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Open research questions in Nonlinear Waves and Solitons

167 unresolved questions extracted from the limitations and future-work sections of 503 Nonlinear Waves and Solitons papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • Further analysis of the asymptotic behavior of nC-HNLS Equation (1) in other regions, - Investigation of the properties of the coupled Painlevé II equation, - Study of the applications of the results in optics and related fields

    Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region · 2026 · DOI

  • The geometric framework developed here extends naturally to perturbed variants of the Kuralay-IIA equation

    Geometric admissibility conditions for travelling-wave solitons in the Kuralay–IIA equation · 2026 · DOI

  • The physical admissibility of travelling-wave solutions is often implicitly assumed. Conventional ansatz-based approaches focus on constructing individual exact solutions, but do not address the structural question of under what conditions travelling waves are physically admissible.

    Geometric admissibility conditions for travelling-wave solitons in the Kuralay–IIA equation · 2026 · DOI

  • The method is limited to compact manifolds or complete non-compact manifolds without boundary, - The method requires dim M > 4, - The method is not applicable for dim Ω d < 4

    A local method for compact and non-compact yamabe problems · 2026 · DOI

  • the work did not present any further information on the solution, - much less is known about sign-changing solutions, - the standard assumption in earlier works is that the ansatzes are Kelvin invariant

    Planar doubling nodal solutions to the Yamabe equation with maximal rank · 2026 · DOI

  • exploring the conformal transformation of the solution constructed in Theorem 2, - analyzing the interaction of the two circles, - extending the construction to higher even dimensions

    Planar doubling nodal solutions to the Yamabe equation with maximal rank · 2026 · DOI

  • Further research on the Yamabe problem for positively curved non-compact spaces, - Extension of the local method to other types of manifolds

    A local method for compact and non-compact yamabe problems · 2026 · DOI

  • Further study of the properties of the solutions obtained. Application of the modified double sub-equation method to other nonlinear partial differential equations.

    Exact solutions for a (2+1)-dimensional B-type Kadomtsev-Petviashvili equation · 2026 · DOI

  • The (2+1)-dimensional BKP equation is a new model that has not been extensively studied. There is a need for new methods to find exact solutions to nonlinear partial differential equations.

    Exact solutions for a (2+1)-dimensional B-type Kadomtsev-Petviashvili equation · 2026 · DOI

  • The linearization of third-order nonlinear ordinary differential equations is a challenging task. The method needs to be able to handle nonlinear terms. The method needs to be able to provide exact analytical solutions.

    UTILIZING THE GENERALIZED SUNDMAN TRANSFORMATION TO LINEARIZE THIRD-ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS · 2026 · DOI

  • The linearization of third-order nonlinear ordinary differential equations is a basic issue in theoretical physics and applied mathematics. There is a need for a methodical approach to linearizing third-order nonlinear ODEs.

    UTILIZING THE GENERALIZED SUNDMAN TRANSFORMATION TO LINEARIZE THIRD-ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS · 2026 · DOI

  • Future research can focus on applying the Bilinear Neural Network Method to other nonlinear PDEs. The method can be used to study other high-dimensional nonlinear wave interactions. The solutions obtained can be used to understand complex behaviors and underlying mechanisms in various fields.

    Exact soliton, lump, and breather solutions of the (3 + 1)-dimensional Jimbo-Miwa equation via the bilinear neural network method · 2026 · DOI

  • The (3+1)-dimensional Jimbo-Miwa equation is a complex equation that requires new methods for solution. The existing methods have limitations and do not provide a systematic construction of broad families of exact analytical solutions.

    Exact soliton, lump, and breather solutions of the (3 + 1)-dimensional Jimbo-Miwa equation via the bilinear neural network method · 2026 · DOI

  • To apply the Aboodh-Adomian Decomposition Method to other partial differential equations. To compare the method with other existing methods. To extend the method to solve other problems in science and engineering.

    Application of the Aboodh Adomian Decomposition Method to Klein-Gordon and Sine-Gordon Equations · 2026 · DOI

  • There is a need for efficient methods to solve Klein-Gordon and Sine-Gordon equations. The existing methods may not provide accurate numerical solutions for linear and nonlinear problems.

    Application of the Aboodh Adomian Decomposition Method to Klein-Gordon and Sine-Gordon Equations · 2026 · DOI

  • The nonlinear nature of the Benjamin-Bona-Mahony equation. The need for a reliable and efficient method to solve the equation. The challenge of handling nonlinear components of the equation.

    Convergence Analysis and Numerical Solution of the BBM Equation using the Kamal-Adomian Decomposition Method · 2026 · DOI

  • The nonlinear nature of the Benjamin-Bona-Mahony equation makes it challenging to obtain exact analytical solutions. Prior methods may not effectively handle nonlinear components of the equation.

    Convergence Analysis and Numerical Solution of the BBM Equation using the Kamal-Adomian Decomposition Method · 2026 · DOI

  • Nonlinear wave phenomena in quantum plasmas is one of the most important areas of research in contemporary physics, but it requires more advanced analytical tools. The (3 + 1)-dimensional extended quantum nonlinear Zakharov-Kuznetsov equation is a more complex level of plasma dynamics in three spatial dimensions and time, but its solutions are not well understood.

    Dynamical analysis and soliton solutions to (3+1)-dimensional extended quantum nonlinear Zakharov–Kuznetsov equation · 2026 · DOI

  • Deriving explicit solution formulas for nonlinear rational difference equations. Ensuring that the denominators remain nonzero within the fractional domain. Analyzing the qualitative behavior of the solutions.

    Algebraic Reduction and Periodic Solvability in a Coupled Ternary Rational System · 2026 · DOI

  • The lack of explicit solution formulas for nonlinear rational difference equations. The need for a systematic methodological framework for solving multidimensional nonlinear difference equations.

    Algebraic Reduction and Periodic Solvability in a Coupled Ternary Rational System · 2026 · DOI

  • The lack of explicit high-order line-soliton solutions for the (2+1)-dimensional nonlinear Schrödinger equation. The need for an asymptotic analysis framework to characterize the dynamical behavior of high-order line-soliton solutions. The importance of understanding the interaction dynamics of high-order line-soliton solutions and anomalous scattering phenomena exhibited by high-order lump solutions.

    High-Order Line-Soliton Interactions and Anomalous Scattering of Lumps in a (2+1)-Dimensional Reverse Space–Time Nonlinear Schrödinger Equation · 2026 · DOI

  • Further study of the Darboux transformation method for other integrable systems. Investigation of the physical properties and dynamic behaviors of the breathers on the periodic background.

    Solitons and breathers for the (2+1)-dimensional Hirota’s system · 2026 · DOI

  • The Darboux transformation method has not been fully explored for the (2+1)-dimensional Hirota's system. There is a need for a comprehensive analysis of the system's dynamical properties.

    Solitons and breathers for the (2+1)-dimensional Hirota’s system · 2026 · DOI

  • There is a lack of explicit quasi-periodic solutions for the GI equation. The previous methods have limitations in deriving solutions for the GI equation.

    Finite-Gap Integration of the Gerdjikov-Ivanov Equation: a Classical Method · 2026 · DOI

  • The study identifies a gap in the understanding of the integrable Shynaray-IIA equation, particularly in the context of soliton solutions. The paper aims to fill this gap by using two semi-analytical techniques, the ESEM and the.G0G0+G+A/-expansion method, alongside the numerical DTM to investigate the equation.

    Analytical and numerical solutions of the integrable Shynaray-IIA equation: solitary wave dynamics in fiber optics · 2026 · DOI

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167 open questions have been extracted from the limitations and future-work passages of 503 Nonlinear Waves and Solitons papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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