A study of the shallow water waves with some Boussinesq-type equations

Kai, Yue; Chen, Shuangqing; Zhang, Kai; Yin, Zhixiang

doi:10.1080/17455030.2021.1933259

Cited by 34 publications

(5 citation statements)

References 32 publications

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“…From Figure 1(b), we can see that trajectory I is a closed orbit with a center inside, which just indicates the existence of the periodic solution, and trajectory II is a homoclinic orbit, which indicates the existence of the bell-shaped soliton solution [36].…”

Section: Dynamic Properties Of Equation (1)mentioning

confidence: 98%

“…The CDSPM is proposed by Liu [25][26][27] and has been successfully applied to a series of integer-order [28,29] and fractional-order equations [30][31][32][33][34]. Then, Kai et al found that this method could also be used to conduct qualitative analysis [35], and combining with the bifurcation method, we can even establish the existence of the soliton and periodic solution [36][37][38].…”

Section: Introductionmentioning

confidence: 99%

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Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Shu-xin

Chen

2022

Advances in Mathematical Physics

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The main idea of this paper is to investigate the exact solutions and dynamic properties of a space-time fractional perturbed nonlinear Schrödinger equation involving Kerr law nonlinearity with conformable fractional derivatives. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely, the Hamiltonian, is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the existences of the soliton and periodic solution are established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for the polynomial method.

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Section: Dynamic Properties Of Equation (1)mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Shu-xin

Chen

2022

Advances in Mathematical Physics

Self Cite

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show abstract

“…1 From Fig. 1b, we can see that trajectory I is a closed orbit with a center inside, which just indicates the existence of the periodic solution, and trajectory II is a homoclinic orbit, which indicates the existence of the bell-shaped soliton solution [21].…”

Section: Introductionmentioning

confidence: 94%

“…The CDSPM is proposed by Liu [10][11][12], and has been successfully applied to a series of integerorder [13][14] and fractional order equations [15][16][17][18][19]. Then Kai found that this method could also be used to conduct qualitative analysis [20], and combining with the bifurcation method, we can even establish the existence of the soliton and periodic solution [21]. The construction of this paper is as follows.…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions and Dynamic Properties of Perturbed Nonlinear Schrodinger Equation Arised From Nano Optical Fibers

Xin¹,

Chen²

2021

Preprint

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The main idea of this paper is to investigate the exact solutions and dynamic properties in optical nanofibers, which is modeled by space-time fractional perturbed nonlinear schr\"odinger equation involving Kerr law nonlinearity with conformal fractional derivative. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely the Hamiltonian is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the prior estimate of the existence of the soliton and periodic solution is established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for polynomial method.

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“…And are widely used by excellent scholars such as Biswas, Gurefe, and Bulut et al to solve nonlinear partial differential equations [38][39][40][41][42][43][44]. Moreover, we rewrite the equation into a dynamic system, and analyze the dynamic properties such as equilibrium points [45][46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions and dynamic properties of the stochastic governing model for embedded solitons with χ2 and χ3 nonlinear susceptibilities

Chen

2023

Preprint

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This paper studies the exact solutions and dynamic properties of the stochastic governing model for embedded solitons with χ 2 and χ 3 nonlinear susceptibilities. We propose a new trial equation method to find the exact solutions of the model. We also analyze the dynamic properties of the model equation by complete discriminant system for polynomial method. In addition, we assign specific values to the parameters to prove the existence of exact solutions, and giving the two-dimensional diagrams to demonstrate the dynamical behavior of the model. The obtained abundant exact solutions are beneficial for better study of the stochastic perturbation model of optical solitons.

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A study of the shallow water waves with some Boussinesq-type equations

Cited by 34 publications

References 32 publications

Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Exact Solutions and Dynamic Properties of Perturbed Nonlinear Schrodinger Equation Arised From Nano Optical Fibers

Exact solutions and dynamic properties of the stochastic governing model for embedded solitons with χ2 and χ3 nonlinear susceptibilities

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A study of the shallow water waves with some Boussinesq-type equations

Cited by 34 publications

References 32 publications

Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Exact Solutions and Dynamic Properties of the Perturbed Nonlinear Schrödinger Equation with Conformable Fractional Derivatives Arising in Nanooptical Fibers

Exact Solutions and Dynamic Properties of Perturbed Nonlinear Schrodinger Equation Arised From Nano Optical Fibers

Exact solutions and dynamic properties of the stochastic governing model for embedded solitons with χ2 and χ3 nonlinear susceptibilities

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Exact Solutions and Dynamic Properties of Perturbed Nonlinear Schrodinger Equation Arised From Nano Optical Fibers