Travelling Wave Solutions for the KdV-Burgers-Kuramoto and Nonlinear Schrödinger Equations Which Describe Pseudospherical Surfaces

Abstract

We use the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudospherical surfaces for the KdV-Burgers-Kuramoto and nonlinear Schrödinger equations with constant Gaussian curvature −1. Travelling wave solutions for the above equations are obtained by using a sech-tanh method and Wu's elimination method.

1. Introduction

The theory of integrable systems has been an active area of mathematics for the past thirty years. Different aspects of the subject have fundamental relations with mechanics and dynamics, applied mathematics, algebraic structures, theoretical physics, analysis including spectral theory and geometry [1–3]. Most differential geometers have some knowledge and experience with finite dimensional integrable systems as they appear in sympectic geometry (mechanics) or ordinary differential equations (ODEs), although the reformulation of part of this theory as algebraic geometry is not commonly known [4].

There are two quite separate methods of extension of these ideas to partial differential equations (PDEs): one based on algebraic constructions and one based on spectral theory and analysis. These are less familiar still to geometers.

Many geometric equations are known to have integrable aspects, especially if one takes into account that most experts do not have a good definition of “integrable” as applied to PDEs, particularly elliptic examples. In addition to those we mention in our historical discussion, the equations for harmonic maps (sigma models) from surfaces into groups [5–7], harmonic tori in symmetric spaces [8], constant mean curvature surfaces in space forms [9], isometric immersions of space forms in other space forms [10, 11], and the theory of affine spheres [12] and affine minimal surfaces [13] are all examples of “elliptic” integrable systems.

The ideas surrounding string theory resulted in a series of deep and not completely understood connections between representation theory of certain algebras and many of the more classical theories of integrable systems in mathematics [14–16]. Most recently, supersymmetric quantum field theories produce in a natural way moduli spaces of vacua or ground states which have new geometry generated by the supersymmetry. Since the supersymmetry generalizes the classical symmetries which produce integrals for the Euler-Lagrange equations via Noether's theorem, the connection with integrability is perhaps not surprising [17–19]. However, this does not explain entirely the use of integrable systems in hyper-Kähler geometry [20], Seiberg-Witten theory [21], special Kähler geometry, and quantum cohomology [22].

The 19th century geometers were mainly interested in the local theory of surfaces in 𝑅³, which we might regard as the prehistory of these modern constructions. The sine-Gordon equation arose first through the theory of surfaces of constant Gauss curvature −1, and the reduced 3-wave equation can be found in Darboux's work on triply orthogonal systems of 𝑅³ [23].

It is well-known that a differential equation (DE) for a real-valued function 𝑢(𝑥,𝑡), or a differential system for a 2-vector-valued function 𝑢(𝑥,𝑡), is said to describe pseudospherical surfaces (pss) if it is the necessary and sufficient condition for the existence of smooth real functions 𝑓_𝑖𝑗, 1≤𝑖≤3, 1≤𝑗≤2, depending only on 𝑢 and a finite number of derivatives, such that the one-forms𝜔_𝑖=𝑓_𝑖1𝑑𝑥+𝑓_𝑖2𝑑𝑡,1≤𝑖≤3, (1.1) satisfy the structure equations of a surface of constant Gaussian curvature −1𝑑𝜔₁=𝜔₃∧𝜔₂,𝑑𝜔₂=𝜔₁∧𝜔₃,𝑑𝜔₃=𝜔₁∧𝜔₂. (1.2)A DE for a real valued function 𝑢(𝑥,𝑡) is kinematically integrable if it is the integrability condition of a one-parameter family of linear problems [24–30]𝜈_𝑥=𝑃(𝜂)𝜈,𝜈_𝑡=𝑄(𝜂)𝜈 (1.3)in which 𝑃(𝜂) and 𝑄(𝜂) are 𝑆𝐿(2,𝑅)-valued functions of 𝑥, 𝑡 and (𝑢 and its derivatives) up to a finite order. Thus, an equation is kinematically integrable if it is equivalent to the zero curvature condition𝜕𝑃(𝜂)−𝜕𝑡𝜕𝑄(𝜂)+𝜕𝑥𝑃(𝜂),𝑄(𝜂)=0, (1.4)where tr𝑃(𝜂)=tr𝑄(𝜂)=0, for each 𝜂 (spectral parameter or eigenvalue). In addition, a DE will be said to be strictly kinematically integrable if it is kinematically integrable and diagonal entries of the matrix 𝑃(𝜂) introduced above are 𝜂 and −𝜂.

The main aim of this paper is to explain the relationships between local differential geometry of surfaces and integrability of evolutionary nonlinear evolution equations (NLEEs). New travelling wave solutions for the KdV-Burgers-Kuramoto (KBK) and non-linear Schrödinger (NLS) equations which describe pss are obtained.

The paper is organized as follows. The correspondence between KBK, NLS equations and their families of pss is established in Section 2. In Section 3, a new exact soliton solution is obtained for the KBK equation by using a sech-tanh method and Wu's elimination method. In Section 4, we construct a new travelling wave solution for the NLS equation by using the same above way. Finally, we give some conclusions in Section 5.

2. The KBK and NLS Equations that Describe pss

The inverse scattering method (ISM) was introduced first for the Korteweg-de Vries equation (KdVE) [18]. Later it was extended by Zakharov and Shabat [31] to a 2×2 scattering problem for the NLS equation and that was subsequently generalized by Ablowitz, Kaup, Newell, and Segur (AKNS) [32] to include a variety of NLEEs. Khater et al. [28] generalized the results of Konno and Wadati [33] by considering 𝜈 as a three-component vector and Ω as a traceless 3×3 matrix one-form. The above definition of a DE is equivalent to saying that the DE for 𝑢 is the integrability condition for the problem𝜈𝑑𝜈=Ω𝜈,𝜈=𝜈¹₂𝜈₃, (2.1)where 𝜈 is a vector and the 3×3 matrix Ω (Ω_𝑖𝑗, 𝑖,𝑗=1,2,3) is tracelesstrΩ=0, (2.2)and consists of a one-paramter (𝜂), family of one-forms in the independent variables (𝑥,𝑡), the dependent variable 𝑢, and its derivatives. Khater et al. [28] introduced the inverse scattering problem (ISP):𝜈_1𝑥=𝑓₃₁𝜈₂−𝑓₁₁𝜈₃,𝜈_2𝑥=−𝑓₃₁𝜈₁−𝜂𝜈₃,𝜈_3𝑥=−𝑓₁₁𝜈₁−𝜂𝜈₂,𝜈_1𝑡=𝑓₃₂𝜈₂−𝑓₁₂𝜈₃,𝜈_2𝑡=−𝑓₃₂𝜈₁−𝑓₂₂𝜈₃,𝜈_3𝑡=−𝑓₁₂𝜈₁−𝑓₂₂𝜈₂. (2.3)The associated integrability conditions for (1.3) or (2.1), which are obtained by cross-differentiation, then take the matrix form𝑑Ω−Ω∧Ω=0, (2.4)or the component form𝑓_12,𝑥−𝑓_11,𝑡=𝑓₃₁𝑓₂₂−𝜂𝑓₃₂,𝑓_22,𝑥=𝑓₁₁𝑓₃₂−𝑓₁₂𝑓₃₁,𝑓_32,𝑥−𝑓_31,𝑡=𝑓₁₁𝑓₂₂−𝜂𝑓₁₂. (2.5)

We will restrict ourselves to the case where 𝑓₂₁=𝜂. More precisely, we say that a DE for 𝑢(𝑥,𝑡) describes a pss if it is a necessary and sufficient condition for the existence of functions 𝑓_𝑖𝑗, 1≤𝑖≤3, 1≤𝑗≤2, depending on 𝑢(𝑥,𝑡) and its derivatives, 𝑓₂₁=𝜂, such that the one-forms in (1.1) satisfy the structure equations (1.2) of a pss. It follows from this definition that for each nontrivial solution 𝑢 of the DE, one gets a metric defined on 𝑀², whose Gaussian curvature is −1.

It has been known, for a long time, that the sine-Gordon (SG) equation describes a pss. In this paper, we extend the same analysis to include the KBK and NLS equations.

Examples: let 𝑀² be a differentiable surface, parametrized by coordinates 𝑥, 𝑡. (a) The KBK Equation
Consider𝜔₁=−121𝑢+𝑔(𝑥,𝑡)𝑑𝑥+2𝑢_𝑥+14𝑢²𝜔+𝑓(𝑥,𝑡)𝑑𝑡,₂1=𝜂𝑑𝑥+2𝜔𝜂𝑢−𝜂𝑔(𝑥,𝑡)𝑑𝑡,₃=−𝜂𝑑𝑥+−12𝜂𝑢+𝜂𝑔(𝑥,𝑡)𝑑𝑡, (2.6) in which the functions 𝑔(𝑥,𝑡) and 𝑓(𝑥,𝑡) satisfy the equations𝑔_𝑥+𝑔²+𝑓=0,𝑓_𝑥−𝑔_𝑡=12𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥𝑥𝑥+𝛾𝑢_{𝑥𝑥𝑥𝑥}. (2.7)Then 𝑀² is a pss if and only if 𝑢 satisfies the KBK equation𝑢_𝑡+𝑢𝑢_𝑥+𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥𝑥𝑥+𝛾𝑢_{𝑥𝑥𝑥𝑥}=0, (2.8)where 𝛼, 𝛽, and 𝛾 are constants.

(b) The NLS Equation
Consider𝜔₁=2𝑤𝑑𝑥+−4𝜂𝑤+2𝑣_𝑥𝜔𝑑𝑡,₂=2𝜂𝑑𝑥+2𝑣²+𝑤²−4𝜂²𝜔𝑑𝑡,₃=−2𝑣𝑑𝑥+4𝜂𝑣+2𝑤_𝑥𝑑𝑡. (2.9) Then 𝑀² is a pss if and only if 𝑢 satisfies the NLS equation𝑖𝑢_𝑡+𝑢_𝑥𝑥+2|𝑢|²𝑢=0,where𝑢=𝑣+𝑖𝑤. (2.10)

3. Travelling Wave Solutions for the KBK Equation

Now we will find travelling wave solutions 𝑢(𝑥,𝑡) for the KBK equation (2.8). The solutions of KBK equation possess their actual physical application; this is the reason why so many methods, such as Wiss-Tabor-Carnevale transformation method [34], tanh-function method [35], truncated expansion method [36], and so on, have been applied to obtain the solution of KBK equation. In this section, we obtain a new travelling wave solution class for KBK equations by using a sech-tanh method [37, 38] and Wu's elimination method [39]. The main idea of the algorithm is as follows. Suppose there is a PDE of the form𝑓𝑢,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,…=0, (3.1)where 𝑓 is a polynomial. By assuming travelling wave solutions of the form𝑢(𝑥,𝑡)=𝜙(𝜌),𝜌=𝜆(𝑥−𝑘𝑡+𝑐), (3.2)where 𝑘, 𝜆 are constant parameters to be determined, and 𝑐 is an arbitrary constant, from the two equations (3.1) and (3.2) we obtain an ODE 𝑓𝜙^′,𝜙^′′,𝜙^′′′,…=0, (3.3)where 𝜙^′=𝑑𝜙/𝑑𝜌. According to the sech-tanh method [37–41], we suppose that (3.3) has the following formal travelling wave solution:𝜙(𝜌)=^𝑛_𝑖=1sech^𝑖−1𝜌𝐵_𝑖sech𝜌+𝐴_𝑖tanh𝜌+𝐴₀, (3.4)where 𝐴₀,…,𝐴_𝑛 and 𝐵₁,…,𝐵_𝑛 are constants to be determined. Then we proceed as follows. (i) Equating the highest-order nonlinear term and highest-order linear partial derivative in (3.3) yields the value of 𝑛.(ii) Substituting (3.4) into (3.3), we obtain a polynomial equation involving tanh𝜌sech^𝑖𝜌, sech^𝑖𝜌 for 𝑖=0,1,2,…,𝑛 (with 𝑛 being positive integer).(iii) Setting the constant term and coefficients of sech𝜌,tanh𝜌,sech𝜌tanh𝜌,sech²𝜌,…, in the equation obtained in (ii) to zero, we obtain a system of algebraic equations about the unknown numbers 𝑘, 𝜆, 𝐴₀, 𝐴_𝑖, 𝐵_𝑖 for 𝑖=1,2,…,𝑛.(iv)Using the Mathematica and Wu's elimination methods, the algebraic equations in (iii) can be solved. These yield the solitary wave solutions for the system (3.3). We remark that the above method yields solutions that include terms sech𝜌 or tanh𝜌, as well as their combinations. There are different forms of those obtained by other methods, such as the homogenous balance method [42, 43]. We assume formal solutions of the form𝑢(𝑥,𝑡)=𝜙(𝜌),𝜌=𝜆(𝑥−𝑘𝑡+𝑐), (3.5)where 𝑘, 𝜆 are constant parameters to be determined later, and 𝑐 is an arbitrary constant. Substituting from (3.5) and (2.8), we obtain an ODE−𝑘𝜙^′+𝜙𝜙^′+𝜆𝛼𝜙^′′+𝛽𝜆²𝜙^′′′+𝛾𝜆³𝜙^′′′′=0. (3.6)

(i) We suppose that (3.6) has the following formal solution: 𝜙(𝜌)=𝐴₀+𝐴₁sech𝜌+𝐵₁tanh𝜌+𝐴₂sech²𝜌+𝐵₂sech𝜌tanh𝜌+𝐴₃sech³𝜌+𝐵₃sech²𝜌tanh𝜌. (3.7)

(ii) From (3.6) and (3.7), we get−𝑘𝜙^′+𝜙𝜙^′+𝜆𝛼𝜙^′′+𝛽𝜆²𝜙^′′′+𝛾𝜆³𝜙^′′′′=−𝐴₁𝐵₁−𝐴₀𝐵₂+𝑘𝐵₂+𝜆𝛼𝐴₁−𝛽𝜆²𝐵₂+𝛾𝜆³𝐴₁+sech𝜌−𝐴₀𝐴₁−𝐵₁𝐵₂+𝑘𝐴₁+𝛼𝜆𝐵₂−𝛽𝜆²𝐴₁+𝛾𝜆³𝐵₂+𝐴sech𝜌tanh𝜌₀𝐵₁−2𝐴₂𝐵₁−2𝐴₁𝐵₂−2𝐴₀𝐵₃−𝑘𝐵₁+2𝑘𝐵₃+4𝛼𝜆𝐴₂+4𝛽𝜆²𝐵₁−8𝛽𝜆²𝐵₃+16𝛾𝜆³𝐴₂sech²𝜌+−𝐴²₁−2𝐴₀𝐴₂+𝐵²₁−𝐵²₂−2𝐵₁𝐵₃+2𝑘𝐴₂−2𝛼𝜆𝐵₁+4𝛼𝜆𝐵₃−8𝛽𝜆²𝐴₂−8𝛾𝜆³𝐵₁+16𝛾𝜆³𝐵₃sech²+𝜌tanh𝜌2𝐴₁𝐵₁−3𝐴₃𝐵₁+2𝐴₀𝐵₂−3𝐴₂𝐵₂−3𝐴₁𝐵₃−2𝑘𝐵₂−2𝛼𝜆𝐴₁+9𝛼𝜆𝐴₃+20𝛽𝜆²𝐵₂−20𝛾𝜆³𝐴₁+81𝛾𝜆³𝐴₃sech³𝜌+−3𝐴₁𝐴₂−3𝐴₀𝐴₃+3𝐵₁𝐵₂−3𝐵₂𝐵₃+3𝑘𝐴₃−6𝛼𝜆𝐵₂+6𝛽𝜆²𝐴₁−27𝛽𝜆²𝐴₃−60𝛾𝜆³𝐵₂sech³+𝜌tanh𝜌3𝐴₂𝐵₁+3𝐴₁𝐵₂−4𝐴₃𝐵₂+3𝐴₀𝐵₃−4𝐴₂𝐵₃−3𝑘𝐵₃−6𝛼𝜆𝐴₂−6𝛽𝜆²𝐵₁+60𝛽𝜆²𝐵₃−120𝛾𝜆³𝐴₂sech⁴𝜌+−2𝐴²₂−4𝐴₁𝐴₃+2𝐵²₂+4𝐵₁𝐵₃−2𝐵²₃−12𝛼𝜆𝐵₃+24𝛽𝜆²𝐴₂+24𝛾𝜆³𝐵₁−240𝛾𝜆³𝐵₃sech⁴+𝜌tanh𝜌4𝐴₃𝐵₁+4𝐴₂𝐵₂+4𝐴₁𝐵₃−5𝐴₃𝐵₃−12𝛼𝜆𝐴₃−24𝛽𝜆²𝐵₂+24𝛾𝜆³𝐴₁−408𝛾𝜆³𝐴₃sech⁵𝜌+−5𝐴₂𝐴₃+5𝐵₂𝐵₃+60𝛽𝜆²𝐴₃+120𝛾𝜆³𝐵₂sech⁵+𝜌tanh𝜌5𝐴₃𝐵₂+5𝐴₂𝐵₃−60𝛽𝜆²𝐵₃+120𝛾𝜆³𝐴₂sech⁶𝜌+−3𝐴²₃+3𝐵²₃+360𝛾𝜆³𝐵₃sech⁶𝜌tanh𝜌+6𝐴₃𝐵₃+360𝛾𝜆³𝐴₃sech⁷𝜌=0. (3.8)

(iii) Setting the coefficients of sech^𝑗𝜌tanh^𝑖𝜌 for 𝑖=0,1 and 𝑗=1,2,…,7 to zero, we have the following set of overdetermined equations in the unknowns 𝐴₀, 𝐴₁, 𝐴₂, 𝐴₃, 𝐵₁, 𝐵₂, 𝐵₃, 𝜆, and 𝑘.

(iv) We now solve the above set of equations (3.8) by using Mathematica and Wu's elimination method, and obtain the following solutions: 𝜆²=(5±19)𝛼64𝛾,𝛽²=16𝛼𝛾,𝑘=5𝛽𝜆²+𝛼𝛽,𝐴4𝛾₀=𝐴₁=𝐵₁=𝐵₂=𝐴₃=0,𝐴₂=15𝛽𝜆²,𝐵₃=−120𝛾𝜆³. (3.9)Substituting (3.9) into (3.7), we obtain𝑢(𝑥,𝑡)=15𝜆²sech²𝜌𝛽−8𝛾𝜆tanh𝜌,where𝜌=𝜆(𝑥−𝑘𝑡+𝑐). (3.10)

4. Travelling Wave Solutions for the NLS Equation

Now we will find travelling wave solutions 𝑢(𝑥,𝑡) for the NLS equation (2.10). Equation (2.10) can be written in the real form 𝑢=𝑣+𝑖𝑤 as follows: 𝑣_𝑡+𝑤_𝑥𝑥+2𝑣²+𝑤²𝑤=0,−𝑤_𝑡+𝑣_𝑥𝑥+2𝑣²+𝑤²𝑣=0. (4.1)We assume formal solutions of the form𝑣(𝑥,𝑡)=𝜙(𝜌),𝑤(𝑥,𝑡)=𝜃(𝜌),𝜌=𝜆(𝑥−𝑘𝑡+𝑐), (4.2)where 𝑘, 𝜆 are constant parameters to be determined later, and 𝑐 is an arbitrary constant. Substituting from (4.2) into (4.1), we obtain two ODEs:−𝑘𝜆𝜙^′+𝜆²𝜃^′′+2𝜙²+𝜃²𝜃=0,𝑘𝜆𝜃^′+𝜆²𝜙^′′+2𝜙²+𝜃²𝜙=0. (4.3)

(i) Equating the highest-order nonlinear term and highest-order linear partial derivative in (4.3) yields 𝑛=1. Then (4.3) has the following formal solutions:𝜙(𝜌)=𝐴₀+𝐴₁sech𝜌+𝐵₁tanh𝜌,𝜃(𝜌)=𝑎₀+𝑎₁sech𝜌+𝑏₁tanh𝜌. (4.4)

(ii) With the aid of Mathematica, substituting (4.4) into (4.3), then we obtain a polynomial equation involving tanh^𝑖𝜌sech^𝑗𝜌 for 𝑖=0,1, 𝑗=0,1,2,3.−𝑘𝜆𝜙^′+𝜆²𝜃^′′+2𝜙²+𝜃²𝜃=2𝑎³₀+2𝑎₀𝐴²₀+4𝑏₁𝐴₀𝐵₁+6𝑎₀𝑏²₁+2𝑎₀𝐵²₁+𝜆²𝑎₁+6𝑎²₀𝑎₁+2𝑎₁𝐴²₀+4𝑎₀𝐴₀𝐴₁+6𝑎₁𝑏²₁+4𝑏₁𝐴₁𝐵₁+2𝑎₁𝐵²₁+sech𝜌2𝑏³₁+6𝑎²₀𝑏₁+4𝑎₀𝐴₀𝐵₁+2𝑏₁𝐴²₀+2𝑏₁𝐵²₁+tanh𝜌𝑘𝜆𝐴₁+12𝑎₀𝑎₁𝑏₁+4𝑏₁𝐴₀𝐴₁+4𝑎₁𝐴₀𝐵₁+4𝑎₀𝐴₁𝐵₁+sech𝜌tanh𝜌6𝑎₀𝑎²₁+4𝑎₁𝐴₀𝐴₁+2𝑎₀𝐴²₁−6𝑎₀𝑏²₁−𝑘𝜆𝐵₁−4𝑏₁𝐴₀𝐵₁−2𝑎₀𝐵²₁sech²𝜌+−2𝜆²𝑏₁+6𝑎²₁𝑏₁−2𝑏³₁+2𝑏₁𝐴²₁+4𝑎₁𝐴₁𝐵₁−2𝑏₁𝐵²₁sech²+𝜌tanh𝜌−2𝜆²𝑎₁+2𝑎³₁+2𝑎₁𝐴²₁−6𝑎₁𝑏²₁−4𝑏₁𝐴₁𝐵₁−2𝑎₁𝐵²₁sech³𝜌=0,𝑘𝜆𝜃^′+𝜆²𝜙^′′+2𝜙²+𝜃²𝜙=2𝑎²₀𝐴₀+6𝐴₀𝐵²₁+2𝐴₀𝐵²₁+4𝑎₀𝑏₁𝐵₁+2𝐴³₀+𝜆²𝐴₁+6𝐴²₀𝐴₁+2𝐴₁𝑏²₁+4𝑎₀𝑎₁𝐴₀+6𝐴₁𝐵²₁+4𝑎₁𝑏₁𝐵₁+2𝐴₁𝑎²₀+sech𝜌2𝐵³₁+6𝐴²₀𝐵₁+4𝑎₀𝐴₀𝑏₁+2𝐵₁𝑎²₀+2𝑏²₁𝐵₁+tanh𝜌−𝑘𝜆𝑎₁+12𝐴₀𝐴₁𝐵₁+4𝑎₁𝐴₀𝑏₁+4𝑎₀𝑎₁𝐵₁+4𝑎₀𝐴₁𝑏₁+sech𝜌tanh𝜌6𝐴₀𝐴²₁+4𝑎₀𝑎₁𝐴₁+2𝐴₀𝑎²₁−6𝐴₀𝐵²₁+𝑘𝜆𝑏₁−4𝑏₁𝑎₀𝐵₁−2𝐴₀𝑏²₁sech²𝜌+−2𝜆²𝐵₁+6𝐴²₁𝐵₁−2𝐵³₁+2𝐵₁𝑎²₁+4𝑎₁𝐴₁𝑏₁−2𝐵₁𝑏²₁sech²+𝜌tanh𝜌−2𝜆²𝐴₁+2𝐴³₁+2𝑎²₁𝐴₁−6𝐴₁𝐵²₁−4𝑏₁𝑎₁𝐵₁−2𝐴₁𝑏²₁sech³𝜌=0. (4.5)

(iii) Setting the constant term and coefficients of tanh^𝑖𝜌sech^𝑗𝜌 for 𝑖=0,1, 𝑗=0,1,2,3, in the equation obtained in (ii) to zero, we obtain a system of algebraic equations about the unknown numbers 𝐴₀, 𝐴₁, 𝐵₁, 𝑎₀, 𝑎₁, 𝑏₁, and 𝑘.

(iv) Now we solve the above set of (4.5) by using Mathematica and Wu's elimination method, and we obtain the following solution:𝐴₀=𝐴₁==𝑎₁=𝑏₁=0,𝑎₀=±𝜆𝐵₁=±𝑖𝜆,𝑘=∓𝑖𝜆. (4.6)Substituting (4.6) into (4.4), we obtain𝑣(𝑥,𝑡)=𝜙(𝜌)=±𝑖𝜆tanh𝜌,𝑤(𝑥,𝑡)=𝜃(𝜌)=±𝜆, (4.7)then the solution of NLS equation (2.10) takes the form𝑢(𝑥,𝑡)=±𝑖𝜆(1+tanh𝜌),𝜌=𝜆(𝑥±𝑖𝜆𝑡+𝑐). (4.8)

5. Conclusions

We find the relationship between KBK, NLS equations and their families of pss. With the help of Mathematica, many travelling solutions for the KBK and NLS equations of the pseudospherical class are obtained by using a sech-tanh method and Wu's elimination method. We obtained some new solitary wave solutions and periodic solutions.