Mixed lump–kink solutions to the KP equation

Zhao, Huandong; Ma, Wen‐Xiu

doi:10.1016/j.camwa.2017.06.034

Cited by 229 publications

(49 citation statements)

References 47 publications

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“…Under the assumption that the solution { u ( x , t ), v ( x , t )} exists, we have showed that the solution can be described in terms of the solution of a 3×3 matrix RH problem formulated in the complex λ‐plane. Furthermore, recent studies on lump and interaction solutions on integrable equations and such solution correspond to special initial values . Indeed, lump to the KP1 can be generated from the IST, and so it would be very interesting to see if any more lump or interaction solutions could be generated by applying the RH problems.…”

Section: Discussionmentioning

confidence: 99%

“…Furthermore, recent studies on lump and interaction solutions on integrable equations and such solution correspond to special initial values. [40][41][42][43][44] Indeed, lump to the KP1 can be generated from the IST, and so it would be very interesting to see if any more lump or interaction solutions could be generated by applying the RH problems.…”

Section: Discussionmentioning

confidence: 99%

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A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

Xia

2018

Math Methods in App Sciences

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In this work, we discuss the coupled modified nonlinear Schrödinger (CMNLS) equation, which describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. By use of the Fokas approach, the initial‐boundary value problem for the CMNLS equation related to a 3×3 matrix Lax pair on the half‐line is to be analyzed. Assuming that the solution {u(x,t),v(x,t)} of CMNLS equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann‐Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions s(λ) and S(λ) are not independent of each other but meet a global relationship.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

Xia

2018

Math Methods in App Sciences

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“…Moreover, there are many ways to solve differential equations such as the trial function method [39], the subequation method [40], the function variable method [41], the first integral method [42], etc. We can also get many other kinds of solutions, such as lump-soliton solutions [43][44][45][46], which are also very important.…”

Section: Introductionmentioning

confidence: 99%

Time–space fractional ( 2 + 1 ) $(2+1)$ dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions

Yang

2018

Adv Differ Equ

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In this article, nonlinear propagation of envelope gravity waves is studied in baroclinic atmosphere. The classical (2 + 1) dimensional nonlinear Schrödinger (NLS) equation can be derived by using the multiple-scale, perturbation method. Further, via the semi-inverse method, the Euler-Lagrange equation and Agrawal's method, the time-space fractional (2 + 1) dimensional nonlinear Schrödinger (FNLS) equation is obtained to describe the envelope gravity waves. Furthermore, the conservation laws of time-space FNLS equation are discussed on the basis of Lie group analysis method. Finally, the exact solutions to the equation are given by employing the exp(-φ(ξ )) method. The results demonstrate that the nonlinear effect caused by the fractional order leads to the change of the propagation characteristics of envelope gravity waves, the construction of fractional model has far-reaching significance for the research of nonlinear propagation of envelope gravity waves in actual atmospheric and ocean movement.

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“…In this paper, using the Hirota bilinear method, we obtain soliton solutions for the new model. Various phenomena can be explained via the application of the solutions given by the above methods [24][25][26]. Additionally, the use of these methods enables a better understanding of various magnetized plasma phenomena.…”

Section: Complexitymentioning

confidence: 99%

Study of Ion‐Acoustic Solitary Waves in a Magnetized Plasma Using the Three‐Dimensional Time‐Space Fractional Schamel‐KdV Equation

Guo

Zhang

et al. 2018

Complexity

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The study of ion-acoustic solitary waves in a magnetized plasma has long been considered to be an important research subject and plays an increasingly important role in scientific research. Previous studies have focused on the integer-order models of ionacoustic solitary waves. With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. The study of fractional calculus has opened a new window for understanding the features of ion-acoustic solitary waves and can be a potentially valuable approach for investigations of magnetized plasma. In this paper, based on the basic system of equations for ion-acoustic solitary waves and using multi-scale analysis and the perturbation method, we have obtained a new model called the three-dimensional(3D) Schamel-KdV equation. Then, the integerorder 3D Schamel-KdV equation is transformed into the time-space fractional Schamel-KdV (TSF-Schamel-KdV) equation by using the semi-inverse method and the fractional variational principle. To study the properties of ion-acoustic solitary waves, we discuss the conservation laws of the new time-space fractional equation by applying Lie symmetry analysis and the RiemannLiouville fractional derivative. Furthermore, the multi-soliton solutions of the 3D TSF-Schamel-KdV equation are derived using the Hirota bilinear method. Finally, with the help of the multi-soliton solutions, we explore the characteristics of motion of ionacoustic solitary waves.

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Mixed lump–kink solutions to the KP equation

Cited by 229 publications

References 47 publications

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

Time–space fractional ( 2 + 1 ) $(2+1)$ dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions

Study of Ion‐Acoustic Solitary Waves in a Magnetized Plasma Using the Three‐Dimensional Time‐Space Fractional Schamel‐KdV Equation

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