KdV Equation with Self-consistent Sources in Non-uniform Media

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1 Commun. Theor. Phys. Beiing China 5 9 pp c Chinese Physical Society and IOP Publishing Ltd Vol. 5 No. 6 June 5 9 KdV Equation with Self-consistent Sources in Non-uniform Media HAO Hong-Hai WANG Guang-Sheng and ZHANG Da-Jun Department of Mathematics Shanghai University Shanghai 444 China Department of Basic Courses Naning Institute of Technology Naning 67 China Received May 9 8 Abstract Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources corresponding to λ t = aλ and its isospectral counterpart is given from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides the soliton solutions for the two equations are obtained by means of Hirota s method and Wronskian technique respectively. Meanwhile the dynamical properties for these solutions are investigated. PACS numbers:.3.ik Key words: non-isospectral KdV equation with self-consistent sources gauge transformation Hirota s method Wronskian technique dynamics Introduction Soliton equations with self-consistent sources SESCSs have received much attention in recent years. [ 8] These equations exhibit richer nonlinear dynamics than soliton equations themselves. Physically the source can result in solitary waves moving with a nonconstant velocity which can describe interactions between different solitary waves and is relevant to some problems of hydrodynamics solid state physics plasma physics etc. [9] Besides this kind of systems also can bring many mathematically interesting treatments. These equations can be solved by the Inverse Scattering Transform the Darboux transformation the bilinear method etc. [ ] Recently we have investigated some SESCSs by means of the Hirota s method and Wronskian technique. [8 3] These equations also admit bilinear forms and N-soliton solutions in Hirota s pertubation and Wronskian form. [] In this paper we aim to investigate the nonisospectral KdV equations with self-consistent sources non-isospectral KdVESCSs. These equations are related to time-dependent spectral parameters and can describe solitary waves in non-uniform media. [ 4] We first derive two non-isospectral KdVESCSs corresponding to different time evolutions of the spectral parameter λ. One is called non-isospectral KdVESCS-I for λ t = aλ and the other is called non-isospectral KdVESCS-II for λ t = 8λ. N- soliton solutions for both of them are obtained in Hirota s form and Wronskian form. For the dynamics we focus on the following three points. First how do the characters of solitons rely on time and sources? Second can the elastic scattering appear in the non-uniform media? Third do the sources lead to special soliton behaviors? We employ asymptotic analysis in the coordinate frame co-moving with a single soliton [4 6] to discuss two-soliton scattering. Besides the ghost soliton behaviors in degenerate two-soliton case are also described in detail. This paper is organized as follows. In Sec. the nonisospectral KdVESCSs are derived and their Lax pairs are given. In Sec. 3 some results for the isospectral KdVESCS are listed out. In Sec. 4 the non-isospectral KdVESCS-I is discussed by means of gauge transformation and bilinear method. In Sec. 5 the non-isospectral KdVESCS-II is investigated through the bilinear method. A conclusion is given in the final section. Non-isospectral KdVESCSs and Lax Intergrability In this section we derive the non-isospectral KdVESCS-I II and give their Lax pairs. Considering the linear problem [7] φ xx = λ uφ a φ t = Aφ + Bφ x. The compatibility condition suggests that b A x + B xx = u t = LB + λ t B x λ 3 L = 3 + u + u x 4 with = / x and = =. Suppose that operator B has following form β meets B = B + n B = b k λ n k β = λ λ k= 5a 5b Lβ = λ β x 6 The proect supported by the National Natural Science Foundation of China under Grant Nos. 377 and 67 the Foundation for Excellent Postgraduates of Shanghai University under Grant No. Shucx87 Corresponding author hao haier@shu.edu.cn

2 99 HAO Hong-Hai WANG Guang-Sheng and ZHANG Da-Jun Vol. 5 therefore we can take β = φ φ xx = λ uφ =...N. 7 Then equation 3 can be rewritten as u t = LB + λ t B xλ + L x λ. 8 In the above equation u t = LB + λ t B xλ reduces to non-isospectral KdV hierarchy when λ t while the rest part yields φ x. 9 When n = b = 4 λ t = aλ a is a real constant we can derive out the non-isospectral KdVESCS-I u t + u xxx + 6uu x + au + xu x u x gt + φ x = a φ xx = λ uφ =...N. b The corresponding A and B are described as A = u x + a φ x λ λ B = 4λ u ax + gt + φ λ λ. When n = b x = 4 λ t = 8λ the non-isospectral KdVESCS-II can be derived u t + xu xxx + 6uu x + 4u xx + 8u + u x u + φ x = a φ xx = λ uφ =...N. b The corresponding A and B are described as A = λ + u + xu x φ x λ λ B = 4xλ xu + u + φ λ λ. 3 3 Some Results for Isospectral KdVESCS We list some results for the isospectral KdVESCS in this section. 3. Bilinear Form and Exact Solutions The isospectral KdVESCS [6] u t + u xxx + 6uu x + φ x = 4a φ xx = λ uφ =...N 4b can be rewritten into the bilinear form [3] D t D x + Dxf 4 f = g 5a D xg f = λ g f λ R =...N 5b by the dependent variable transformations u = lnf xx 6a φ = g =...N 6b f D is the well-known Hirota bilinear operator defined as [7] The N-soliton solution [3] f = µ= D m x D n t a b = x x m t t n ax tbx t x =xt =t. 7 N exp µ ζ + g h = k h β h te ζ h µ= <l N N exp h µ µ l A l 8a µ ζ + A h + <l N l h µ µ l A l 8b λ = k ζ = k x 4k 3 t e A l = k k l k + k l β zdz + ζ 9 k ζ k < k < < k N are all real constants k β t is an arbitrary non-negative continuous function of t defined on + and the sum over µ = refers to each of the µ = =...N. Besides the N-soliton solution also has the Wronskian form. Theorem 3. [3] The bilinear isospectral KdVESCS 5 admits the following Wronskian solution f = Wφ φ...φ N = N a g h = G h t N τ h h =...N b τ h = δ h δ h...δ hn T h G h t = N kh β h t kh k l l= φ = e ζ + e ζ l=h+ kl k h a b ζ is defined by 9 and we also assume that k < k <

3 No. 6 KdV Equation with Self-consistent Sources in Non-uniform Media 99 < k N. 3. Conservation Laws The conservation law of KdV isospectral evolution hierarchy is ω t = A + ikb + Bω x ω = φ x /φ ik is determined by Riccati equation ω x + ω + ikω + u =. 3 For the isospectral KdVESCS we have ω t = u x φ x λ λ Noting that + 4λ u + λ λ k = λ φ ω + ik. 4 λ λ x ±λk n 5 λ n= we can get infinitely many conservation laws ω t = J x 6 ω are conserved densities and J associated fluxes. Obviously the isospectral KdVESCS has the same conserved densities but different associated fluxes with the KdV equation. The first three non-trivial conserved densities are ω = u ω = u xx u 7a ω 3 = u xxxx 5u x 6uu xx u Dynamics of Isospectral KdVESCS 7b i One-soliton characteristics From Eqs. 6 and 8 we get one-soliton and the corresponding source u = k sech ζ φ = k β tsech ζ 8a 8b ζ is given by 9. Equation 8a provides a soliton travelling with the amplitude k and top trace xt = 4k3 t + β zdz ζ. 9 k For any fixed time t this top trace is a straight line. The source or the role that β t plays is to change the velocity of the soliton but not the shape. Because of k β t is an arbitrary non-negative continuous function of t lead to x t > so we cannot find the stationary soliton in the real area. Figure describes the shape and motion of one-soliton with time dependent top trace. Fig. Shape and motion of one-soliton of the isospectral KdVESCS. A moving soliton given by 8a for t [ 44] x [ 33] k =.5 β t = cost and ζ =. ii Two-soliton scattering When N = equations 8 and 9 provide the twosoliton solution. Let us call these two solitons ζ -soliton and ζ -soliton for convenience. The solitons can scatter elastically under some conditions as shown in Fig.. Fig. Two-soliton scattering of the isospectral Kd- VESCS. a Shape and motion of two-soliton solution u given by 6 and 8a for k =.5 k =.55 β t =.5 β t = cos t and ζ = ζ = ; b Density plot of the two-soliton solution given by 6 and 8a for k =.5 k =.55 β t =.5 β t = cos t and ζ = ζ =. We first suppose that k > k > and [ β t β ] t + 4k k k dt k as t ± 3 then investigate the behavior of two-soliton. We also straighten the travelling traectories of ζ -soliton and in-

4 99 HAO Hong-Hai WANG Guang-Sheng and ZHANG Da-Jun Vol. 5 troduce X = x Y = x β t + 4k k β t + 4k k dt + ζ k 3 dt + ζ k. 3 In the coordinate frame X t if the frame co-moves with ζ -soliton i.e. ζ = k X stays constant we have [ β t ζ = k X + k β ] t + 4k k k dt k + ζ k ζ k as t ± 33 which suggests k sech ζ t + k sech ζ + ln k k t. k + k 34 Thus we have extracted out the initial and final states of ζ -soliton and it then follows that ζ -soliton gets a leftward phase shift lnk k /k + k with respect to the frame X t after interaction. Similarly in the coordinate frame Y t we let ζ stay constant and t ±. We have k sech ζ t. k sech ζ + ln k k t + k + k 35 That means ζ -soliton gets a rightward phase shift lnk k /k + k with respect to the frame Y t after interaction. 4 Conservation Laws and Exact Solutions for Non-isospectral KdVESCS-I 4. Gauge Transformation Exact Solutions and Conservation Laws There exists a gauge transformation between the e t [e at 3at t Ω e at x + 3a e = x [e 3at 3at t J e at x + 3a [ ax gt ] e e at 3at Ω e at x + 3a non-isospectral KdVESCS-I and the isospectral counterpart.we describe it by the following theorem. Theorem 4. By the transformation S = e at u X = e at x + gze az dz T = e 3at η = λe at ψ = e at φ 36a 3a η = λ e at ψ = e at φ =...N 36b the non-isospectral KdVESCS-I can be transformed into isospectral KdVESCS S T + S XXX + 6SS X + ψ x = 37a ψ XX = η Sψ =...N 37b and the Lax pair for non-isospectral KdVESCS-I is also transformed to the Lax pair of isospectral one with ψ XX = η Sψ ψ T = Aψ + Bψ X 38 B = 4η S + A = S X ψ η η ψ X η η. 39 The proof can be finished by direct verifications. Employing the gauge transformation we can easily get conservation laws of the non-isospectral KdVESCS-I from the known results 37. If T ΩT X S S X... XS... = X JT X S S X... XS... 4 is a conservation law for the isospectral KdVESCS 37 Ω is a density and J a flux then ] gze az dz e at u e 3at u x...e +at xu... gze az dz e at u e 3at u x... e +at xu... is a conservation law for the non-isospectral KdVESCS-I. In fact by noting that equation 4 can directly be derived from 4. Thus if gze az dz e at u e 3at u x...e +at xu... ] 4 X = e at x T = e 3at [ t + [ax gt] x ] 4 Q = + ΩT X S S X... XS...dX 43 is a conserved quantity for the isospectral KdVESCS 37 so + e Q = e at 3at t Ω e at x + gze az dz e at u e 3at u x...e +µt 3a xu... dx 44

5 No. 6 KdV Equation with Self-consistent Sources in Non-uniform Media 993 is a conserved quantity for the non-isospectral KdVESCS- I. As a result from Eq. 7 the first three non-trivial conserved densities for the non-isospectral KdVESCS-I are ω = e at u ω = e 3at u + u xx 45a ω 3 = e 5at u xxxx + 5u x + 6uu xx + u Bilinear Approach 45b In this subsection we solve the non-isospectral KdVESCS-I through Hirota s method and Wroskian technique. By the transformation 6 the equation can be transformed into the bilinear form D t D x + D 4 x + ax gtd xf f + af x f = g 46a D xg f = λ tg f =...N. 46b Employing the standard Hirota s procedure one can work out one-soliton solution ft x = + e ξ g = b e at β te ξ 47 ξ = b e at x + w t β zdz + ξ 48 b and ξ are arbitrary real constants b e at β t is an arbitrary non-negative continuous function of t w t satisfies the identity w t = b e at gt 4b 3 e 3at and we have taken λ t = k t = b e at in 46b. Two-soliton solution can be described as ft x = + e ξ + e ξ k t k t e ξ + +ξ k t + k t 49a g = k tβ te ξ + k tβ t k t k t k t + k t eξ +ξ 49b g = k tβ te ξ k tβ t k t k t k t + k t eξ +ξ λ t = k t = b e at in 46b and ξ is defined as 48 but with subscript instead of. Generally the N-soliton solution can be described as f = N exp µ ξ + µ µ l A l 5a µ= g h = k h tβ h te ξ h µ= <l N N exp h µ ξ + A h + <l N l h 49c µ µ l A l h =...N 5b k t = b e at ξ = b e at x + w t e A l = k t k l t k t + k l t 5a β zdz + ξ 5b with arbitrary real constants b and ξ arbitrary nonnegative continuous t-dependent function k tβ t and w t satisfies the identity w t = b e at gt 4b 3 e 3at. Besides the non-isospectral KdVESCS-I also admits solutions in Wronskian form. Theorem 4. The following Wronskian solutions f = N = Wφ φ... φ N 5a g h = G h t N τ h h =... N 5b solve the bilinear non-isospectral KdVESCS-I 46 h G h t = N kh tβ h t kh t k l t kl t k h t φ = e ξ + e ξ 53 l= l=h+ τ h = δ h δ h...δ hn T k t and ξ are defined as 5a and 5b respectively and λ t = k t in 46b we also assume that b < b < < b N. Proof Using property of determinate we have f x = N N f xx = N 3 N N + N N + 54a f xxx = N 4 N N N + N 3 N N + + N N + f xxxx = N 5 N 3 N N N + 3 N 4 N N N + 54b + N 3 N N N 3 N N + + N N c

6 994 HAO Hong-Hai WANG Guang-Sheng and ZHANG Da-Jun Vol. 5 g hx = G h t N 3 N τ h g hxx = G h t N 4 N N τ h + G h t N 3 N τ h. 55 Noting that φ satisfies and the identity φ xx = k tφ =...N 56 N k h tg h f = ki g tf N h By a direct substitution and complicate calculation Eq. 46b yields Besides φ satisfies i= i=i ki tg h f. 57 N 3 N τ h N N 3 N τ h N N + N 3 N N N τ h =. 58 for each =...N which leads to φ t = 4φ xxx [ax gt]φ x β t k t φ x 59 f t = 4 N 4 N N N N 3 N N + + N N + [ ax gt ] N N an N N β t +l l e ξ e ξ A l 6 f tx = 4 N 5 N 3 N N N N 3 N N + + N N + 3 [ ax gt ] l= N 3 N N + N N + a N N an N N N β t [ N +l l e ξ e ξ B l + +N N e ξ e ξ ] A l 6 l= A l is the cofactor of f B l is the cofactor of f x and A N = B N. By employing the same treatment for β t in the isospectral KdVESCS given in Ref. [3] the left side of 46a can be written into β t N + h= N N l= h= N+h k h+n l+h k l+h N t[ h l ]A l B h l= t[ h N ]A N B h + N k N ta N l+n k l+n t[ l + N ]A l B N. 6 Then by referring to the proof for the Wronskian solution to the isospectral one in Ref. [3] we immediately reach the right hand side of 46a if taking g h to be defined as 5b. 4.3 Dynamics for Non-isospectral KdVESCS-I i One-soliton characteristics From 6 and 47 we have one-soliton u = b e at sech ξ 63 ξ is given by 48. Equation 63 provides a soliton travelling with a timedependent amplitude b e at and top trace xt = b e at w t β zdz + ξ. 64 The function β t plays the role of source and it changes the velocity of the soliton but not the shape. We can have variety of travelling traectories by choosing different β t. One special case is that when β t = b e at gt 4b 3 e 3at +aξ e at we will have a stationary soliton with top line x ξ /b shown in Fig. 3a. ii Two-soliton scattering Firstly let us call these two solitons as ξ -soliton and ξ -soliton for convenience equation 6b provides the corresponding sources.although the non-isospectral KdVESCS-I describes solitons in non-uniform media its solitons can scatter elastically under some conditions as shown in Fig. 4. We first suppose that b > b > and [ β t β t + 4b b b b e 3at] dt as t ±. 65 We also straighten the travelling traectories of ξ -soliton and introduce X = e at β t x e az gz b

7 No. 6 KdV Equation with Self-consistent Sources in Non-uniform Media b e 3az dz + ξ 66 b Y = e at β t x e az gz b + 4b e 3az dz + ξ. 67 b Fig. 3 Shape and motion of one-soliton for the nonisospectral KdVESCS-I. a A stationary soliton u given by 63 for gt = 4b e at a =. b = β t = b e at gt 4b 3 e 3at + aξ e at and ξ = ; b A moving soliton given by 63 for gt = 4b e at a =. b = β t = cost and ξ =. Fig. 4 Two-soliton scattering of the non-isospectral KdVESCS-I. Shape and motion of two-soliton solution u given by 6 and 49a for gt = 4e at a =. b =.8 b =.8 β t =. β t = sint and ξ = ξ =. Similar to the discussion about isospectral KdVESCS if the frame co-moves with ξ -soliton i.e. X stays constant in the coordinate frame X t we have b e at sech ξ t + b e at sech ξ + ln b b t. b + b It follows that ξ -soliton gets a leftward phase shift /b e at lnb b /b + b with respect to the frame X t after interaction. And in the coordinate frame Y t co-moving with ξ -soliton we have b e at sech ξ + ln b b t + b + b b e at sech ξ t. That means ξ -soliton gets a rightward phase shift with respect to the frame Y t after interaction. Similar results hold when [ β t β t + 4b b b e 3at] dt ± b as t ± b > b >. 68 In the following we consider the degenerate case b = b of two-soliton interactions. In this case f = + e ξ + e ξ. 69 We discuss the special case for two different sources. Figures 5a and 5b describe the special behaviors of such degenerate two-soliton solutions from the two solitons travel first with their original sources and then suddenly with other different sources. Corresponding to the ghost solitons of the Hirota Satasuma equation [45] in our case the soliton u also shows ghost behaviors. Let us give more details in the following suppose that β t = satisfy [ β t β t ] dt as t ± 7 and we consider the following coordinate frame co-moving with ξ -soliton X = e at β t x e az gz + 4b e 3az dz + ξ t 7 b ξ stays constant but due to 7 ξ = b X + b β t β tdt + ξ ξ as t ±. 7 It then follows that b e at sech ξ t + t which means ξ -soliton does not exist initially but appears finally. Similarly under the frame Y = e at β t x e az gz + 4b e 3az dz b + ξ t 73 b which co-moves with ξ -soliton we have t + b e at sech ξ t

8 996 HAO Hong-Hai WANG Guang-Sheng and ZHANG Da-Jun Vol. 5 which means ξ -soliton exists initially but disappears finally. Similar results can be obtained for β t satisfying β t β tdt ± as t ±. 74 Fig. 5 Degenerate two-soliton interactions b = b of the non-isospectral KdVESCS-I. a Shape and motion of the solution given by 6a and 69 with gt = 4e at a =.5 b = b =.6 β t =.5 β t = 3.5 and ξ = ξ = ; b Shape and motion of the solution with gt = 4e at a =. b = b =.6 β t =.3 β t = sint and ξ = ξ =. Thus we have shown how the sources play roles in the degenerate two-soliton case. 5 Non-isospectral KdVESCS-II In this section we investigate the non-isospectral KdVESCS-II via Hirota method and Wronskian technique. 5. Bilinear Form and Multisoliton Solutions The non-isospectral KdVESCS-II can be transformed into the bilinear form D t D x + xd 4 xf f + 4D x f xx f + fh = D 4 xf f = D x h f g 75a 75b D xg f = λ tg f =...N 75c with the transformation 6. Similar to the discussion about the non-isospectral KdVESCS-I the multisoliton solutions for 75 can be obtained through a standard Hirota s perturbation too. For N = one-soliton can be described through f = + e θ g = k tβ te θ 76 4 θ = k tx β z + dz + θ 8z + c k t = t > c 8t + c 8 77 with arbitrary real constants c > and θ and arbitrary non-negative continuous time-dependent function β t and λ t = k t. For N = we can get two-soliton solution f = + e θ + e θ k t k t e θ + + θ 78a k t + k t g = k tβ te θ + k t k t k t + k t e θ 78b g = k tβ te θ k t k t k t + k t e θ 78c λ t = k t and 4 θ = k tx β z + dz + θ 8z + c k t = 8t + c t > c 8 79 θ with arbitrary real constants c > and and arbitrary non-negative continuous time-dependent function β t =. The N-soliton solution can be obtained by taking λ t = k t in 75c and f = µ= N exp µ θ + g h = k h tβ h te θ h µ= <l N N exp µ µ l A l 8a h µ θ + A h + <l N l h µ µ l A l h =...N 8b θ is defined as 79 and e A l = [k t k l t]/[k t + k l t] and we set c c c N > without loss of generality and also set t > c /8 =...N to avoid singularities. 5. Solutions in Wronskian Form Theorem 5. The bilinear non-isospectral KdVESCS-II 75 admits the following Wronskian solutions f = N = Wφ φ...φ N g h = G h t N τ h h =...N 8

9 No. 6 KdV Equation with Self-consistent Sources in Non-uniform Media 997 h G h t = N kh tβ h t kh t k l t kl t k h t φ = e θ + e θ 8 l= l=h+ and τ h = δ h δ h...δ hn T λ t = k t in 75c and θ is defined as θ = k tx k t = 8t + c β zdz + θ t > c 8 =...N 83 with arbitrary real constants c > and θ and arbitrary non-negative continuous time-dependent function β h t and we also assume that c c c N >. To achieve the proof one should notice that φ satisfies which further implies t l xφ = φ l and this leads to φ t = 4xφ xxx β t k t φ x 84 t = 4xφl+3 4lφ l+ β t k t φl+ 85 f t = 4x N 4 N N N N 3 N N + + N N + + 4N N 3 N N 4N N N + β t +l l l= e θ e θ A l 86 A l is the cofactor of f. Then the rest part of the proof is very similar to the bilinear non-isospectral KdVESCS-I so we skip the details here. 5.3 Dynamics i One-soliton characteristics We would like to consider the Wronskian solution for -soliton in this part. In this case we can easily get It follows from 6 that u = f = e θ + e θ. 87 8t + c sech θ 88 θ is defined by 83. Equation 88 provides a soliton travelling with a decay amplitude /8t + c and time-dependent top trace xt = 8t + c β zdz θ 89 the function β t also plays the role of source. The stationary soliton can be obtained when we take c > θ and β t = 4θ c 8t + c 3/. In this case xt θ c and x t. Figure 6 describes shape and motion of one-soliton. Fig. 6 Shape and motion of one-soliton for the nonisospectral KdVESCS-II. a A stationary soliton given by 88 for c = 5 β t = 4θ c 8t + c 3/ and θ = ; b Density plot of a moving soliton given by 88 for c = 5 β t = sint and θ = x [ 66] t [ 5 5] and plot range [.5.]. Grey area denotes zero value and bright strap denotes positive soliton. ii Two-soliton scattering We consider two-soliton solution in Hirota s form in this part i.e. f is given by 78. Supposing that c > c > and t > c /8. In this case it is not easy to analytically investigate two-soliton interactions. but they do have elastic scattering. The analysis about two-soliton scattering in this case is quiet same to the one shown in ii of Subsec. 4.3 so we skip the detail but give a figure. Figure 7a exhibits the elastic interactions of two solitons with decay amplitudes. For comparison we give the two corresponding single solitons in Figs. 7b and 7c. For the degenerate case c = c > we have f = + e θ + e θ 9 θ = x 8t + c =. 4 β z + dz + θ 8z + c

10 998 HAO Hong-Hai WANG Guang-Sheng and ZHANG Da-Jun Vol. 5 Similarly under the frame Y = θ x t = β z+ 4 dz+ θ 8t + c 8z + c t 95 which co-moves with θ -soliton letting θ stays constant we have t Thus we conclude that for the final state of two solitons involved in the degenerate case one exists but another disappears under the condition 9. Obviously similar result holds when β t β tdt + as t Fig. 7 Two-soliton interactions for the non-isospectral KdVESCS-II. a Density plot of the two-soliton solution given by 6 and 78 for c = 8 c = 6 β t =.4e.4t β t =.5e.5t θ = θ = 6 x [ 8] t [ 7 ] and plot range [.5.]; b Density plot of the corresponding one-soliton for c = 8 β t =.4e.4t θ = x [ 8] t [ 7] and plot range [.5.]; c Density plot of the corresponding one-soliton for c = 6 β t =.5e.5t θ = 6 x [ 8] t [ 7] and plot range [.5.]. Grey area denotes zero value and bright straps denote positive solitons. Figures 8a and 8b describe that a single soliton is disturbed by an invisible ghost soliton. In Fig. 8c a soliton travels first with its original source and then suddenly with another different source. To obtain more details for such degenerate case we also investigate the asymptotic behaviors in the light of β t β tdt as t + 9 and the two solitons involved in the degenerate case are called θ -soliton and θ -soliton respectively. We consider the coordinate frame co-moving with θ -soliton X = θ = x 8t + c + θ stays constant but θ = θ + β z + 4 dz 8z + c θ t 9 β z β zdz + θ θ as t + 93 due to 9. This further gives sech θ 8t + c t Fig. 8 Degenerate two-soliton solution c = c for the non-isospectral KdVESCS-II. a Density plot of the degenerate two-soliton solution given by 6 and 9 for c = c = 8 β t =.5e.5t β t =.5e.5t θ = θ = 6 x [ 86] t [ 7] and plot range [.5.]; b Density plot of the degenerate two-soliton solution with the same parameters as a except θ = instead of 6. Thus one can see the existence of θ -soliton by comparing a and b; c Density plot of the degenerate two-soliton solution for c = c = 8 β t =.e.4t β t = sint θ =.5 θ =.5 x [ 86] t [ 68] with same plot range. 6 Conclusion In the paper the N-solution solutions for the nonisospectral KdVESCS-I and KdVESCS-II are derived through Hirota s method and Wronskian technique respectively. We also have investigated non-isospectral dynamics and sources effects for these equations. It shows that the sources can provide a time-dependent phase shift for each line-soliton but not change the shape of the soliton. More some special sources can also lead to ghost solitons. The typical character for this case is because there exists an

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