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Recently, economical, psychological, and physical abuse [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] was reported. It has evolved against the background of structures that systematically devalue and control the feminine. Such structures effect on the individuals of the respective groups and engrave into their bodies and minds. Thus, in a cultural-insensitive, unreflected transfer of knowledge of Buddhist philosophy and respective training techniques, the culturally ingrained ways in dealing with the feminine and its unconscious attitudes were absorbed. A visualization concept of females merely forming the passive counterparts to male characters, their devaluation in comments and acquired behavior patterns as well as misunderstood visualization techniques leading to narcissistic self-aggrandizing patterns contribute to confusion. This disorientation in turn not only forms the basis for developing self-devaluation and uncertainty regarding the individuals social positions in the groups, but also causes a double bind-based inability to act. Thus, one option seems to be the proximity to the spiritual master, the identifying of women with the male master, which somehow is supposed to define their position and seemingly would enhance their spiritual progress towards enlightenment as well. The undefined social positions of women in such groups, which encourage projecting undesirable aspects onto them, and sexualized aspects, such as the role of a secret lover, pose a cultural challenge for them. However, besides uncertainties due to a lack of cultural socialization, and the opportunities for honest communication on this issue, usually prevented by vows of secrecy, damage to group members is incurred by those regarding themselves as Vajrayāna practitioners, even preaching Vajrayāna, for whom gender equality ought to be standard in their training. Thus, in neglecting individual and institutional betrayal trauma [13], the implicit traumatization of people, despite being undeniable injuries, has been successfully silenced for decades. And control of the feminine has also been achieved by projecting the shadow, one’s own unwanted aspects, at the women.
Moreover, the narrative of Buddhism being a panacea for mental health not only misinterprets its spiritual methods to be psychological or medical ones. Damage is also caused by seducing and misleading sick individuals and by misinterpreting any trauma of those severely injured by the leadership or group members. Thus, the shadows behind the devaluation of females and seducing some of them into neglected social positions reveal a blind spot in Tibetan Buddhism and its seminar-, meditation-, and retreat-centers in the West, where it has continued to grow over decades. In rationalizing misconduct of the leadership and their helpers to be ‘crazy wisdom’ [1, 2, 5, 8] or even so-called ‘karma purification’ [1, 2], it was possible to even cover the injury to individuals and structural defects.
Due to the ongoing efforts to whitewash, denigrate those concerned, and silence the required discourse on leadership responsibility and accountability in the respective international contexts and head organizations, it is now crucial to broaden the perspective and consider the societal impact of such offences which, even as Buddhism and spirituality, have already been injuring and traumatizing many people. In the following subchapter, the descriptions of eleven probands are presented.
Qualitative data of the research project TransTibMed from eleven participants in different international Tibetan Buddhist groups are presented covering the following eleven questions: Which kind of abuse or violence have you experienced in Buddhist groups; how did you react to abuse or violence; how have your experiences in Buddhist groups had an impact on you; which kind of indoctrination, abuse, or violence against others have you witnessed; how have you reacted in the face of this indoctrination, violence, or abuse towards others; how did the experiences in Buddhist groups change you; how did the experiences in Buddhist groups change your relationships and your ways of relating to others; how did your experiences in Buddhist groups change the way you work; and have you parted with one or more Buddhist groups or teachers? If so, why?
The questions when you think of abuse in Buddhist groups - the abuse of people who have been manipulated and of minors, including sexual abuse - how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what and why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality were replied to by others and one person who also replied to the above.
Probands were recruited through information at university, the German Buddhist Union (DBU), and indirectly through (former) group members informing others at social networks.
All probands presented wrote about their experiences in international Tibetan Buddhist organizations. Some of them were in several groups. All together, these statements refer to eleven different international Buddhist organizations. In order to protect the individuals, the names of the respective organizations are not provided here.
The following results are structured as follows: All answers of one person are presented in one subchapter. The first nine questions were replied to by #1, #2, #3, #4, and #8 and the last two questions were replied to by #3, #5, #6, #7, #9, #10, and #11. The first six persons (#1–#6) are female and #7–#11 are male.
Questions are written in italics and the replies in straight letters. All answers of probands in German were translated by the author.
Which kind of abuse or violence have you experienced in Buddhist groups?: “Psychological abuse - gaslighting (stating everything is simply in my mind), manipulation to accept abuse by the abuser himself stating that he was my teacher, that everything ‘is empty’ therefore he couldn’t be an abuser. Psychological abuse through an email attack to my workplace claiming I am having a nervous breakdown” (#1).
How did you react to abuse or violence: “At the time of the individual abuse within […] I felt inadequate, I did not acknowledge that the perpetrator should be able to take some responsibility for their behaviors. I felt abandoned and inadequate because of the way the group reacted to the abuse - they did not acknowledge it had happened and when I disclosed it they did not view it as abuse. I was minimized by the ‘spiritual director’ when I disclosed it to her - she said that my ‘practice was enduring suffering’. At the time this made me feel more inadequate. Later I understood this as a minimizing activity, and I became very angry about six months later after reading testimonies of other former members. Following my workplace receiving an email from […] management using a fake identity I experienced rage, could not concentrate, my boundaries were poor, I turned to alcohol to cope, I experienced suicidality. I felt paranoid, I did not understand how much danger I might be in or not. I did not know who could help, I felt abandoned by people who did not understand and could not help me. I had digestion problems due to the anxiety and shame for several months. I had little energy and could not exercise, experienced stiffness and rigidity in my face, I had slow speech and slow processing speed. I dissociated frequently e.g. not remembering where I had put things, where I had parked my car. I had nightmares about being murdered by the leadership” (#1).
How have your experiences in Buddhist groups had an impact on you: “Severe post traumatic stress, anxiety, shame, and post traumatic growth. I am now connected to people who I find kind and who listen to me and appreciate my perspective on the groups. I therefore feel heard and understood, and no longer gaslighted” (#1).
Which kind of indoctrination, abuse, or violence against others have you witnessed: “I witnessed long term members so highly indoctrinated that they repeated teachings verbatim all the time, they did not have vocabulary from outside the group anymore. I saw people force the teachings on each other to invalidate the other persons feelings and dismiss their needs. I have read testimonies and seen evidence of threats against former members - threats to ruin peoples reputations and livelihoods. I heard people with mental health problems be spoke about disrespectfully” (#1).
How have you reacted in the face of this indoctrination, violence, or abuse towards others: “At the time I believed what was said about former members - that they had ‘lost patience in their spiritual path’ or had ‘developed an angry mind and were blaming the centre’. When I came to realize that this was gaslighting of whistleblowers (after leaving) I became very distressed that the centre could avoid taking responsibility for people’s wellbeing by blaming their ‘mind’ for everything. When I saw the threats and character assassination of former members I went into shock, and then experienced righteous anger” (#1).
How did the experiences in Buddhist groups change you: “I now understand spiritual abuse and am very sensitive to coercion. I am suspicious of all spiritual teachers. I only trust survivors of abuse and a few friends and family. I have more highly developed critical thinking skills and enjoy being disobedient. I now struggle to follow rules I believe are unnecessary, I trust my intuition more” (#1).
How did the experiences in Buddhist groups change your relationships and your ways of relating to others: “I recognise my needs and when someone is not good for me, I do not automatically put other people first. I do not practice accepting everything, I feel more in touch with righteous anger. I try to communicate my emotions more. I trust few people, especially religious people” (#1).
How did your experiences in Buddhist groups change the way you work: “I am suspicious of meditation and mindfulness, I read all about the under reported adverse effects. I understand abusive systems of power and abusive group dynamics. I don’t trust cognitive therapies so much anymore, I only trust interventions that are also systemic” (#1).
Have you parted with one or more Buddhist groups or teachers? If so, why: “Yes both […] after reading testimonies of abuse and reading cultic studies literature, I understand that they are personality cults and that they do not understand trauma. They claim to help you with stress and teach you mindfulness when they actually believe you should be practising dissociation from your emotions. They have obscured and whitewashed serious abuse, appointed only internal safeguarding officers who are invested in the continuation of the movements, and who are interested in saving their reputations” (#1).
Which kind of abuse or violence have you experienced in Buddhist groups: “Sexual assaults, however without violence, we were servile, so there was no need for it. Men were instrumentalized, delegated by relocation, assigned as secretaries of centers, to work for the Lama was an honour, sometimes for little or no payment, many did not longer pursue their own lives, partnerships were interrupted, separations and uniting were arranged” (#2).
How did you react to abuse or violence: “I did not perceive it as such. It was only when my best friend revealed herself to me that I could see it as abuse on her and then on me as well” (#2).
The questions how have your experiences in Buddhist groups had an impact on you and which kind of indoctrination, abuse, or violence against others have you witnessed were not answered by #2.
How have you reacted in the face of this indoctrination, violence, or abuse towards others: “later I also learned about sexual assaults against very young women” (#2).
How did the experiences in Buddhist groups change you: “have become more suspicious, but also more adult and autonomous” (#2).
How did the experiences in Buddhist groups change your relationships and your ways of relating to others: “Sometimes one gets arrogant, considering oneself as something better, but one also becomes more compassionate. That we are all essentially the same and can simply be on equal level with each other has reached me and I have internalized it to this day. In a therapeutic context it has helped to make encounters on equal level, but also endangered for much closeness and confluence” (#2).
The questions how did your experiences in Buddhist groups change the way you work and have you parted with one or more Buddhist groups or teachers? If so, why were not answered by #2.
Which kind of abuse or violence have you experienced in Buddhist groups?: “spiritual, psychological, emotional” (#3).
How did you react to abuse or violence: “Anger, helplessness, eventually some deference to my teacher for survival and to maintain my position in the group. Basically you have to tell him he is right and you are wrong to preserve your standing as a student” (#3).
The question how have your experiences in Buddhist groups had an impact on you was not answered by #3.
Which kind of indoctrination, abuse, or violence against others have you witnessed: “Gaslighting, shaming, slander, emotional manipulation, violation of confidentiality, outright lies, telling the Executive Director to fire someone because ‘she was too neurotic’, psychologizing every single student, punishment sending people away from group retreats into solitary or banning people from retreats” (#3).
How have you reacted in the face of this indoctrination, violence, or abuse towards others: “I was often not aware or stood by quietly and watched” (#3).
How did the experiences in Buddhist groups change you: “Sustained attention, strength to do multi day ritual, I am not afraid to be alone in the woods, I accept pain and negative experiences as just part of life. I feel a tremendous union with the divine available in my body. I regularly experience space that feels between life and death, no self, and primordial. I will also never ever become a student of a male spiritual teacher again. I will never give my power away to any human teacher ever again. I have learned a lot about this and still feel very hurt” (#3).
The questions how did the experiences in Buddhist groups change your relationships and your ways of relating to others, how did your experiences in Buddhist groups change the way you work and have you parted with one or more Buddhist groups or teachers? If so, why were not answered by #3.
When you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what: “Angry. Sad. Helpless. The teachers are responsible for the abuse, the board of directors is responsible for not intervening and for allowing a spiritual director who is also paid employee to have veto power over the board (absolute control of the whole organization, which is legally questionable and the board knew it)” (#3).
Why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality: “We all participate in patriarchy. Tibetan Buddhism encourages mind control, submissiveness, and patriarchal hierarchy. I believe that we allow it because we are so disconnected from our own power that we are willing to let some guy and some exotic tradition tell us what to do. We are eager to hand the reigns over, and Tibetan Buddhism in its sleek ways explains why we should. I believe that men have been suing the tradition for their own agendas for so many centuries that perhaps it has only become a vehicle for patriarchal abuse and I seriously question if the Vajrayana is of benefit in the world now” (#3).
#4 did not answer the question: which kind of abuse or violence have you experienced in Buddhist groups?
How did you react to abuse or violence: “I got super angry with myself, blaming myself very badly for not being good enough” (#4).
How have your experiences in Buddhist groups had an impact on you: “Lack of trust, no direction in life, I cannot see my future, I’m afraid of making decisions, I feel helpless and good for nothing” (#4).
Which kind of indoctrination, abuse, or violence against others have you witnessed: “Later on I heard for a community member that the senior teacher had sexual abuse on different women. Even one of the women contacted me and [was] telling me her story. Also I saw people leaving abruptly and getting very confused. Some left with anger. And I heard in the community that there was a woman who got crazy and left. Whatever that means. I guess mentally ill” (#4).
How have you reacted in the face of this indoctrination, violence, or abuse towards others?: “I was blaming myself and asking help from others. I saw everyone else right and I thought they are the best people on Earth, I wanted to get back to the community. I was scared of my family. The community members were immaculate and wise. They even sent me pictures, messages, poems I couldn’t understand because I so much wanted to figure out the messages behind” (#4).
How did the experiences in Buddhist groups change you: “I doubt myself, I had better periods but I keep falling back. I lost trust in life, I lost trust in myself and others, I have a lot of tension and fear in me. I cannot imagine to be capable of anything. Study or move away from my parents” (#4).
How did the experiences in Buddhist groups change your relationships and your ways of relating to others: “I can be still opened but deep down I have fear, doubt and it is not easy to trust people in a deeper level. I feel alone” (#4).
How did your experiences in Buddhist groups change the way you work: “I cannot imagine myself to have a job with responsibility. I don’t work much a week and sometimes I just sleep for the whole day” (#4).
Have you parted with one or more Buddhist groups or teachers? If so, why: “Yes, two groups, because I had psychosis and got sick” (#4).
When you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what: “I think leaders of Buddhist groups are individuals and it is unavoidable to find some ‘black sheep’ among them. Certainly I regret such incidents and would feel a serious betrayal of trust if I were to experience this from someone at the centre where I frequently visit. Once someone in the group hears about such, I believe it is their responsibility to communicate to stop it” (#5).
Why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality: “I would not allow this, but have not had any experience with it, usually I would take action or at least distance myself in case this would not be possible” (#5).
When you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what: “Abuse is unethical and illegal and we always have an obligation to stop it. I don’t think most groups are capable of policing themselves. They are too insular and secretive and outside law enforcement should be contacted. I tried to contact the Dalai Lama and other leaders repeatedly, but they never respond” (#6).
Why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality: “Because some people actually believe these men have special rights and powers and don’t have the balls to stand up for people being exploited. People are blinded by robes and spiritual authority and it’s dangerous” (#6).
When you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what: “This should not happen at all and ought to lead to an immediate criminal report. The person abusing is responsible, however, depending on the circumstances, also the group involved and the structures prevailing within it. The veneration of the teacher in Tibetan Buddhism unfortunately also creates conditions for abuse, be it physical or authoritative, even if the teachings clearly do not provide for that. In my opinion, Tibetan Buddhism needs some effort towards adjustment to at least make abuse more difficult” (#7).
Why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality: “Teachers are often held sacrosanct, and criticism of them is mostly unwanted. Sometimes a strong relationship of superiority/subordination can be seen, which facilitates abuse” (#7).
Which kind of abuse or violence have you experienced in Buddhist groups: “Humiliation, exposure, psychological violence, repression, deprivation of healthy self-esteem, intrigues, defamation campaigns, systematic manipulation and lies. Theft: do not take what is not given = misuse of tied donations” (#1) [cited in 2].
How did you react to abuse or violence: “denial, reality negation, rationalization, first insights, perception, evaluation, attempts to communicate, departure” (#8).
How have your experiences in Buddhist groups had an impact on you: “I no longer conform to any group consensus I consider inappropriate. I insist on discussing openly without individual participants manipulating others first. I advocate transparency and truthfulness, even if this is not popular, including financial issues, and try to convey democratic values. My position is that ethics could make a difference, before talking about philosophy and religion” (#8).
Which kind of indoctrination, abuse, or violence against others have you witnessed: “Anyone at […] should have seen quite a lot. For me, these subtle constant manipulations have been the worst, and that behind the scenes the violence was brutal” (#8).
How have you reacted in the face of this indoctrination, violence, or abuse towards others: “First I rationalized. I found many ‘old students’ so unpleasant that I considered more drastic measures to possibly be an option. Today I look at it differently. I do not count the eight letter writers among the ‘old students’, nor all students of […]. After realizing that all of this at […] is a dead end, and being frustrated to find out my fellow sanghaists lacked understanding, I left. They were in a ‘tunnel’. Particularly the many people afraid of losing their ‘dzogchen’, whatever that might mean to the individual” (#8).
How did the experiences in Buddhist groups change you: “Now I look at the world more openly, in its entirety, with all of its problems and opportunities” (#8).
How did the experiences in Buddhist groups change your relationships and your ways of relating to others: “I live quite isolated, beyond my professional and family life. I keep considerable distance from Buddhists” (#8).
How did your experiences in Buddhist groups change the way you work: “I take care of myself and my environment as much as I can. People who want to preach to me, I send away” (#8).
Have you parted with one or more Buddhist groups or teachers? If so, why: “Of all of them. My disgust with Buddhist groups is an impulse way too strong to be ignored. It either will subside or it doesn’t. I am a Buddhist who pretends he is not” (#8).
When you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what: “I feel angry and a huge sense of injustice about it, because it totally goes against the spirit of Buddhism, which is compassion, kindness, and bringing people to enlightenment. The teacher and the Buddhist institution the teacher is affiliated with is responsible” (#9).
Why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality: “It is an issue to do with the dominant patriarchal society we are part of today, where men are in positions of authority. This, unfortunately has extended to religion” (#9).
When you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what: “Every single person has his share of responsibility, even if he acted in blind trust. These are mitigating circumstances, of a sort. In […]: The higher the position, the greater is the degree of blindness. And the fish is always stinking from its head. Those at the top of an authoritarian structure, and who, due to their position as lama per se are endowed with particular abilities, ought to take responsibility for the consequences of their actions. Responsibility in groups is usually given to older students, who might have been accordingly indoctrinated and bring the relevant skills. Sociological studies might show 80% of leadership in Western Buddhist groups are female, except for the top positions. A psychologist and Buddhist, who I know well, even claims that women are more susceptible to manipulating behaviors in these positions” (#10).
Why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality: “Ignorance, lack of awareness, lack of ethical conscience. This is true for all genders, although it is mainly women who are currently victims of male offenders. Mostly all is well, as long as one isn’t victimized oneself” (#10).
When you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group? Who is responsible for what: “Disgusted and annoyed. Everyone is responsible in particular the people who know directly about it and keep silent. However I know some people did try to bring abuse to light in the group I knew many years ago. Everything was ignored. They kept files they later gave to an investigation. Yet the group or the lama has still not admitted abuse” (#11).
Why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality: “Because it happens behind closed doors and unless the women themselves speak out it is difficult to prove. Also Tibetan Buddhism like most religions use fear of hell and damnation to silence. Also women are told they are dakini and as such special with special merit when they die” (#11).
The current issues in so-called Buddhist centers unfold against the background of oversimplifications of terms and concepts of Buddhist philosophy as well as of damaging neologisms. In particular, however, traditional application-oriented learning processes crucial for understanding and teaching are missing. The individual effects of these substantial and structural deficiencies are evident as health damage, specifically mental health, and traumatization of group members. Unreflected decontextualized use of Buddhist terminology and concepts and the neologisms in these contexts thus negatively impact on group dynamics and the health of a group and its members. Since the indoctrination described by the testimonies and the traumatization is still attempted to be denied by the mainstream within these organizations as well as the victims silenced [2], the latter are not compensated in any ways by precisely these organizations which have often previously enriched themselves on them. Furthermore, by propagating Buddhism a panacea for mental diseases, even persons with mental disorders are misled in highly irresponsible ways. Although oaths of secrecy associated with seemingly Buddhist concepts render it difficult for those affected to talk, testimonies regarding indoctrination and systematic abuse causing trauma and mental illness are available now. However, a high number of unrecorded cases of those, who for various reasons are unable to communicate, ought to be assumed.
The kind of abuse described by the probands covers psychological abuse (#2, #3), especially those aspects of the students’ submission, which serve as a basis for sexual assault (#2), the indoctrinative interpretations that are suggesting the abuse itself would take place in the mind of the student (#1) only, as well as psychological violence (#8). This questioning of one’s own autonomous understanding of situations and even one’s perception leads to a continuous state of disorientation, which initially may contribute to psychological damage and is particularly important to consider in later processes of treatment. By means of neologisms such as ‘karma purification’ and ‘crazy wisdom’, which serve to rationalize any misconduct of group leaders and entourage towards the goal of the students’ enlightenment, a form of dependence is constructed, in which they even bind themselves to the insulting leadership. In such ways their identification processes [14] are enhanced and self-reliance and autonomy reduced.
Emotional abuse (#3) was presented as humiliation (#8), exposure (#8), and repression (#8), which explains the very reduction in self-esteem (#8) of students in such groups. However, by using misunderstood visualizations of the Vajrayāna, also narcissistic tendencies get encouraged, resulting in severe arrogance, particularly among those seeking to establish themselves in positions of authority in such contexts, and in such ways forming a kind of spiritual narcissism. This, in turn, may explain their lack of empathy and compassion towards the victims, even whilst still proclaiming Buddhist compassion.
The abuse of authority for slandering any current or former students was reported as interfering with lies even at the workplace, outside the so-called Buddhist organization (#1). The lies (#8), intrigues (#8), and defamation campaigns (#8) mentioned have a huge impact on the individuals, particularly when used to irritate or destroy their reputation, social networks, and income. Since one has grown accustomed to such situations in these contexts, it seems necessary to emphasize that such is far from the basic ethical values taught within traditional Tibetan Buddhism.
The indoctrination saying the abuser would be allowed for the abuse, because of being a spiritual teacher (#1) already indicates an utterly inappropriate way of defining spiritual authority. Groups propagating such dangerous concepts concerning their spiritual teacher, devoid of ethics and adherence to national legislation, together with any authorities imposing their authoritative rules at will, endanger group members as well as visitors.
Any manipulation of students, interpreting the work for a lama, who is regarded a spiritual authority in the group, being an honor (#2), which in turn meant sexual abuse of servile (female) students (#2) and the instrumentalization of men for all kinds of little or unpaid services (#2), as well as systematic manipulation (#8), were also described as abuse (#2). Interpreting work for a spiritual authority as honor already indicates narcissistic group dynamics, where autonomous authentic training of Buddhist ethics and meditation has been replaced by rivalry over physical proximity to the leadership, which in turn has been redefined and also implies power over the group through identification processes. Thus, such concepts shape not only the group’s structures and behavior patterns but also its economic handling, that is expecting donations for certain group members. Furthermore, manipulation is not only for complying with financial and other service expectations, but often people are played off against each other and exchanged after their exploitation at will. Over the years of affiliation, however, these kinds of interactions, which are accompanied by psychological and financial dependencies as well as mental identification processes, can cause serious damage. Theft in the sense of misuse of tied donations (#8) was also mentioned to be abuse. And the manipulation with the conceptual distortion of the Buddhist concept of emptiness used to explain away abuse and abuser (#1) severely devaluates Buddhist philosophy and undermines sound and informed terminological and conceptual discourse.
The testimony of indoctrination, abuse, or violence against others was described as follows: subtle constant manipulations (#8), highly indoctrinated long-term members lacking vocabulary from outside the group (#1), and repeating verbatim all the time (#1). For so-called Buddhist seminar- and retreat-participants the following methods were described: psychologizing every single student (#3), emotional manipulation (#3), violation of confidentiality (#3), gaslighting (#3), shaming (#3), slander (#3), and outright lies (#3). Someone also testified disrespectful speech about people with mental health problems (#1). Furthermore, blackmailing of the management was described, such as telling the executive director to fire someone because of being diagnosed neurotic (#3) by nonprofessionals. Furthermore, there is testimony on punishment in the form of sending people away from group into solitary retreats or banning people from retreats (#3) and brutal violence behind the scenes (#8). Someone interpreted dismissing people’s needs by invalidating their feelings (#1) through forcing what was called ‘teachings’ on them. There is also testimony on threats to ruin people’s reputations and livelihoods (#1) and on sexual abuse of different women by a senior teacher (#4). Whereas sexual abuse of minors is a clearly defined a criminal offense, currently also severely manipulated and indoctrinated adults, who have mostly initially made their way up to the ‘inner circle’ [2, 5] and have then moved on to physical proximity or agreed to the position of secret lovers, claim the same for themselves. Thus, the overloaded phrase of ‘sexual abuse’ in current contextual discourses ought to be further differentiated, taking into account unequal balance of power, indoctrination, seduction, untrue promises, coercive control, self-responsibility of adults, as well as victimizing collaborators and witnesses. Moreover, the dynamics of controlling the feminine, which have been adopted unreflectively from Tibetan Buddhism, manifest in seemingly impersonal attitudes towards the women concerned, which thus results in a high number of victims.
Three successive states of individual reaction to indoctrination, abuse or violence in the respective Buddhist group showed: continuous close affiliation, critical reflection, and departure. For the period of close affiliation to a group helplessness (#3), denial (#8), feeling of inadequacy (#1), not acknowledging that the perpetrator should be able to take some responsibility for their behaviors (#1), feeling abandoned and inadequate because of the reactions of the leadership and group to the abuse (who did not view it as abuse) (#1), oneself not perceiving abuse and violence as such, but only understanding it when seeing it on someone else first (#2) and anger (#3) were reported. One person severely turned against herself and reported getting seriously angry at herself (#4) with putting herself down (#4). Another person in high position in her group and dependent on financial terms told she has taken over the opinion of the teacher to preserve her own standing and maintain her position in the group (#3). Thus, these replies also may show some implications and limitations of holding positions in such groups. In phase two, the period of reflection and doubts, there were communication attempts before departure (#8). In phase three, after departure from the group, there were feelings to be abandoned by those who did not understand and could not help (#1), anxiety and shame for several months causing digestion problems (#1), frequent dissociation (#1), which might have been accompanied by stiffness and rigidity in the face (#1), little energy (#1), and slow speech and slow processing speed (#1). Furthermore, doubts regarding the dangers (#1) with nightmares about being murdered by the leadership (#1), lack of concentration (#1), poor boundaries (#1), turning to alcohol to cope (#1), and suicidality (#1) were reported. Reading testimonies of former members (#1) was reported having caused anger. And the workplace receiving an email from the so-called Buddhist organization’s management, who was even using a fake identity (#1), caused rage (#1).
The above three successive states of individual reaction were reported to be accompanied by two inner processes when observing indoctrination, abuse, or violence of others: the identifying and indoctrinating phase and the reflection phase. These observations show concepts and behavior far from the attitude of traditional Buddhist practice of having the individual’s autonomy, awareness and self-reflection as its basis. At the period of being indoctrinated, identifying and merging with the leadership, management or group expectations reactions such as lack of one’s own awareness of it (#3), standing quietly (#3) and watching (#3), which could be interpreted as a kind of freezing in shock about what was seen, as well as rationalizing (#8) and even believing what was said about former members (#1) were reported. Someone did not respond to the question directly, but stated sexual assaults against very young women (#2). Yet another person told she was even blaming herself (#4) for the above behavior of other group members while regarding everyone else right. At the period of reflection, when interpreting the issues leading to decisions, someone described realizing all of it a dead end (#8) and his frustration about his fellows narrowing their understanding like in a ‘tunnel’ (#8) before he left. Others responded to the manipulation and indoctrination tactics in the group with shock and later righteous anger (#1), because of the threats and character assassination of former members (#1) as well as with high distress about the center avoiding to take responsibility by blaming people’s ‘mind’ for anything (#1).
The question on how the person was changed him- or herself by their experiences in these groups was replied to with some positive aspects such as sustained attention (#3), strength to do multi-day ritual (#3), being not afraid to be alone in the woods (#3), and the feeling of union with the divine (#3). However, the positively connotated aspect of accepting pain and negative experiences as just a part of life (#3) may already endanger this individual in highly manipulative contexts. Aspects with negative connotations, such as doubting oneself (#4), not being able to imagine to be capable of anything (#4), having a lot of tension and fear (#4), as well as having lost trust in oneself and others (#4) and in life (#4), were reported as well. People also described having become more suspicious (#2), particularly of all spiritual teachers (#1), being sensitive to coercion (#1) and understanding spiritual abuse (#1), the latter of which may not only protect the individual but also enable them to protect others from unreasonable nonsense and misuse. The phrase spiritual abuse might be interpreted as people being seduced into giving up their own strength as if dependent on a master providing them access to their own spirituality. Furthermore, with reporting greater autonomy (#2) and trusting one’s own intuition more (#1), an individual process of change was addressed, which is also found in phrases such as becoming more adult (#2), having developed critical thinking skills (#1), and probably even in enjoying being disobedient (#1) after experiencing abuse in highly authoritative structures to some extent. Also, the result of a process, such as looking at the world more openly (#8), was referred to, and decisions were shared such as never becoming a student of a male spiritual teacher again (#3) and never ever giving one’s power away to any human teacher (#3).
The scope of the personal impact of these experiences in Buddhist groups was described with adverse effects such as severe self-devaluation (#4), posttraumatic stress (#1), and feelings of helpless (#4), shame (#1), and anxiety (#1). Furthermore, lack of trust (#4) was described at the interpersonal level, which might impact considerably on future relationships and work decisions. Although the fear of making decisions (#4) and loss of direction in life (#4) may come up in an early period of leaving an abusive group, they sometimes continue for even years. The refusal to conform to any group consensus considered inappropriate (#8) is often part of the separating process and might stay as a lesson learned. Furthermore, there are positive connotations for a time period after having left the group and making new personal and working relationships, which were called new connections to kind people (#1) and posttraumatic growth (#1). Also, the values of the person were described to be discussing openly without individual participants manipulating others first (#8) and advocating transparency and truthfulness (#8), even if this is not popular, including financial issues (#8), as well as trying to convey democratic values (#8) and ethics (#8). The reflection of absorbed values and integration of one’s own value system are aspects of the process of separation, which often occur long after the physical separation. Sometimes this also encompasses referring to traditional basic Buddhist ethics. In therapeutic processes, it takes place in the phase of dealing with introjects and in the phase of integration. The impact on relationships and ways of relating to others was described as getting arrogant sometimes (#2) for the period in the group, which is regarding oneself higher or more spiritually advanced than others. Negative impact of fear and doubt with feeling lonely and difficulties to trust people (#4) was described as well. Furthermore, it was said recognizing one’s own needs (#1), which seems to be a key issue to reconnect to oneself, not practicing accepting everything (#1) and not automatically putting other people first (#1), trusting few people (#1), as well as trying to communicate emotions more (#1), which may show as becoming more compassionate (#2) or feeling in touch with righteous anger (#1). The making of encounters on equal level at a therapeutic context with dangers for too much confluence (#2) was an impact for a professional therapist. Someone also shared keeping a considerable distance from Buddhists (#8) and living quite isolated beyond professional and family life (#8).
The impact on one’s work was described as taking care to the best of one’s abilities of oneself and surrounding (8), understanding abusive systems of power and abusive group dynamics (#1) and also as suspiciousness of meditation and mindfulness (#1). A young person shared a highly negative impact for her life saying she could not imagine herself at a job with responsibility (#4), she would not work much (#4) and sometimes sleep for the whole day (#4).
The following two questions refer to open letters and ongoing investigations widely discussed in the context. They were asked to understand not only the persons’ opinion but also their position in the group and their behavior. Not everyone replied to the questions: when you think of abuse in Buddhist groups—the abuse of people who have been manipulated and of minors, including sexual abuse—how do you think and feel about it? How do you evaluate the responsibility in the group?Who is responsible for what? Whereas someone told abuse to be unethical and illegal (#6) and in this way referred to basic values, another person simply stripped the leadership of its exaggerated superiority saying the leaders of Buddhist groups were individuals and it would be unavoidable to find some ‘black sheep’ among them (#5). #7 judged that it should not happen at all and ought to lead to an immediate criminal report. Someone replied the veneration of the teacher in Tibetan Buddhism unfortunately would also create conditions for abuse (#7). Reports on one’s own feelings range from helpless (#3), sadness (#3), and anger (#3) to disgust and annoyance (#11). Furthermore, someone told to regret such incidents (#5) and feel a serious betrayal of trust (#5). Someone reported feeling anger and a huge sense of injustice about it (#9) and gave reasons for such behavior going against the spirit of Buddhism, which would be defined as compassion, kindness, and bringing people to enlightenment (#9). Regarding the assignment of responsibilities, one woman said people always have an obligation to stop abuse (#6), and similarly another person told it would be the responsibility of anyone hearing about abuse to communicate to stop it (#5). A man told the person abusing is responsible (#7), however, depending on the circumstances, also the group involved, and the structures prevailing within (#7). Another man told that the teacher and the Buddhist institution the teacher is affiliated with were responsible (#9) and someone else that every single person had his share of responsibility, even if they acted in blind trust (#10), and that the latter would be mitigating circumstances (#10). Furthermore, one woman regarded the leadership and its teachers to be responsible for the abuse (#3) and the board of directors for not intervening (#3) and she analyzed challenging power structures allowing for abuse (#3). #6 assumed most of these so-called Buddhist groups would not be capable of policing themselves (#6) arguing them to be too insular (#6) and secretive (#6) and outside law enforcement (#6). This woman also reported having tried to contact the Dalai Lama and other leaders repeatedly without ever having received any response (#6). Someone else told that Tibetan Buddhism would need some effort towards adjustment to at least make abuse more difficult (#7), and #10 analyzed the degree of higher position in the group to positively correlate with the degree of blindness (#10). He continued to elaborate that responsibility in groups were usually given to older students, who might have been be accordingly indoctrinated and bring the relevant skills (#10). He argued they ought to take responsibility for the consequences of their actions (#10). Yet another person replied that everyone would be responsible, in particular those people who knew but kept silent (#11). This person then reported to know some people who were first ignored when they tried to bring abuse to light in the group (#11) and that neither the lama nor his group would have admitted abuse (#11), even after they later have given the files to an investigation (#11). Despite singularizing individuals with shifting the issues on them, the persons in the groups acting in such ways are “agents in violent betrayal” [15].
The question why do we allow male Buddhist teachers to not only denigrate but also exploit women for their own (mostly hidden) agendas in the name of spirituality was replied to as follows: Someone telling to have no experience with it assumes she would not allow this (#5) by taking action (#5) or at least distancing herself (#5). A man called it ignorance, lack of awareness, and lack of ethical conscience (#10) and elaborated this to be true for all genders, although currently mainly women were victims of male offenders (#10) and that as long as one is not victimized oneself (#10) people would perceive everything to be fine (#10). Another man analyzed the secrecy behind closed doors (#11) telling unless the women themselves speak out, it would be difficult to prove (#11). He described the fear of hell used in Tibetan Buddhism for silencing (#11) and the seduction with calling the women a dakini promising them special merit at the time of their death (#11). Another man analyzed it as an issue to do with the dominant patriarchal society (#9), where men are in positions of authority having extended to religion (#9). A man analyzed that teachers would be often held sacrosanct (#7), telling criticism of them were mostly unwanted (#7). Furthermore, he argued the strong relationship of superiority facilitating abuse (#7). One women responded with participation in patriarchy (#3) accused what she thinks is Tibetan Buddhism encouraging mind control, submissiveness, and patriarchal hierarchy (#3) and interprets people disconnected from their own power (#3) being willing to let someone tell them what to do (#3). She accused men suing the tradition for their own agendas for many centuries (#3) and regards what she has learned to be so-called Vajrayana even a vehicle for patriarchal abuse (#3). Another woman argued with attributing special rights and powers to certain men (#6), saying people were blinded by robes and spiritual authority, judging this to be dangerous (#6). She also attributes lack of courage to protect those exploited (#6).
Thus, the opening of any closed elitist circles to the wider societal discourse, the democratization processes required within the strictly hierarchical structures of Tibetan Buddhist centers and a supervised training of group members in traditional interpretations of Buddhist terms, concepts and ethics as well as the urgently needed cultural discourses are a vital condition for differentiated, linguistically and scientifically sound processes of knowledge transfer and exchange. However, whereas these concerns contribute to overcoming structural and conceptual inadequacies, the view of and behavior towards women remains a key issue. Particularly, the tacit transfer of projecting the shadow onto the female and the implicit mechanisms of their control ought to be reflected and addressed. In this context, the role and function of a teacher’s secret lover and unreflective copying culturally shaped structures and behavior with regards to her are crucial. Given the lack of reference within their own cultural context, this raises questions about the psychological effects of such kind of social isolation of these women as well as about their secret position of power, which is even defined highly desirable in some circles. This issue raises further questions about leadership responsibility with regard to the group dynamics induced by it and the effects on the women concerned and their self-responsibility. The societal challenge of correcting individual damages that have been recurring throughout generations, which negatively impact on the knowledge of Buddhism itself, however, is directly opposed to the current mainstream of secrecy and individualizing these issues as so-called ‘bad karma’ [1, 2] in such contexts. But although those who still draw their individual profits from these structures try to influence their followers with refusing to listen to the victims and authoritative and confusing definitions, such as trauma being even the aggression of the victim, and in such ways continue to manipulate seminar participants and publicly ridicule and slander victims, now the wider social surrounding has started to counteract the strategy of individualizing issues.
Lacking intercultural exchange of scientific knowledge and its integration into the education in Buddhist philosophy and Tibetan medicine resulted in oversimplification and misguided practice.
Furthermore, the collective projection of the shadow onto the feminine reveals through the current individual and structural challenges in these contexts. This pattern being passed down through generations and unreflectively transferred to Western centers has contributed to forming a subculture spreading internationally under Buddhist cover, with its own secrecy rules, use of language in rationalizing neologisms for the silencing of trauma and the traumatized [2] and trauma dynamics. Ignoring the unconscious in teacher-student relationships and group dynamics has led to the superelevation of persons, even loosing ethics, which form the basis of every Buddhist tradition. Thus, based on the demands for identification with the spiritual master and his entourage, mental diseases were induced.
For individuals coming from Buddhist contexts, it is therefore important to replace the convenient ways of handing over responsibility, which are leading to patterns of dependency and serving exploitation, with freedom to autonomy along with self-responsibility regarding their own training and ethics. The autonomy in one’s own spirituality can never be substituted by an intermediary who is regarded as the sole access to one’s own resource. And the role of the lineage holder in Vajrayāna is not meant for such.
Society, which provides care for the shamelessly exploited, diseased, and traumatized people, now faces several challenges. Apart from prevention through dismantling common idealizations and providing information about the circumstances, terminology involved in indoctrination and the dangers in hypnotising techniques or dissociation sold for so-called Buddhist meditation, it is crucial to develop treatment concepts for those with complex diseases to provide therapeutic care. Furthermore, broadening of the perspective is needed to enable improvement of these closed systems with their seminar-, meditation-, and retreat-centers, particularly concerning education as well as conveying human rights and national legislation to those group members who assume their spiritual masters and entourage are above legislation. For preserving the knowledge of Buddhist philosophy and practice, curing the transgenerational patterns of traumatization of individuals, compensating victims and deciding on how to deal with perpetrators, accomplices and collaborators are crucial.
This research was funded by the German Federal Ministry of Education and Research, grant reference number 01UL1823X.
In the field of physics and applied mathematics research getting an exact solution of a nonlinear partial differential equation is very important. The elaboration of many complex phenomena in fluid mechanics, plasma physics, optical fibers, biology, solid-state physics, etc. is possible if analytical solutions can be obtained. Most of the differential equation arises in these field has no explicit solution as popularly known. This problem creates hindrances in the study of nonlinear phenomena and makes it time-consuming in the research of nonlinear models in the plasma and other science. However recent researches in nonlinear differential equations have seen the development of many approximate analytical solutions of partial and ordinary differential equations.
\nThe history behind the discovery of soliton is not only interesting but also significant. In 1834 a Scottish scientist and engineer—John Scott-Russell first noticed the solitary water wave on the Edinburgh Glasgow Canal. In 1844 [1] in “Report on Waves” he accounted his examinations to the British Association. He wrote “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished and after a chase of one or two miles I lost it in the windings of the channel. Such in the month of August 1834 was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.” He coined the word “solitary wave.” The solitary wave is called so because it often occurs as a single entity and is localized. The most important characteristics of solitary waves were unearthed after thorough study along with extensive wave-tank experiments. The following are the properties of solitary waves:
\n(a) These localized bell-shaped waves travel with enduring form and velocity. The speed of these waves are given by \n
John Scott-Russell’s study created a stir in the scientific community. His study not only initiated a debate with the prevailing knowledge of the theories of waves but also challenged the antecedent knowledge of waves. The previous study claimed that a periodic wave of finite amplitude and permanent shape are feasible only in deep water unlike Russell’s observation that the permanent profile is also possible in shallow water. Finally the stable form of solitary waves was received in scientific community with the aid of nonlinearity and dispersion. An ideal equilibrium between nonlinearity and dispersion can generate such waves.
\nDiederik Johannes Korteweg in 1895 [2] along with his PhD student Gustav De Vries obtained an equation from the primary equation of hydrodynamics. This equation explains shallow water waves where the existence of solitary waves was mathematically recognized. This equation is called KdV equation which is of the form \n
This chapter will discuss the analytical solitary wave solution of the KdV and KdV-like equations. In the study of nonlinear dispersive waves, these equations are generally seen. The KdV equation, a generic equation, is important in the study of weakly nonlinear long waves. This equation consists of a single humped wave characterized by several unique properties. The Soliton solutions of the KdV equation have been quite popular but it also not devoid of problems. The problems not only restrict to dispersion but also dissipation and interestingly these are not dominated by the KdV equation. The standard KdV equation fails to explain the development of small-amplitude solitary waves in case the particles collide in a plasma system. KdV equation with an additional damping term or the damped Korteweg-de Vries (DKdV) equation becomes handy in explaining this issue of elaborating the character of the wave. But in the presence of any critical physical situation (critical point) nonlinearity of the KdV equation disappears and the amplitude of the waves reaches infinity. To control this situation, a new nonlinear partial differential equation has to be derived that can explain the system at that critical point. This is known as the modified Korteweg-de Vries (MKdV) equation. In the presence of collisions, this equation is not also adequate and a damped MKdV equation is necessary. Also in the presence of force source term then the equation will be further modified and become DFKdV/DFMKdV.
\nNow we will derive the KdV equation from a classic plasma model, in which we consider a collision-free unmagnetized plasma consists of electrons and ions, in which ions are mobile and electrons obey the Maxwell distribution. The basic equation will be given as:
\nwhere the electrons obey Maxwell distribution, i.e., \n
To write Eqs. (1)–(3) in dimensionless from we introduce the following dimensionless variables
\nwhere \n
To linearized (5)–(7), let us write the dependent variable as sum of equilibrium and perturbed parts, so that we write \n
neglecting the nonlinear term \n
which is the linearized form of Eq. (5).
\nPutting \n
Neglecting the nonlinear term from (10), we get
\nThis is the linearized form of Eq. (6).
\nPutting \n
Hence Eqs. (9), (11), (12) are the linearized form of Eq. (5)–(7) respectively.
\nTo get dispersion relation for low frequency wave let us assume that the perturbation is proportional to \n
So,
\nPutting these value in Eqs. (9), (11) and (12), we get,
\nSince the system (22)–(24) is a system of linear homogeneous equation so for nontrivial solutions we have
\nThis is the dispersion relation.
\nFor small \n
The phase velocity as
\nso that \n
In this case, we have \n
For long-wave as \n
The \n
Now the phase of the waves can be written as
\nHere \n
Let us consider a new stretched coordinates \n
where \n
where x and t are function of \n
We can express (5)–(7) in terms of \n
Substituting the Eqs. (31)–(34) in Eqs. (38)–(40) and collecting the lowest order \n
Integrating Eqs. (41)–(43) and all the variables tend to zero as \n
From Eq. (44)–(46) we get the phase velocity as
\nSubstituting the Eqs.(31)–(34) in Eqs. (38)–(40) and collecting order \n
Differentiating Eq. (50) With respect to \n
Eq. (51) is known as KdV equation. \n
The steady-state solution of this KdV equation is obtained by transforming the independent variables \n
The steady state solution of the KdV Eq. (51) can be written as
\nwhere \n
Solitary wave solution of Eq. (52) for the parameter value \n\nt\n=\n1\n,\n\nu\n0\n\n=\n0.2\n\n.
Let us consider an unmagnetized collisional dusty plasma that contains cold inertial ions, stationary dusts with negative charge and Maxwellian electrons. The normalized ion fluid equations which include the equation of continuity, equation of momentum balance and Poisson’s equation, governing the DIAWs, are given by
\nwhere \n
where \n
The normalized electron density is given by
\nWe introduced the same stretched coordinates use in Eq.(31). The expansion of the dependent variables also considered as (32)–(34) with
\nSubstituting (31)–(34) and (58)–(59) along with stretching coordinates into Eqs. (53)–(55) and equating the coefficients of lowest order of \n
Taking the coefficients of next higher order of \n
where \n
It has been noticed that the behavior of nonlinear waves changes significantly in the presence of external periodic force. It is paramount to note that the source term or forcing term due to the presence of space debris in plasmas may be of different kind, for example, Gaussian forcing term [4], hyperbolic forcing term [4], (in the form of \n
which is termed as damped and forced KdV (DFKdV) equation.
\nIn absence of \n
where \n
In this case, it is well established that
\nis a conserved. For small values of \n
where \n
Differentiating Eq. (64) with respect to \n
Again,
\nUsing Eq. (66) and (67) the expression of \n
Therefore, the solution of the Eq. (62) is
\nwhere \n
Solitary wave solution of Eq. (62) in the absence of damping(\n\n\nν\n\nid\n0\n\n\n=\n0\n\n) and external force(\n\n\nf\n0\n\n=\n0\n\n) with the parameter value \n\n\nM\n0\n\n=\n0.2\n,\nω\n=\n1\n,\nτ\n=\n1\n,\nμ\n=\n0.2\n\n.
In Figure 3, the soliton solution of the damp force KdV equation is plotted from Eq. (65) for different values of the strength of the external periodic force \n
Variation of solitary wave from Eq. (62) for the different values of \n\n\nf\n0\n\n\n with \n\n\nM\n0\n\n=\n0.2\n,\nω\n=\n1\n,\nτ\n=\n1\n,\nμ\n=\n0.2\n,\n\nν\n\nid\n0\n\n\n=\n0.01\n\n.
Variation of solitary wave from Eq. (62) for the different values of collisional frequency \n\n\nν\n\nid\n0\n\n\n\n with \n\n\nM\n0\n\n=\n0.2\n,\nω\n=\n1\n,\nτ\n=\n1\n,\nμ\n=\n0.2\n,\n\nf\n0\n\n=\n0.01\n\n.
To obtain damped KdV Burgers equation we considered an unmagnetized collisional dusty plasma which contains cold inertial ions, stationary dusts with negative charge and Maxwellian distributed electrons. The normalized ion fluid equations are as follows
\nwhere \n
Here normalization is taken as follows
\n\n\n
To obtain damped KdV burger we introduced the same stretched coordinates use in Eq.(31). The expansion of the dependent variables are also considered same as (32)–(34) with
\nSubstituting the above expansions (32)-(34) and (73)–(74) along with stretching coordinates (31) into Eqs. (69)–(71) and equating the coefficients of lowest order of \n
Taking the coefficients of next higher order of \n
where \n
In absence of \n
where amplitude of the solitary waves \n
It is well established for the KdV equation that,
\nis a conserved quantity [7].
\nFor small values of \n
where the amplitude \n
Differentiating Eq. (78) with respect to \n
where,
\nand
\nSubstituting Eq. (81) and (82) into Eq. (80), we obtain
\nwhich is the Bernoulli’s equation, where \n
The solution of the Eq. (83) is
\nTherefore, the slow time dependence form of the ion acoustic solitary wave solution of the DKdVB Eq. (76) is given by (79)where.
\n\n\n
Let us consider an unmagnetized collisional dusty plasma that contains cold inertial ions, stationary dusts with negative charge and Maxwellian distributed electrons. The normalized ion fluid equations which include the equation of continuity, equation of momentum balance and Poisson’s equation, governing the DIAWs, are given by
\nwhere \n
where \n
The normalized \n
Phase velocity and nonlinear evolution equation
\nWe introduced the same stretched coordinates use in Eq. (31). The expansion of the dependent variables also considered same as (32)–(34) and (58)–(59). Substituting (31)–(34) and (58)–(59) along with stretching coordinates into Eqs. (84)–(86) and equating the coefficients of lowest order of \n
with \n
where \n
Now at the certain values, for example \n
and expand of the dependent variables same as Eqs. (32)–(34) with
\nNow substituting Eq. (32)–(34) and (91)–(93) into the basic Eqs. (84)–(86) and equating the coefficients of lowest order of \n
Equating the coefficients of next higher order of \n
Equating the coefficients of next higher order of \n
where \n
From Eq. (94)–(96), one can obtain the Phase velocity as \n
where \n
It has been noticed that the behavior of nonlinear waves changes significantly in the presence of external periodic force. For simplicity, we assume that \n
Such a form of this source function is observed in experimental situations or conditions for a particular device. Eq. (104) is termed as damped force modified Korteweg-de Varies (DFMKdV) equation.
\nIn absence of \n
The slow time dependence form of the ion acoustic waves solution of the DFMKdV Eq. (104) is given by,
\nwhere \n
The amplitude and width are as follows:
\nwhere
\nLet us consider a plasma model [9] consisting of cold ions, Maxwellian electrons in the presence of dust particles and the external static magnetic field \n
The normalized electron density is given by
\nwhere \n
Here the normalization is done as follows:
\nHere \n
To obtain the DFZK equation we introduce the new stretched coordinates as
\nwhere \n
Substituting the equations (112)-(119) into the system of Eqs. (106)-(110) equating the coefficient of \n
Equating the coefficient of \n
Considering the coefficient of \n
Comparing the coefficients of \n
Using the relationships (120)–(124), one can obtain the linear dispersion relation as
\nExpressing all the perturbed quantities in terms of \n
where
\nChoudhury et al. [5] studied analytical electron acoustic solitary wave (EASW) solution in the presence of periodic force for an unmagnetized plasma consisting of cold electron fluid, superthermal hot electrons and stationary ions. Motivated by the these works, here we consider the source term as \n
where \n
where \n
where, \n
The analytical solitary wave solution of the Eq. (134) as obtained in (68), is
\nwhere \n
It is clear from the structure of the solitary wave solution of the DFKdV, DFMKdV and DFZK that the soliton amplitude and width depends on the nonlinearity and dispersion of the evolution equations, which are the function of different plasma parameter involve in the consider plasma system. Also evident from the structure of the approximate analytical solution that the amplitude and the width of the soliton depends on the Mach number \n
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