Nonautonomous soliton solutions of variable-coefficient fractional nonlinear Schrödinger equation

Wu, Gang-Zhou; Dai, Chao-Qing

doi:10.1016/j.aml.2020.106365

Cited by 66 publications

(24 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, Xu et al [23] proposed a new non-integral-order derivative called two-parameter fractional derivative. Dai et al [24][25][26] studied stochastic, non-autonomous and optical solitons in fractional models.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Zhang

2021

Symmetry

View full text Add to dashboard Cite

Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.

show abstract

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Zhang

2021

Symmetry

View full text Add to dashboard Cite

show abstract

“…Exact solutions of nonlinear models play an important part in the explanation and description of engineering and physical issues, including optics, fluid dynamics, electromagnetism, plasma physics, and other fields including nonlinear wave propagation phenomenon 1–3 . Fractional nonlinear models (FNMs) were also widely utilized in different domains of engineering, 4,5 physics, 6–9 biology, 10–14 and mathematics 15,16 .…”

Section: Introductionmentioning

confidence: 99%

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

Wang

Dai

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

A (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation is solved via fractional Riccati method and fractional bifunction method, and exact traveling wave solutions including soliton solution and combined soliton solutions are constructed based on Mittag–Leffler function. A series of fractional orders is used to demonstrate the graphical representation and physical interpretation of the resulting solutions. The role of the fractional order is revealed.

show abstract

“…Different analytical methods have been developed and used to obtain the solutions of the WBK model and the soliton solutions of nonlinear wave equations, such as the improved Riccati equations method [6][7][8][9][10] , the first integral method [11] , the Backlund transformation method [12] , the optimal homotopy asymptotic method [13] , the hyperbolic function method [14] , the homogeneous balance method [15] , the parity-time symmetric potential method [16] , the Darboux method [17] , the variable-coefficient fractional Y-expansion method [18] , and the fractional F-expansion method [19] .…”

Section: Introductionmentioning

confidence: 99%

New groups of solutions to the Whitham-Broer-Kaup equation

Wang

Sun

2020

Appl. Math. Mech.-Engl. Ed.

View full text Add to dashboard Cite

The Whitham-Broer-Kaup model is widely used to study the tsunami waves. The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method. 20 sets of solutions are obtained of which, to the best of the authors’ knowledge, some have not been reported in literature. Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.

show abstract

Nonautonomous soliton solutions of variable-coefficient fractional nonlinear Schrödinger equation

Cited by 66 publications

References 15 publications

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

New groups of solutions to the Whitham-Broer-Kaup equation

Contact Info

Product

Resources

About