Reductive Perturbation Method of Super KdV Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＳｕｐｅｒＫｄＶｅｑｕａｔｉｏｎｓａｒｅｕｔ-ｂｕｕｘ 3ｈｈｘ ｕｘｘｘ =0 ,ｈｔ-ｂｕｈｘ-ａｈｕｘ ｃｈｘｘ =0 ,( 1 )ｗｈｅｒｅａ ,ｂ,ｃ(ｃ≠ 0 )ａｒｅｃｏｎｓｔａｎｔｓ.Ｔｈｅｓｉｍｉｌａｒｓｏｌｕｔｉｏｎｓｏｆ ( 1 )ｗｅｒｅｇｉｖｅｎｉｎ [1 ]ｂｙｄｉｒｅｃｔｍｅｔｈｏｄｐｒｅｓｅｎｔｅｄｂｙＣｌａｒｋｓｏｎａｎｄＫｒｕｓｋａｌ.Ｉｎｔｈｉｓｐａｐｅｒ,( 1 )ａｒｅｃｈａｎｇｅｄｉｎｔｏｏｒｄｉｎａｒｙＫｄＶｅｑｕａｔｉｏｎｓｂｙｒｅｄｕｃｔｉｖｅｐｅｒｔｕｒｂａｔｉｏｎｍ…     

Bilinear forms and soliton interactions for two generalized KdV equations for nonlinear waves

Korteweg–de Vries (KdV)-type equations can describe the nonlinear waves in fluids, plasmas, etc. In this paper, two generalized KdV equations are under investigation. Bilinear forms of which are constructed with the Bell polynomials and an auxiliary variable. \(N\) -soliton solutions are given through the Hirota direct method. Via the asymptotic analysis, the soliton interactions of the first generalized KdV equation are analyzed, which turn out to be elastic. Singular breather solutions have been derived from the two-soliton solutions. The collision between soliton and singular breather appears to be elastic, and the bound states of soliton and singular breather are exhibited. Unlike the first one, the other generalized KdV equation can only support the bound states of solitons, for the regular and singular solitons alike.     

Explicit solutions to the coupled KdV equations with variable coefficients

1　IntroductionandtheProblemPresentedSeekingtheexplicitsolutionofthenonlinearpartialdifferentialequation (NPDEs)isanimportantsubjectinsolitontheoryanditsapplication .Formanyyears,themainattentionwaspaidtotheconstantcoefficientNPDEs[1～ 7],manypowerfulmethodshavebeenproposedanddeveloped.Inrecentyears,moreandmoreattentionshavebeenpaidtovariablecoefficientNPDEs[8～ 13].Manymethodssuchassimilarityreductionmethod ,truncatedexpansionmethodandhomogeneousbalancemethodhavebeenextendedtosolvevaria…     

Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).     

Exact solutions of a KdV equation with variable coefficients via Exp-function method

In this paper, the Exp-function method is used to obtain generalized solitonary solutions and periodic solutions of a KdV equation with five arbitrary functions. The results show that the Exp-function method with the help of symbolic computation provides a very effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

New exact solutions to KdV equations with variable coefficients or forcing

Jacobi elliptic function expansion method is extended to construct the exact solutions to another kind of KdV equations, which have variable coefficients or forcing terms. And new periodic solutions obtained by this method can be reduced to the soliton-typed solutions under the limited condition.     

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.     

Exact solutions to the KdV-Burgers'' equation

This paper presents two different methods for the construction of exact solutions to the KdVB equation. The first is a direct one based on a combination of solutions to the KdV equation and Burgers' equation. In this approach a number of unknown constants are involved, and it is shown that the equations leading to their determination are properly determined and are capable of solution.

The second method involves a series, and is essentially an extension of Hirota's method. This approach is capable of solving the KdVB equation exactly, and also of generalization to higher order equations with a KdVB-type nonlinearity.     

Analytical solutions for some nonlinear evolution equations

IntroductionItiswell_knownthatmanyimportantdynamicsprocessescanbedescribedbyspecificnonlinearpartialdifferentialequations .Whenanonlinearpartialdifferentialequationisusedtodescribeaphysicalparameterthatshowssomekindsofpropagationoraggregationproperties,oneofthemostimportantphysicalmotivationsistosolvethepartialdifferentialequationwithacertaintypeoftravellingwavesolution .Inthepastseveraldecades,therehavebeenmanyattemptsinthisfieldbothbymathematiciansandphysicists[1]- [16 ],however,duetothecomp…     

Solitary wave and similarity solutions of the combined KdV equation

In this paper, we discuss a property of solitary wave solutions of the combined KdV equation. Meantime, we point out that the combined KdV equation can be reduced to the Painlevé equation. Furthermore, utilizing special transformations of similarity variables, we derive a kind of new partial differential equations.     

Lump,soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [1] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation’s dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas.

     

Travelling wave solutions for a second order wave equation of KdV type

The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type.In different regions of the parametric space,sufficient conditions to guarantee the existence of solitary wave solutions,periodic wave solutions,kink and anti-kink wave solutions are given.All possible exact explicit parametric representations are obtained for these waves.     

Preventing Blow up by Convective Terms in Dissipative PDE’s

We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger’s type equations, convective Cahn–Hilliard equation, generalized Kuramoto–Sivashinsky equation and KdV type equations. The following common scenario is established: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in a finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similar to the case, when the equation does not involve convective term. This kind of result has been previously known for the case of Burger’s type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.     

Multi-soliton solutions of a variable-coefficient KdV hierarchy

In this paper, Hirota’s bilinear method is extended to a new KdV hierarchy with variable coefficients. As a result, one-soliton solution, two-soliton solution and three-soliton solutions are obtained, from which the uniform formula of \(N\) -soliton solution is derived. Thanks to the arbitrariness of the included functions, these obtained solutions possess rich local structural features like the ridge soliton and the concave column soliton. It is shown that the Hirota’s bilinear method can be used for constructing multi-soliton solutions of some other hierarchies of nonlinear partial differential equations with variable coefficients.     

Analysis of lumps,single-stripe,breather-wave,and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method

In this paper, we implement the Hirota’s bilinear method to extract diverse wave profiles to the generalized perturbed-KdV equation when the test function approaches are taken into consideration. Several novel solutions such as lump-soliton, lump-periodic, single-stripe soliton, breather waves, and two-wave solutions are obtained to the proposed model. We conduct some graphical analysis including 2D and 3D plots to show the physical structures of the recovery solutions. On the other hand, this work contains a correction of previous published results for a special case of the perturbed KdV. Moreover, we investigate the significance of the nonlinearity, perturbation, and dispersion parameters being acting on the propagation of the perturbed KdV. Finally, our obtained solutions are verified by inserting them into the governing equation.

     

Generation and evolution of oblique solitary waves in supercritical flows

It is considered that a thin strut sits in a supercritical shallow water flow sheet over a homogeneous or very mildly varying topography. This stationary 3-D problem can be reduced from a Boussinesq-type equation into a KdV equation with a forcing term due to uneven topography, in which the transverse coordinate Y plays a same role as the time in original KdV equation. As the first example a multi-soliton wave pattern is shown by means of N-soliton solution. The second example deals with the generation of solitary wave-train by a wedge-shaped strut on an even bottom. Whitham's average method is applied to show that the shock wave jump at the wedge vertex develops to a cnoidal wave train and eventually to a solitary wavetrain. The third example is the evolution of a single oblique soliton over a periodically varying topography. The adiabatic perturbation result due to Karpman & Maslov (1978) is applied. Two coupled ordinary differential equations with periodic disturbance are obtained for the soliton amplitude and phase. Numerical solutions of these equations show chaotic patterns of this perturbed soliton.     

A Simple Fast Method in Finding Particular Solutions of Some Nonlinear PDE

1　ＡＴｒｉａｌＦｕｎｃｔｉｏｎａｎｄａＲｏｕｔｉｎｅｔｏＦｉｎｄＡｎａｌｙｔｉｃａｌＳｏｌｕｔｉｏｎｏｆＴｗｏＴｙｐｅｓｏｆＮｏｎｌｉｎｅａｒＰＤＥ　　Ｗｅｔｒｅａｔｔｈｅｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎ ,ｗｈｉｃｈｉｓｆｏｒｍｅｄｂｙａｄｄｉｎｇｈｉｇｈｏｒｄｅｒｄｅｒｉｖａｔｉｖｅｔｅｒｍｓａｎｄｎｏｎｌｉｎｅａｒｔｅｒｍｓｔｏｔｈｅＢｕｒｇｅｒｓｅｑｕａｔｉｏｎ ｕ ｔ ｕ ｕ ｘ … ｕｐ ｕ ｘｑ α1 ｕ ｘ … αｎ ｎｕ ｘｎ =0 ,( 1)ｗｈｉｃｈｐ ,ｑ ,ｎａｎｄαｉ(ｉ =1,2…     

Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations

We prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized Korteweg-de Vries equations in the subcritical case (this includes in particular the KdV equation and the modified KdV equation). This result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation, which has its own interest. The proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized KdV equation. Accepted December 1, 2000?Published online April 3, 2001