the integrability of the resulted model. As a result, multiple soliton solutions of newly discovered model are formally obtained.