Nonlinear Dynamics, ISSN 0924-090X, 10/2016, Volume 86, Issue 1, pp. 667 - 675

The Hirota bilinear method and Painleve-Backlund transformation are used to discuss the soliton solutions of the (3 + 1)-dimensional generalized shallow water...

Multiple-soliton solutions | Painlevé–Bäcklund transformation | Symbolic computation | Generalized shallow water equation | Hirota bilinear method | TRAVELING-WAVE SOLUTIONS | MECHANICS | KP | Painleve-Backlund transformation | ENGINEERING, MECHANICAL | Estuaries | Shallow water equations | Hyperbolic functions | Oceans | Trigonometric functions | Solitary waves | Solitons | Mathematical models | Transformations

Multiple-soliton solutions | Painlevé–Bäcklund transformation | Symbolic computation | Generalized shallow water equation | Hirota bilinear method | TRAVELING-WAVE SOLUTIONS | MECHANICS | KP | Painleve-Backlund transformation | ENGINEERING, MECHANICAL | Estuaries | Shallow water equations | Hyperbolic functions | Oceans | Trigonometric functions | Solitary waves | Solitons | Mathematical models | Transformations

Journal Article

Optik - International Journal for Light and Electron Optics, ISSN 0030-4026, 06/2016, Volume 127, Issue 12, pp. 4970 - 4983

This paper studies the exact solutions with parameters and optical soliton solutions of the (2 + 1)-dimensional hyperbolic nonlinear Schrödinger equation which...

Bright–dark-singular soliton solutions | Hyperbolic nonlinear Schrödinger equation | The soliton ansatz method | Modified simple equation method | Exp-function method | Bright-dark-singular soliton solutions | TRAVELING-WAVE SOLUTIONS | EVOLUTION | Hyperbolic nonlinear Schrodinger equation | OPTICS | TANH-FUNCTION METHOD | MEW | Mathematical analysis | Exact solutions | Solitons | Nonlinearity | Evolution | Schroedinger equation | Trigonometric functions | Solitary waves

Bright–dark-singular soliton solutions | Hyperbolic nonlinear Schrödinger equation | The soliton ansatz method | Modified simple equation method | Exp-function method | Bright-dark-singular soliton solutions | TRAVELING-WAVE SOLUTIONS | EVOLUTION | Hyperbolic nonlinear Schrodinger equation | OPTICS | TANH-FUNCTION METHOD | MEW | Mathematical analysis | Exact solutions | Solitons | Nonlinearity | Evolution | Schroedinger equation | Trigonometric functions | Solitary waves

Journal Article

Modern Physics Letters B, ISSN 0217-9849, 02/2018, Volume 32, Issue 6, p. 1850082

In this paper, we consider the (3 + 1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili (mKdV-KP) equation, which can be used to describe the...

bright soliton solution | explicit power series solution | A (3 + 1)-dimensional mKdV-KP equation | solitary wave ansatz | travelling wave solution | PHYSICS, CONDENSED MATTER | BREATHER WAVES | PHYSICS, APPLIED | LUMP SOLUTIONS | INFINITE CONSERVATION-LAWS | NONLINEAR SCHRODINGER-EQUATION | PHYSICS, MATHEMATICAL | A (3+1)-dimensional mKdV-KP equation | NONAUTONOMOUS ROGUE WAVES | BACKLUND TRANSFORMATION | QUASI-PERIODIC WAVES | MODULATION INSTABILITY | BILINEAR EQUATIONS | RATIONAL CHARACTERISTICS

bright soliton solution | explicit power series solution | A (3 + 1)-dimensional mKdV-KP equation | solitary wave ansatz | travelling wave solution | PHYSICS, CONDENSED MATTER | BREATHER WAVES | PHYSICS, APPLIED | LUMP SOLUTIONS | INFINITE CONSERVATION-LAWS | NONLINEAR SCHRODINGER-EQUATION | PHYSICS, MATHEMATICAL | A (3+1)-dimensional mKdV-KP equation | NONAUTONOMOUS ROGUE WAVES | BACKLUND TRANSFORMATION | QUASI-PERIODIC WAVES | MODULATION INSTABILITY | BILINEAR EQUATIONS | RATIONAL CHARACTERISTICS

Journal Article

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Full Text
Abundant soliton solutions for the Kundu–Eckhaus equation via tan(ϕ(ξ))-expansion method

Optik - International Journal for Light and Electron Optics, ISSN 0030-4026, 07/2016, Volume 127, Issue 14, pp. 5543 - 5551

In this paper, the improved -expansion method is proposed to seek more general exact solutions of the Kundu–Eckhaus equation. Being concise and...

Periodic and rational solutions | Improved [formula omitted]-expansion method | Kink | Kundu–Eckhaus equation | Solitons | Kundu-Eckhaus equation | Improved tan (Φ(ξ))-expansion method | Improved tan (Phi(xi)/2)-expansion method | BISWAS-MILOVIC EQUATION | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | OPTICAL SOLITONS | BACKLUND TRANSFORMATION | TRAVELING-WAVE SOLUTIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | COEFFICIENTS | SYSTEMS | OPTICS | Methods | Differential equations | Partial differential equations | Searching | Mathematical analysis | Nonlinear differential equations | Exact solutions | Hyperbolic functions | Trigonometric functions

Periodic and rational solutions | Improved [formula omitted]-expansion method | Kink | Kundu–Eckhaus equation | Solitons | Kundu-Eckhaus equation | Improved tan (Φ(ξ))-expansion method | Improved tan (Phi(xi)/2)-expansion method | BISWAS-MILOVIC EQUATION | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | OPTICAL SOLITONS | BACKLUND TRANSFORMATION | TRAVELING-WAVE SOLUTIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | COEFFICIENTS | SYSTEMS | OPTICS | Methods | Differential equations | Partial differential equations | Searching | Mathematical analysis | Nonlinear differential equations | Exact solutions | Hyperbolic functions | Trigonometric functions

Journal Article

Optik - International Journal for Light and Electron Optics, ISSN 0030-4026, 07/2017, Volume 140, pp. 730 - 734

By using the dynamical system method, the exact travelling wave solutions of a reduced model for (3 + 1)-dimensional spatiotemporal optical solitons in...

Bifurcation | Bright soliton solution | Periodic solution | Dynamical system method | Nonlocal nonlinear media | TRAVELING-WAVE SOLUTIONS | OPTICS | EQUATION | EXPLICIT | Sustainable development

Bifurcation | Bright soliton solution | Periodic solution | Dynamical system method | Nonlocal nonlinear media | TRAVELING-WAVE SOLUTIONS | OPTICS | EQUATION | EXPLICIT | Sustainable development

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 01/2014, Volume 15, Issue 1, pp. 140 - 148

In this paper, the method of bifurcation of planar dynamic systems is used to investigate bounded traveling wave solutions for the two-component...

MATHEMATICS, APPLIED | CAMASSA-HOLM | INTEGRABLE EQUATION | SHOCK-WAVES | Nonlinear dynamics | Breaking | Mathematical analysis | Solitons | Traveling waves | Mathematical models | Dynamical systems | Solitary waves

MATHEMATICS, APPLIED | CAMASSA-HOLM | INTEGRABLE EQUATION | SHOCK-WAVES | Nonlinear dynamics | Breaking | Mathematical analysis | Solitons | Traveling waves | Mathematical models | Dynamical systems | Solitary waves

Journal Article

International Journal of Non-Linear Mechanics, ISSN 0020-7462, 05/2019, Volume 111, pp. 95 - 105

The paper deals with different classes of non-linear reaction–diffusion equations with variable coefficients that admit exact solutions. The direct method for...

Equations with variable coefficients | Generalized traveling-wave solutions | Functional separable solutions | Non-linear reaction–diffusion equations | Exact solutions in implicit form | EXPLICIT SOLUTIONS | SIMILARITY REDUCTIONS | CONVECTION | BOUNDARY-LAYER EQUATIONS | MECHANICS | POROUS-MEDIUM EQUATION | GROUP CLASSIFICATION | Non-linear reaction-diffusion equations | NONCLASSICAL SYMMETRY REDUCTIONS | CONDITIONAL SYMMETRIES | DELAY | CONSTRAINTS METHOD | Differential equations | Nonlinear equations | Boundary value problems | Approximation | Functionals | Partial differential equations | Exact solutions | Linear equations | Reaction-diffusion equations | Diffusion

Equations with variable coefficients | Generalized traveling-wave solutions | Functional separable solutions | Non-linear reaction–diffusion equations | Exact solutions in implicit form | EXPLICIT SOLUTIONS | SIMILARITY REDUCTIONS | CONVECTION | BOUNDARY-LAYER EQUATIONS | MECHANICS | POROUS-MEDIUM EQUATION | GROUP CLASSIFICATION | Non-linear reaction-diffusion equations | NONCLASSICAL SYMMETRY REDUCTIONS | CONDITIONAL SYMMETRIES | DELAY | CONSTRAINTS METHOD | Differential equations | Nonlinear equations | Boundary value problems | Approximation | Functionals | Partial differential equations | Exact solutions | Linear equations | Reaction-diffusion equations | Diffusion

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 07/2018, Volume 93, Issue 2, pp. 349 - 360

Lie group analysis is applied to carry out the similarity reductions of the -dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. We obtain generators of...

Similarity solutions | (3 + 1) -Dimensional Calogero–Bogoyavlenskii–Schiff | Generators of infinitesimal transformations | Lie symmetries | Similarity transformations method | FORMS | TRAVELING-WAVE SOLUTIONS | MECHANICS | REDUCTION | (3+1)-Dimensional Calogero-Bogoyavlenskii-Schiff equation | DE-VRIES EQUATION | CONSERVATION-LAWS | ENGINEERING, MECHANICAL | Algebra | Differential equations | Partial differential equations | Brackets | Mathematical analysis | Group theory | Lie groups | Transformations | Generators | Commutators | Symmetry

Similarity solutions | (3 + 1) -Dimensional Calogero–Bogoyavlenskii–Schiff | Generators of infinitesimal transformations | Lie symmetries | Similarity transformations method | FORMS | TRAVELING-WAVE SOLUTIONS | MECHANICS | REDUCTION | (3+1)-Dimensional Calogero-Bogoyavlenskii-Schiff equation | DE-VRIES EQUATION | CONSERVATION-LAWS | ENGINEERING, MECHANICAL | Algebra | Differential equations | Partial differential equations | Brackets | Mathematical analysis | Group theory | Lie groups | Transformations | Generators | Commutators | Symmetry

Journal Article

Optik, ISSN 0030-4026, 04/2018, Volume 158, pp. 391 - 398

In this paper, we consider the nonlinear Schrödinger governing equation in fibers, which can be used to describe the propagation properties of optical soliton...

Stability analysis | Nonlinear Schrödinger equation | Bright soliton | Gaussian solution | Dark soliton | Traveling wave solution | BREATHER WAVES | SYMMETRIES | Nonlinear Schrodinger equation | DARK | ROGUE WAVES | SOLITARY WAVES | OPTICS | RATIONAL CHARACTERISTICS | G'/G-EXPANSION | BRIGHT

Stability analysis | Nonlinear Schrödinger equation | Bright soliton | Gaussian solution | Dark soliton | Traveling wave solution | BREATHER WAVES | SYMMETRIES | Nonlinear Schrodinger equation | DARK | ROGUE WAVES | SOLITARY WAVES | OPTICS | RATIONAL CHARACTERISTICS | G'/G-EXPANSION | BRIGHT

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 01/2017, Volume 73, Issue 2, pp. 220 - 225

Based on the Hirota bilinear form of the (3+1)-dimensional Jimbo–Miwa equation, ten classes of its lump-type solutions are generated via Maple symbolic...

Jimbo–Miwa equation | Lump-type solution | Quadratic function | RATIONAL SOLUTIONS | MATHEMATICS, APPLIED | TRAVELING-WAVE SOLUTIONS | SOLITONS | Jimbo-Miwa equation

Jimbo–Miwa equation | Lump-type solution | Quadratic function | RATIONAL SOLUTIONS | MATHEMATICS, APPLIED | TRAVELING-WAVE SOLUTIONS | SOLITONS | Jimbo-Miwa equation

Journal Article

Pramana, ISSN 0304-4289, 5/2018, Volume 90, Issue 5, pp. 1 - 20

In this research, we apply two different techniques on nonlinear complex fractional nonlinear Schrödinger equation which is a very important model in...

new auxiliary equation method | optical solitary travelling wave solutions | Astrophysics and Astroparticles | novel $$\left( {G'}/{G}\right) $$ G ′ / G -expansion method | kink and antikink | Physics, general | Nonlinear complex fractional Schrödinger equation | Physics | Astronomy, Observations and Techniques | novel (G | G) -expansion method | BOUSSINESQ EQUATION | PHYSICS, MULTIDISCIPLINARY | BIFURCATIONS | novel (G '/G)-expansion method | FIBERS | TRAVELING-WAVE SOLUTIONS | EVOLUTION | Nonlinear complex fractional Schrodinger equation | GINZBURG-LANDAU EQUATION | BRIGHT | Quantum theory | Methods

new auxiliary equation method | optical solitary travelling wave solutions | Astrophysics and Astroparticles | novel $$\left( {G'}/{G}\right) $$ G ′ / G -expansion method | kink and antikink | Physics, general | Nonlinear complex fractional Schrödinger equation | Physics | Astronomy, Observations and Techniques | novel (G | G) -expansion method | BOUSSINESQ EQUATION | PHYSICS, MULTIDISCIPLINARY | BIFURCATIONS | novel (G '/G)-expansion method | FIBERS | TRAVELING-WAVE SOLUTIONS | EVOLUTION | Nonlinear complex fractional Schrodinger equation | GINZBURG-LANDAU EQUATION | BRIGHT | Quantum theory | Methods

Journal Article

Optical and Quantum Electronics, ISSN 0306-8919, 3/2018, Volume 50, Issue 3, pp. 1 - 20

In this study, some new traveling wave solutions for fractional partial differential equations (PDEs) have been developed. The time-fractional Burgers...

Burgers equations | Fractional modified Reimann-Liouville derivative | ( $$\frac{G^{'}}{G^{2}}$$ G ′ G 2 )-Expansion method | Whitham Broer Kuap equations | Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Computer Communication Networks | Physics | Traveling wave solution | Electrical Engineering | Biological population model | (G′G2)-Expansion method | QUANTUM SCIENCE & TECHNOLOGY | 1ST INTEGRAL METHOD | EXP-FUNCTION METHOD | FUNCTIONAL VARIABLE METHOD | OPTICS | (G '/G)-Expansion method | ENGINEERING, ELECTRICAL & ELECTRONIC | Water waves | Mortality | Methods | Differential equations

Burgers equations | Fractional modified Reimann-Liouville derivative | ( $$\frac{G^{'}}{G^{2}}$$ G ′ G 2 )-Expansion method | Whitham Broer Kuap equations | Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Computer Communication Networks | Physics | Traveling wave solution | Electrical Engineering | Biological population model | (G′G2)-Expansion method | QUANTUM SCIENCE & TECHNOLOGY | 1ST INTEGRAL METHOD | EXP-FUNCTION METHOD | FUNCTIONAL VARIABLE METHOD | OPTICS | (G '/G)-Expansion method | ENGINEERING, ELECTRICAL & ELECTRONIC | Water waves | Mortality | Methods | Differential equations

Journal Article

Pramana, ISSN 0304-4289, 7/2018, Volume 91, Issue 1, pp. 1 - 4

In this paper, the canonical-like transformation method and trial equation method are applied to $$(2+1)$$ (2+1) -dimensional Chaffee–Infante equation, and...

Chaffee–Infante equation | canonical-like transformation method | Astrophysics and Astroparticles | travelling wave solutions | trial equation method | Physics, general | Physics | Astronomy, Observations and Techniques | 03.65.Vf | 05.45.Yv | 02.30.Jr | SYSTEM | TRAVELING-WAVE SOLUTIONS | PHYSICS, MULTIDISCIPLINARY | GORDON EQUATION | Chaffee-Infante equation | CLASSIFICATION | NONLINEAR EVOLUTION-EQUATIONS

Chaffee–Infante equation | canonical-like transformation method | Astrophysics and Astroparticles | travelling wave solutions | trial equation method | Physics, general | Physics | Astronomy, Observations and Techniques | 03.65.Vf | 05.45.Yv | 02.30.Jr | SYSTEM | TRAVELING-WAVE SOLUTIONS | PHYSICS, MULTIDISCIPLINARY | GORDON EQUATION | Chaffee-Infante equation | CLASSIFICATION | NONLINEAR EVOLUTION-EQUATIONS

Journal Article

Pramana, ISSN 0304-4289, 9/2019, Volume 93, Issue 3, pp. 1 - 7

In this paper, a generalised $$(3+1)$$ (3+1) -dimensional Kadomtsev–Petviashvili (KP) equation is considered. By transforming it into the bilinear form, one-,...

Grammian solution | Astrophysics and Astroparticles | soliton solution | Wronskian solution | Physics, general | Hirota bilinear form | Physics | Astronomy, Observations and Techniques | 02.30.Ik | TRAVELING-WAVE SOLUTIONS | PHYSICS, MULTIDISCIPLINARY | RENORMALIZATION METHOD | Aerospace engineering

Grammian solution | Astrophysics and Astroparticles | soliton solution | Wronskian solution | Physics, general | Hirota bilinear form | Physics | Astronomy, Observations and Techniques | 02.30.Ik | TRAVELING-WAVE SOLUTIONS | PHYSICS, MULTIDISCIPLINARY | RENORMALIZATION METHOD | Aerospace engineering

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 11/2017, Volume 104, pp. 607 - 612

The objectives of this paper are construct the conservation laws and the corresponding conserved quantities for the generalized coupled Zakharov–Kuznetsov...

Multiplier | Soliton solutions | Simplest equation method | Periodic solutions | Generalized coupled Zakharov–Kuznetsov equation | TRAVELING-WAVE SOLUTIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Generalized coupled Zakharov-Kuznetsov equation | PHYSICS, MATHEMATICAL | Environmental law

Multiplier | Soliton solutions | Simplest equation method | Periodic solutions | Generalized coupled Zakharov–Kuznetsov equation | TRAVELING-WAVE SOLUTIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Generalized coupled Zakharov-Kuznetsov equation | PHYSICS, MATHEMATICAL | Environmental law

Journal Article

Optik, ISSN 0030-4026, 05/2016, Volume 127, Issue 10, pp. 4222 - 4245

We analytically study the Schrödinger type nonlinear evolution equations by improved tan(Φ( )/2)-expansion method. Explicit solutions are derived, which...

Trigonometric function solution | Hyperbolic function solution | Improved tan(Φ(ξ)/2)-expansion method | Schrödinger equation | Exponential solution and rational solution | Improved tan(Pdbl(ξ)/2)-expansion method | TRAVELING-WAVE SOLUTIONS | Improved tan(Phi(xi)/2)-expansion method | Schrodinger equation | PARTIAL-DIFFERENTIAL-EQUATIONS | OPTICS | Searching | Mathematical analysis | Nonlinear differential equations | Solitons | Nonlinear evolution equations | Mathematical models | Schroedinger equation | Hyperbolic functions

Trigonometric function solution | Hyperbolic function solution | Improved tan(Φ(ξ)/2)-expansion method | Schrödinger equation | Exponential solution and rational solution | Improved tan(Pdbl(ξ)/2)-expansion method | TRAVELING-WAVE SOLUTIONS | Improved tan(Phi(xi)/2)-expansion method | Schrodinger equation | PARTIAL-DIFFERENTIAL-EQUATIONS | OPTICS | Searching | Mathematical analysis | Nonlinear differential equations | Solitons | Nonlinear evolution equations | Mathematical models | Schroedinger equation | Hyperbolic functions

Journal Article

Physics Letters A, ISSN 0375-9601, 2008, Volume 372, Issue 4, pp. 417 - 423

The ( )-expansion method is firstly proposed, where satisfies a second order linear ordinary differential equation (LODE for short), by which the travelling...

Travelling wave solutions | Hirota–Satsuma equations | KdV equation | Variant Boussinesq equations | ( [formula omitted])-expansion method | Homogeneous balance | Solitary wave solutions | mKdV equation | Hirota-Satsuma equations | G))-expansion method | frac(G | EXPANSION METHOD | F-EXPANSION | NONCOMPACT STRUCTURES | VARIANTS | PHYSICS, MULTIDISCIPLINARY | SCHRODINGER-EQUATION | SUB-ODE METHOD | (G '/G)-expansion method | PARTIAL-DIFFERENTIAL-EQUATIONS | solitary wave solutions | homogeneous balance | travelling wave solutions | variant Boussinesq equations | EXPLICIT

Travelling wave solutions | Hirota–Satsuma equations | KdV equation | Variant Boussinesq equations | ( [formula omitted])-expansion method | Homogeneous balance | Solitary wave solutions | mKdV equation | Hirota-Satsuma equations | G))-expansion method | frac(G | EXPANSION METHOD | F-EXPANSION | NONCOMPACT STRUCTURES | VARIANTS | PHYSICS, MULTIDISCIPLINARY | SCHRODINGER-EQUATION | SUB-ODE METHOD | (G '/G)-expansion method | PARTIAL-DIFFERENTIAL-EQUATIONS | solitary wave solutions | homogeneous balance | travelling wave solutions | variant Boussinesq equations | EXPLICIT

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 08/2017, Volume 40, Issue 12, pp. 4350 - 4363

An application of the G ′ / G ‐expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for...

the | exact traveling wave solutions | solitary wave and periodic wave solutions | expansion method | Burger equations | variants of the KdV–Burger | K(n,n)–Burger equations | the G′/G-expansion method | MATHEMATICS, APPLIED | the G '/G-expansion method | 1-SOLITON SOLUTION | SHALLOW-WATER WAVES | K(n,n)-Burger equations | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | KADOMTSEV-PETVIASHVILI EQUATION | OPTICAL SOLITON-SOLUTIONS | BACKLUND TRANSFORMATION | VARIABLE-COEFFICIENTS | variants of the KdV-Burger | PARTIAL-DIFFERENTIAL-EQUATIONS | TANH-FUNCTION METHOD | Cytokinins | Methods | Nonlinear evolution equations | Nonlinear programming | Partial differential equations | Nonlinear differential equations | Exact solutions

the | exact traveling wave solutions | solitary wave and periodic wave solutions | expansion method | Burger equations | variants of the KdV–Burger | K(n,n)–Burger equations | the G′/G-expansion method | MATHEMATICS, APPLIED | the G '/G-expansion method | 1-SOLITON SOLUTION | SHALLOW-WATER WAVES | K(n,n)-Burger equations | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | KADOMTSEV-PETVIASHVILI EQUATION | OPTICAL SOLITON-SOLUTIONS | BACKLUND TRANSFORMATION | VARIABLE-COEFFICIENTS | variants of the KdV-Burger | PARTIAL-DIFFERENTIAL-EQUATIONS | TANH-FUNCTION METHOD | Cytokinins | Methods | Nonlinear evolution equations | Nonlinear programming | Partial differential equations | Nonlinear differential equations | Exact solutions

Journal Article